How to weed out candidates who will be dissatisfied with these IOLs, even after a perfect surgery.
If you choose to offer presbyopia-correcting intraocular lenses, you're quickly faced with the problem of matching the IOL to each patient's lifestyle, which isn't always easy. In this article, we'll take a look at who isn't a suitable candidate for premium IOLs like the Symfony, ReSTOR, Tecnis Multifocal and Crystalens, and how to successfully avoid creating unhappy postop patients.
In an ideal world, all patients could get the procedures they want. For presbyopic IOLs, however, the perfect patient can be difficult to come by. Zaina Al-Mohtaseb, MD, an assistant professor of ophthalmology at Baylor College of Medicine, discusses her ideal candidate. "Patients with excellent ocular surfaces are the first indication of a great candidate," she says. "They should also have no co-existing ocular pathology and would like to be glasses-independent for most things, but understand they might need glasses for small print.
A 71-year old man presented with a history of cataract. Scans showed dry eye, and the physician prescribed Xiidra before considering IOL implantation.
Dr. Loden also describes his ideal candidate for these premium IOLs. "In terms of personality type, you're going to want to look for someone who can manage their expectations—they're a realist. There is likely going to be some follow-up that they'll have to be on board for, and that's important to identify from the get-go.
• Fuchs' dystrophy. "Most patients with co-existing corneal pathology won't be an ideal candidate," Dr. Mohtaseb says.
• Advanced dry eye. "Any pre-existing conditions need to be treated prior to surgery to ensure good outcomes," she adds.
"Ignoring these signs of bad candidates will cost you too much chair time for suboptimal results," according to Dr. Loden. "Any ocular surface disease has to be corrected preoperatively. These patients won't be satisfied with their outcomes otherwise, no matter how perfect the surgery and implantation is. You have to be cautious and judicious with those outcomes.
Along with the obvious medical signs of a bad candidate for these premium IOLs, there are other red flags that visual tests won't detect. The personality types of your patients can also be a good indication of whether or not a premium IOL is right for them.
Dr. Loden also shares his experiences with evaluating the personalities of his patients to see if they'll be a fit for these lenses. "There are definitely some psychological signs to keep an eye out for, and you'll learn most of it from just talking with your patients. Evaluate their visual needs. Ask if they mind wearing glasses. Do they consider themselves to be obsessive or a perfectionist? What are they hoping to gain from this procedure in terms of visual acuity? Would they prefer distance or near visual acuity? It's also important to evaluate their state of mind as best you can, too. If you have a depressed patient, you probably shouldn't implant one of these lenses, as adjusting to the new, imperfect visual acuity might be stressful. It's important to take the time to talk with them, and use your gut to make the call. There are plenty of other options that might better suit your patient. It's okay to go in a different direction," he says.
Along with a patient's psychological makeup, his or her career should also be taken into account when looking at implanting a premium IOL. For example, say your patient is an editor who needs closer visual acuity to edit his work. You'll definitely want to keep that in mind while evaluating whether or not to implant a premium IOL.
Despite the many benefits of premium IOLs, there are plenty of patients who just aren't suited for them. Surgeons advise exploring other lens options. Even with perfect surgery and implantation, a patient not fit for these lenses will still be dissatisfied, leaving you and him frustrated. Beyond the medical contraindications, make sure your patient's psychological makeup can adapt to these lenses.
Dr. Loden is a consultant for LENSAR, J & J vision and Lenstec. Dr. Mohtaseb is a consultant for Alcon and Allergan.


text:
Lucas Oil Stadium Indianapolis
Tennessee Titans vs Indianapolis Colts Pick & Prediction, Sun Oct 31 01:00 PM
NFL | FOOTBALL
Sun Oct 31 01:00 PM
TV: CBS
OPENNER
49o -110
-3 -110
Preview and Prediction
Last Recap
EXPERT NFL TOTAL
Jimmy Liu
WIN 8, LOSE 4, PUSH 0 + $327
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It's an AFC South showdown between the two top teams. The Tennessee Titans head to Indianapolis for their second game of the year. In their first meeting, the Titans won the game 25-16. This second game could be tougher for the Titans though, especially because the Colts are on a roll. Can the Titans sweep the Colts in this Sunday morning matchup?
Sports Bookies
The Indianapolis Colts have won two straight games after their loss to the Ravens. After beating the Texans and the 49ers, the Colts prepare to take on the Titans. Tennessee isn't an easy team to beat though, and many have tried. Slowing down the freight train, Derrick Henry is something every defense struggles with. The question is can the Colts shut him down on Sunday?
Titans vs. Colts Point Spread, Total and Money Line Odds
MONEY LINE: Titans (-122) vs. Colts (+102)
SPREAD: Titans +1.5 (-110) vs. Colts +1.5 (-110)
GAME TOTAL: Over 49 (-110) vs. Under 49 (-110)
NFL Week 8 Odds
Titans vs. Colts Betting Preview for NFL Week 8
The Titans have yet again found a way to be one of the most dominant forces in the NFL. Derrick Henry is only a mere 131 yards away from rushing for 1,000 yards already. We're only in Week 8 and Henry is terrorizing opposing defenses each week. The best part is that the rest of the offense is getting in on it too. Ryan Tannehill even scored a rushing touchdown last week.
The Colts' defense is slightly tougher to rush against than the Chiefs though. Tennessee will likely need to rely on their passing just a little more. This shouldn't be too big of a problem because Julio Jones should be returning. If the Titans get the passing game going, the Colts will have no chance to stop Derrick Henry.
Heading into this game, the Colts need to understand that the end goal for the Titans is running the ball. If they rush heavy early on to stop the run, Tennessee will just beat them in the second half. The good news for the Colts is that they've seen this happen plenty of times. In nearly every game for the Titans, the plan remains the same.
On the offensive side of the ball, the Colts need to take advantage of the Titans' secondary. They have a lot of injuries to defensive backs that the Colts can exploit. Using Michael Pittman Jr. to stretch the defense will definitely help them out. If they can turn this into a shootout, Derrick Henry will become a non-factor.
NFL Week 8 Predictions
Titans vs. Colts Prediction
Titans 28 - Colts 24
It's hard to think of a team that can beat the Titans right now. They just finished with an incredible win over the Chiefs and their running game is impeccable. The only question is whether they can successfully stick to the game plan. They did so against the Bills and Chiefs, so we can't doubt them here.
The Colts' problem is that they have no answer for Derrick Henry. Even if he's shut down early in the game, he just gets better as the game goes on. For the Titans to lose it takes an incredibly strong rushing defense. The Colts allow an average of 111 yards rushing per game, this falls right into Tennessee's plans.
1. The Titans average 34:08 minutes of the time of possession.
2. In their last three meetings, the Tennessee Titans are 2-1.
3. The Tennessee Titans are 5-1 ATS in their last six games.
Best Bet: Tennessee Titans -1.5
Tennessee Titans vs. Indianapolis Colts Recap OCT 31TH 2021
Randy Bullock converted a 44-yard field goal with 4:03 left in overtime Sunday, lifting the Tennessee Titans to a 34-31 win over the Indianapolis Colts in Indianapolis.
Bullock's kick came four plays after Kevin Byard intercepted Carson Wentz and returned it to the Colts' 32 with 5:48 remaining. Bullock's second game-winner of the year improved Tennessee to 6-2 and gave it a three-game lead over the Colts (3-5) in the AFC South, plus the tiebreaker of a season sweep.
Wentz completed 27 of 51 passes for 231 yards with three touchdowns and two picks. His first interception in 213 passes occurred with 1:26 remaining in regulation as he tried to avoid a safety and threw a left-handed pass that Elijah Molden grabbed and returned two yards for a score.
But Indianapolis forced overtime when Jonathan Taylor scored on a 1-yard run with 22 seconds left in regulation, one play after Tennessee committed a 42-yard pass interference penalty in the end zone.
Titans quarterback Ryan Tannehill hit on 23 of 33 passes for 265 yards with three touchdowns and two interceptions. NFL rushing leader Derrick Henry was held to 68 yards on 28 carries.
Indianapolis nearly buried Tennessee in the first quarter. It started the game with a 14-play, 82-yard drive that Wentz capped with a 2-yard touchdown pass to Pittman. Then the duo connected again from the 7, one play after Tannehill tossed an interception.
But the Titans found traction after their defense came up with a fourth-down stop at their 46 late in the first. A 60-yard drive ended with Tannehill's 6-yard scoring strike to Geoff Swaim at the 12:07 mark of the second quarter.
Another Tannehill interception set up the tying Tennessee score. That's because Tyquan Lewis fumbled at the end of the return and tight end Anthony Firkser recovered. On the next play, a short pass to A.J. Brown turned into a 57-yard touchdown as he made a man miss and raced down the right side.
The Colts regained the lead on the half's last play as Michael Badgley converted a 34-yard field goal.
The teams traded touchdowns in the third quarter before Bullock nailed a 46-yard field goal to tie the game at 24-24 early in the fourth.
W/L/P
Luke Moore Wilson 14/5/1 +782
Elijah Davis 14/6/0 +604
Ryan Davis 14/6/0 +441
James Anderson 16/4/0 +957
Mario DeLuca 15/5/0 +688
Seth Cohen 10/9/1 +32
William Taylor 13/7/0 +498
Ethan Wilson 3/2/0 +51
Seth Cohen 11/9/0 +119
Jimmy Liu 13/7/0 +509
Anthony Addison 13/7/0 +411
Diego Garcia 12/7/1 +436
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Words reflect workplace culture and can reinforce negative stereotypes around gender, race, and sexual orientation. Learn to stop using common words and phrases that harm the advancement of women, people of color, and LGBTQI people in the workplace.
The information in this project, in both English and French, serves to help you flip the script on harmful language, providing alternatives that don't have such a high cost to your co-workers.
Learn how to communicate so transgender colleagues know they are valued members of their teams.
This webinar provides an overview of common barriers to LGBT workers' advancement and the actions individuals and organizations can take to build an inclusion.
Stop using common words and phrases that undermine men's ability to bring their whole authentic selves to work.
Même avec les meilleures intentions, nous pouvons dire ou faire des choses qui sont vexantes et blessantes.


Bernardo Bellotto, The Zwinger Complex in Dresden, 1751–52, oil on canvas (Gemäldegalerie Alte Meister, Staatliche Kunstsammlungen Dresden, photo by Elke Estel/Hans-Peter Klut).
Bernardo Bellotto (1722–1780) is recognized as one of the greatest view painters in history, acquiring his fame in mid-18th-century Dresden as the court painter for the elector of Saxony, Frederick Augustus II—who was also King Augustus III of Poland. Over the course of a decade, Bellotto produced dozens of breathtaking depictions of the city and its environs, most measuring over eight feet in width. The success and renown of these grand, expansive works would earn Bellotto prestigious commissions at prominent courts throughout Europe.
Bellotto's magnificent paintings of Dresden are now in the collection of the Gemäldegalerie (Picture Gallery) of the Dresden State Art Collections and will be on loan to the Kimbell Art Museum for the special exhibition The Lure of Dresden: Bellotto at the Court of Saxony. They will be accompanied by portraits and allegories of the elector and his queen, as well as view paintings of Venice and Saxony by Bellotto's uncle and teacher Antonio Canaletto and Dresden court painter Alexander Thiele. Visitors to the exhibition will have the unique opportunity to view the majesty that was Dresden in the 1700s. One of the greatest cities of 18th-century Europe, it is only now, following its near-total destruction in the Second World War, being rebuilt to its former glory—with the aid of Bellotto's pictorial legacy.
This exhibition is organized by the Staatliche Kunstsammlungen Dresden in cooperation with the Kimbell Art Museum.


In line with the objectives of the Panel Paintings Initiative (a collaborative project of the Getty Conservation Institute, the J. Paul Getty Museum and the Getty Foundation), an upcoming Summer Institute aims to ensure that critical knowledge about the historical production of panel paintings as well as methods and recent theories and the best practice in the structural conservation of panel paintings will be transferred from current experts to a new generation of conservators and curators. The course instructors will be experienced panel painting conservators, museum curators, wood specialists and conservation scientists from renowned museums and institutes in the USA and Europe. We believe the best way to foster improvement of the structural conservation of panel paintings is to exchange current aesthetic notions and material knowledge of art works, and to fuse the latest developments in modern sciences with shared professional experiences and observations in treatment options. A two-week intensive Summer Institute could offer a new professional network for conservators, curators and wood scientists from Western, Central European and Russian institutions holding panel paintings; a network that could prove essential for keeping the field of panel painting conservation alive and well for the future.
• Is there a divergence between Northern and Southern European treatment approaches, and if so what is the rationale behind these differences?
• Is the structural treatment of a panel painting primarily the responsibility of a panel conservator/wood specialist?
• The presentation of panel paintings often requires adaptations to the frame (microclimate enclosure, extended rebate, glass), which in turn influences the overall aesthetics of the work. What is most important when making such a decision?
Jørgen Wadum is keeper of conservation at the National Gallery of Denmark, Copenhagen. He trained as a flower painter, an art historian and as a paintings conservator. Since the 1980s he has specialized in the painting techniques of 16th and 17th-century Dutch and Flemish artists. He has published and lectured extensively internationally on a multitude of subjects related to this and other issues of importance for the understanding and preservation of our cultural heritage. Wadum holds positions in several international organizations and committees. He is the co-chair of the Panel Paintings Advisory Group, Getty Conservation Institute, Los Angeles, and a member of the Jan Van Eyck Advisory Board, Ghent.


A contemporary garden for a venture capitalist and avid collector of modern art. The brief was for a 'very contemporary' garden for entertaining to include a fire pit. The zig-zag design adds drama to the garden: raised beds with sides of Venetian polished plaster and oak tops which act as seats.
One novel feature of the design is an 'outside-inside' water feature – two distinct units that appear to be joined by using design and landscaping 'links'. The inside water feature comprises a space 40cm below floor level with water bubbling over a layer of white marble pebbles – all covered by a sheet of toughened glass that can be walked on. Lit from below, it forms a glowing and distinctive feature of the extension to this magnificent house.
The border hedging has diagonal spaces cut into it so allowing views through to the private park beyond and 'borrowing' this landscape to make the garden appear larger without loosing any privacy.


text:
Home > NewsRelease > Greater Prairie Business Consulting, Inc. CEO Wins Two-(2) Gold Stevie Awards-
Greater Prairie Business Consulting, Inc. CEO Wins Two-(2) Gold Stevie Awards-
Institute of Management Consultants USA (IMC USA)
James J Talerico, Jr, CEO
Greater Prairie Business Consulting, Inc.
320 Decker Drive, Suite 100
Irving, Texas 75062
E-Mail: jjtalericojr@GPBusinessSolutions.com
[Irving, Texas]: James J. Talerico, Jr, CEO of Greater Prairie Business Consulting, Inc. received a "Gold Stevie" Award for both "Thought Leadership," and as a "Media Hero of the Year During Covid" at the 20th Annual American Business Awards in New York City. ® The competition received more than 3,700 nominations and is the premier accolade for excellence in US business honoring organizations of all sizes and industries. Winners were determined by the average scores of more than 230 professionals over three months of judging. Stevie Awards competitions receive more than 12,000 nominations each year from organizations of every type in more than 70 nations.
At the event, James J. Talerico, Jr. stated that he has been a Stevie Award judge since 2019, because he gets to learn about the incredible innovation and market leadership by the companies like those in the room and went on to thank the committee and company's founder for his two awards.
The event was held on Monday, June 10th at the Marriott Marquis on Times Square, and was hosted by Cheryl Casone from Fox Business.
For more information about James J. Talerico, Jr., CMC ® and Greater Prairie Business Consulting, Inc., go to www.greaterprairiebusinessconsulting.com.
About the Institute of Management Consultants
IMC USA is a founding member of the International Council of Management Consulting Institutes, with over 64,000 consultants and 8,200 Certified Management Consultants (CMC(R)) worldwide. Members of IMC USA advance their skillsets by getting certified as a Certified Management Consultant (CMC(R)) and learn from their peers and from industry thought leaders on online webinars and networking sessions, local chapter meetings and Consult-Con, the IMC USA Annual Conference. For more information, please visit www.imcusa.org.
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text:
One day before its official unveiling, the next Pebble watch leaks online
That iThingy You're Wearing
by Bryan M. Wolfe
The first image of the next-generation Pebble smart watch has made an appearance online. The image, which was first uncovered by 9to5Mac, appears on Pebble's servers and the leak has occurred just a day before the company is expected to officially announce its next wearable device.
As you can see by the image below, the new Pebble watch is likely to have a larger bezel and smaller buttons, when compared to its predecessor. It also appears to have a colored e-paper display for the first time.
According to 9to5Mac:
On the inside, the next Pebble is going to have a Cortex M4 processor and a 6-axis gyroscope. Unlike the Apple Watch, the new Pebble won't have a touch screen, and its resolution is definitely on the low side compared to Apple's upcoming device.
Back in January, Pebble CEO Eric Migicovsky announced that the company has sold more than 1 million smart watches. He also promised that new software and hardware would be unveiled sometime in 2015.
The first-generation Pebble arrived after a successful Kickstarter project in 2013. One year later, the Pebble Steel launched. Both models have the following features: black and white e-paper display, programmable CPU, memory, storage, Bluetooth, magnetometer, ambient light sensor, and an accelerometer.
It will be interesting to see what the next Pebble turns out to be. With the Apple Watch launch just weeks away, a refresh was certainly necessary.
Currently, you can purchase the original Pebble for $99. The higher-end Pebble Steel is $199.
The AppAdvice week in review: An iOS beta program for non-developers and more on the Apple Watch
Apple columnist John Gruber offers an interesting new perspective on Apple Watch pricing


text:
\section{Introduction}\label{s:intro}

The many-normal-means problem concerns simultaneous inference
about a collection of unknown normal means.
More precisely, one has the observed data $X_1, ..., X_n$ that are
assumed to have the sampling model
\begin{equation}\label{eq:association-01}
	X_i = \theta_i + Z_i\qquad(Z_i \stackrel{iid}{\sim} N(0, 1);
	\;i=1,...,n)
\end{equation}
This problem has proven to be fundamental in statistics.
The most noticeable contribution, perhaps, dates back to
\cite{Stein1956},
on the remarkable discovery that
the usual maximum likelihood estimates $\hat{\theta}_i=X_i$
are inadmissible under quadratic loss for $n\ge 3$.
Understanding of and developing methods for this
problem have been resulting in successful research in multi-parameter or
high-dimensional problems, including, for example,
the James-Stein shrinkage method \citep{js1961},
the empirical Bayes method of \cite{efron1972empirical},
the wavelet shrinkage method of \cite{donoho1995adapting}
for non-parametric regression, and
LASSO of \cite{tibshirani1996regression} for linear regression.
The literature on the many normal means problem itself and
methods built up on shrinkage is massive and is, thereby, impossible to
provide a complete review in this note.

\newcommand{\mbox{\rm sort}}{\mbox{\rm sort}}

We refer to the problem setting given above as is as the {\it first or classic} kind,
reflecting Stein's discovery.
It is perhaps somewhat surprising that the understanding
of the Stein phenomenon has not been very satisfactory, at least not
to a level that would help develop efficient methods based on a sound reasoning
with uncertainty, as
\cite{stigler19901988} wrote:
\begin{quote}
	\it
	The best heuristic explanation that has been offered is a Bayesian argument ...
Another explanation that has been offered is that
[the James-Stein estimator]
can be viewed as a relative of a ``pre-test'' estimator ...
	But neither of these explanations is fully satisfactory (although
	both help render the result more plausible);
	the first because it requires special {\it a priori}
	assumptions where Stein did not, the second because it corresponds
	to the result only in the loosest qualitative way. The difficulty
	of understanding the Stein paradox is compounded by the fact that
	its proof usually depends on explicit computation of the risk function
	or the theory of complete sufficient statistics, by a process that
	convinces us of its truth without really illuminating the reasons
	that it works.
\end{quote}
Attributing back to Francis Galton's knowledge in the 1880's and
viewing Stein estimation as an inverse regression problem $E(\theta|X)$,
\cite{stigler19901988} provided a nice interpretation of
the James-Stein (shrinking toward zero) and Efron-Morris
(shrinking toward the sample mean $\bar{X}$) estimators of
$\theta$.
The fundamental problem of this classic kind has however remained unsolved,
due to the difficulty as recognized by \cite{stigler19901988}:
\begin{quote}\it
	With no distributional assumptions about the $\theta$'s,
	we are of course prevented from looking for an optimal
	estimate of ``$E(\theta|X)$''.
\end{quote}
Our innovative formulation in Section \ref{s:classic}
via introducing a latent permutation in addition to unobserved $Z$'s in
\eqref{eq:association-01}
leads to an inferentially valid $E(\theta|X)$
and thus offers a new understanding,
shedding light on this challenging problem from an Inferential
Models perspective.

A popular variant of the classic kind is obtained by modeling
the unknown $\theta_i$'s as an unobserved sample from a non-parametric
distribution $\mathbb{G}$ with the cumulative distribution function (cdf) $G(.)$. In this case, integrating out the unobservable
$\theta_i$'s leads to inference about the mixture distribution
\begin{equation}\label{eq:conv}
	F_G(x) = \int \phi(x-\theta)dG(\theta)
\end{equation}
from the observed $X$'s. Note that in this case, $X$'s is necessarily a
sample from the convoluted distribution $F_G(x)$.
Inference about $\theta$'s is then carried out
with their conditional distribution, given $X_i$'s and estimates of $G$.
The central problem in this case is thus to estimate $G(.)$.
There is a considerable literature on this deconvolution problem,
as in \cite{laird1978nonparametric},
\cite{carroll1988optimal},
\cite{stefanski1990deconvolving},
\cite{fan1991optimal},
\cite{zhang1997empirical},
\cite{hall2007ridge},
\cite{butucea2009adaptive},
and
\cite{efron2014two,efron2016empirical}.
For convenience,
we refer to inference via this non-parametric partial Bayes model
as the {\it second or (non-parametric) empirical-Bayes} kind, in contrast to the classic kind.

Here, we consider exact and efficient inference for the two kinds of
the many-normal-means problems
by taking the Inferential Models (IMs) approach,
which is to be briefly reviewed in Section \ref{s:IMs}.
The empirical-Bayes kind and classic kind are
investigated in Sections
\ref{s:partial-bayes} and \ref{s:classic}, respectively.
The development of IMs for the empirical-Bayes kind
appears relatively straightforward because of the nice
independence structure imposed.
Nevertheless,
following the discretization idea in \cite{efron2014two,efron2016empirical}
but not going further for $g$-modeling, our approach is to make
inference via over-parameterization or numerical deconvolution without
effectively having to impose more restrictions on $\mathbb{G}$.
It should be noted that unlike traditional Frequentist and Bayesian
methods, IMs can provides honest inference
even if the unknown parameters for $\mathbb{G}$ are unidentifiable.

For the classic kind, we introduce a sorted standard normal sample
with a random permutation as the so-called auxiliary variable.
This novel approach, with a full IMs argument, leads to
a prior-free probabilistic inference, for example, by making use of
the quantity $E(\theta|X)$ for point estimation.
As a result, it sheds a new light on a satisfactory understanding Stein
estimation.

Predictive random sets for IMs inference are constructed to produce
probabilistic inference in terms of belief and plausibility.
It is well known that valid probabilistic inference says more
than but can be used for point and interval estimations.
As a method for point estimation,
maximum plausibility estimation is discussed in detail.
For interval estimation in the more familiar frequentist inference framework,
exact but conservative interval estimation is obtained based on plausibility.
A Monte Carlo approach is proposed as an adaptive adjustment method to
create shorter confidence intervals with targeted coverage.
This adaptive method is so straightforward in the context of
frequency calibration that it is simply presented as a technical
remark; see Remark \ref{remark:adaptive-adjustment}.
This is illustrated in Section \ref{ss:real-data-example} for a real-data example.

To evaluate the performance of the proposed methods,
a simulation study is conducted in
Section \ref{s:simu} with the computational methods provided
in Appendices.
The numerical results show that the proposed methods outperform
the traditional James-Stein and Efron's g-modeling \citep{Narasimhan2020},
and are comparable to the auto-modeling method proposed recently by
\cite{jiang2022estimation}.
Section \ref{s:discussion}
concludes the paper with a few remarks,
including a brief discussion of other kinds of the
many-normal-means problem,
future potential developments,
and extensions.

\section{Inferential models: a brief review}
\label{s:IMs}

IMs was proposed in \cite{martin2013inferential} as an inferential framework
for prior-free probabilistic inference.
It makes use of the so-called unobserved but predictable auxiliary random variables, or simply,
auxiliary variables $U$ that has a known distribution
${\mathbf P}_U$ on ${\mathbb U}$.
The setup for IMs is a system of equations, called association, representing
a statistical model with unknown parameter $\theta\in\Theta\subseteq
{\mathbb R}^p$ for the observed data $X \in {\mathbb X}$ via
an auxiliary variables $U$.
For the sake of clarity, we write this association as
\begin{equation}\label{eq:association-basic}
        X = a(U, \theta)\qquad
        (X \in {\mathbb X}, \theta\in\Theta, U \sim {\mathbf P}_U)
\end{equation}
IMs relies on so-called valid or frequency-calibrated predictive random sets
(PRS) $\mathcal{S}$ for predicting the unobserved $U$ to
produce valid uncertainty quantification on unknown quantities of interest.
Formally, a PRS $\mathcal{S}$ is said to be valid for $U$ if
it is independent of $U$ and
\begin{equation}\label{eq:valid-prs}
        \mbox{Prob}(\mbox{Prob}(\mathcal{S} \not\ni U) \ge 1-\alpha)
        \le \alpha
\end{equation}
for all $\alpha \in (0,1)$.
Typically, $\mathcal{S}$ is chosen in such a way that
$\mbox{Prob}(\mathcal{S} \not\ni U) \sim \mbox{Uniform}(0,1)$,
the standard uniform distribution.
For example, the $\mathcal{S}(U)=\{u: |u| \leq |U|\}$,
$U\sim \mbox{N}(0,1)$, is valid for predicting
an unobserved realization from $\mbox{N}(0,1)$.

Given a PRS $\mathcal{S}$ for predicting $U$ in \eqref{eq:association-basic},
the inverse set-valued mapping
\begin{equation}\label{eq:theta-prs}
        \Theta_X(\mathcal{S}) =\{\theta: X=a(u, \theta) \,\mbox{ for some }\,
        u \in \mathcal{S}\}
\end{equation}
defines a random set on $\Theta$.
For simplicity, here we assume $\Theta_X(\mathcal{S})\neq \emptyset$
for all values of $\mathcal{S}$
({\it c.f.} \cite{leaf2012inference} for the case
where $\mbox{Prob}(\mathcal{S} = \emptyset)>0$).
IMs produces uncertainty assessments on assertions of interest,
$A \subset \Theta$, by computing the following lower and upper probabilities
\begin{equation}\label{eq:lower-prob}
        \mbox{bel}_X(A) = \mbox{Prob}(
           \Theta_X(\mathcal{S}) \subseteq A
        )
\end{equation}
and
\begin{equation}\label{eq:upper-prob}
        \mbox{pl}_X(A) = 1 - \mbox{bel}_X(A^c) = \mbox{Prob}(
           \Theta_X(\mathcal{S})\not \subseteq A^c
        )
\end{equation}
The probabilities \eqref{eq:lower-prob} and \eqref{eq:upper-prob}
are also
known as the belief function at $A$ and plausibility of $A$, respectively.
A recent discussion of these IMs lower and upper probabilities
and other existing ones is given in
\cite{liu2021comment}; see also references therein.

As described above, IMs is rooted in
R. A. Fisher's fiducial argument in the sense to
make prior-free probabilistic inference about parameters
by using auxiliary variable to represent the sample model.
Avoiding the difficulties of R. A. Fisher's fiducial argument
and the related theories \citep{zhang2011dempster,martin2010dempster,liu2015frameworks},
however,
IMs emphasizes frequency-calibrated/valid and efficient
probabilistic inference.
Together with Conditional IMs and marginal IMs
(\cite{martin2015conditional} and
\cite{martin2015marginal}), subject to further developments,
this theory, if so desired, also provides as a
new mathematical tool for developing exact and efficient frequentist methods.
For example,
for the familiar Neyman-Pearson school of thought,
it offers its plausibility regions for unknown parameters as exact Neyman-Pearson confidence regions \citep{martin2013inferential}, and  for the Fisherian school of thought, its plausibility for assertions of
interest provides a plausibility interpretation of
or a probabilistic alternative to
Fisher's P-value \citep{martin2014note}.

Nevertheless, research on foundations of statistical
inference has been challenging.
Like other existing approaches other than the standard
Bayes and frequentist schools of thought, including
generalized fiducial \citep{hannig2009generalized, cui2022demystifying},
the method of confidence distribution \citep{xie2011confidence,schweder2016confidence},
and the Dempster-Shafer theory of belief functions
\citep{shafer1976mathematical,shafer1992dempster,dempster2008dempster,martin2010dempster},
IMs is subject to scrutinization and acceptance.
This is expected to take time.

\section
IMs for MNMs of the empirical Bayes kind
}
\label{s:partial-bayes}

\subsection{The sampling model and IMs association}

Without real loss of generality, our jumping-off point is to take a numerical approach
to approximate $G(.)$ in \eqref{eq:conv} via discretization;
see also \cite{efron2014two,efron2016empirical}.
Denote by $\vartheta_1 < ... < \vartheta_K$ the known space
of $\theta$, and denote by $\gamma_k$ the unknown probability mass at
$\vartheta_k$, where
\begin{equation}\label{eq:numeric-G}
	\gamma \equiv (\gamma_1,...,\gamma_K)
	\in \left\{(\gamma_1,...,\gamma_K):
	\gamma_k \geq 0 \mbox{ for all $k=1,...,K$;
	$\sum_{k=1}^K \gamma_k = 1$}\right\}
\end{equation}
\begin{remark}
Take $\vartheta_{k}-\vartheta_{k-1}=\delta$ for
all $k=1,...,K$. This discrete approximation can also
be view as numerical approach to mixed distributions,
including both discrete and continuous distributions.
\end{remark}
In this scenario, $G(.)$ is represented by the $K$-dimensional $\gamma$
vector in \eqref{eq:numeric-G}, that is, in terms of cdf
\[ G(\theta) =
\sum_{\vartheta_k \le \theta}\gamma_k
	\]
and
$X_1,...,X_i$ is a sample from the convolution of
$G(.)$ and $N(0,1)$:
\begin{equation}\label{eq:numeric-C}
	F_{\vartheta, \gamma}(x) = \sum_{k=1}^K \gamma_k \Phi(x-\vartheta_k)
\end{equation}
where $\Phi(.)$ denotes the cdf of the standard norm distribution $N(0,1)$.
Furthermore, given $\gamma$ and $X$,
$\theta_i$'s are independent of each other with
\begin{equation}\label{eq:numeric-T}
	\mbox{Pr}(\theta_i = \vartheta_j | \gamma, X)
	= \frac{\gamma_k\phi(X_i-\theta_j)}
	{\sum_{k=1}^K\gamma_k\phi(X_i-\vartheta_k)}
	\qquad(j=1,...,K)
\end{equation}
where $\phi(.)$ stands for the probability density function of the standard norm distribution $N(0,1)$.

From \eqref{eq:numeric-C} and \eqref{eq:numeric-T}, we
see that inference about $\theta_i$s is reduced to inference
about $\gamma_k$'s.
It is known that $F_{\vartheta, \gamma}(X_{i})$'s form a sample
of size $n$ from $F_{\vartheta, \gamma}(x)$.
For an IMs approach, we take a sorted uniforms of size of $n$,
denoted by $U_{(1)} < ..., < U_{(n)}$, as auxiliary variables
and specify the association as
\begin{equation}\label{eq:numeric-association}
	U_{(i)} = F_{\vartheta, \gamma}(X_{(i)})
	= \sum_{k=1}^K \gamma_k \Phi(X_{(i)}-\vartheta_k)
\end{equation}

\newcommand{B}{B}

\subsection{A PRS for sorted uniforms}

Let $\mathbb{S}_n$ denote
the sample space of $U_{(i)}$'s,
the simplex space of size $n$:
\[
	\mathbb{S}_n = \{(u_1,...,u_n): 0 < u_1 < ... < u_n < 1\}
	\]
Intuitively, we aim to constructing a PRS for
$U_{(i)}$'s that should work well for the well-known
non-parametric one-sample test, where it is known that
the cdf values at the observations serve as
the sufficient statistics and form a sample of uniforms.
For simplicity, we use a type of predictive random set
similar to that of
\cite{liu2014probabilistic} and
\cite{zhang2010statistical}
for $(\Phi(Z_{(1)}), ..., \Phi(Z_{(n)}))$,
\begin{equation}\label{eq:prs}
	\mathcal{U} = \{u: u\in \mathbb{S}_n, B(u)\le B(U)\}
\end{equation}
where $U=(U_{(1)},..., U_{(n)})$, the vector of sorted sample of size $n$
from $\mbox{Uniform}(0,1)$ and
\begin{equation}\label{eq:boundary}
	B(u) = -\sum_{i=1}^n\left[ a_i\ln(u_i) + b_i\ln(1-u_i) \right]
\end{equation}
with
$a_i = \frac{1}{n-i+c_n}$,
$b_i=\frac{1}{i-1+c_n}$, and
$c_n \approx \frac{2}{3}$.
\cite{zhang2010statistical} considered
this PRS with $c_n=0.7$ as an IMs approach to the non-parametric
one-sample problem and showed that
the resulting method is either comparable to
or better than traditional methods such as
the Kolmogorov-Smirnov test.

Recall the familiar results on sorted uniforms
regarding the marginal distribution of $U_{(i)}$. We have
\(
        U_{(i)}\sim \mbox{Beta}(i, n-i+1)
        \)
which has its mean and variance given by
\(\frac{i}{n+1}\) and
\(\frac{i(n-i+1)}{(n+1)^2(n+2)}\).
The PRS
defined by \eqref{eq:prs} and \eqref{eq:boundary}
is constructed as a weighted logarithms of the marginal densities
with a minor modification so that the medians are approximately most likely
\citep[{\it c.f. \rm}][for the choice of $c_n=\frac{2}{3}$]{kerman2011closed}.
The weights are set to be about inversely proportional to their variances and,
thereby, pay more attentions to the tail areas of the underlying sampling distribution.
This leads to
a slightly more general class of PRSs is to use alternative weight
schemes:
\begin{equation}\label{eq:prs-general}
 a_i = w_i(i-1+c_n),\;
	b_i = w_i(n-i+c_n)
\end{equation}
with
\begin{equation}\label{eq:prs-weights}
	w_i =
	\frac{

	{[i(n-i+1)]^{-\nu/2}}
	}{
		\sum_{j=1}^
		{[j(n-j+1)]^{-\nu/2}}
	}
\end{equation}
for $i=1,...,n$, where $\nu \in[0,2]$.
The PRS \eqref{eq:prs-general} with $\nu=2$ is
used in the numeric examples in Section \ref{s:simu}.

\subsection{Induced random sets in the $\gamma$ space}

A realization of random set \eqref{eq:prs} induced the
random sets in the $\gamma$ space
through the convolution \eqref{eq:numeric-association}:
\begin{equation}\label{eq:numeric-association}
	\Gamma(U) =
	\left\{(\gamma_1, ..., \gamma_K):\;
	\mbox{$
	B\left(
	 \sum_{k=1}^K \gamma_k \Phi(X_{(1)}-\vartheta_k) ,...,
	 \sum_{k=1}^K \gamma_k \Phi(X_{(n)}-\vartheta_k)\right)
	 \leq B(U)
	 $}
		\right\}
\end{equation}
This random set for $\gamma$ defines the set of
numerical approximations to the unknown distribution $G(.)$.
The usual IMs uncertainty quantification about
$G(.)$ can be carried out as prescribed in Section \ref{s:IMs}.
In the context of multiple testing, for example,
inference may be made about fractional number of outliers.
For MNMs problem, inference about $\theta_i$'s is often of interest
and obtained by making use of \eqref{eq:numeric-association} and
\eqref{eq:numeric-T}.
This is discussed in detail below in next two subsections.

\subsection{Maximum plausibility estimation}
\label{s:MPE}

A straightforward application of the above IMs is to produce point estimation.
Given an estimate of $G$, the usual Bayes posterior mean
\begin{equation}\label{eq:theta-mean}
	\hat{\theta}_i(\gamma) =
	\frac{\sum_{k=1}^K \gamma_k\phi(X_i-\vartheta_k) \vartheta_k}
	{\sum_{k=1}^K \gamma_k\phi(X_i-\vartheta_k)}
\end{equation}
provides the most efficient estimator \citep{brown1971admissible},
in terms of mean square error (MSE):
\begin{equation}\label{eq:mse}
	\mbox{MSE}(\hat{\theta}) =
	\frac{1}{n}\sum_{i=1}^n(\hat{\theta}_i-\theta_i)^2
\end{equation}

Here we consider MPE of $\gamma$.
The smallest PRS with the nested focal elements,
obtained with the method of (elastic) for validity,
is given by
\begin{equation}\label{eq:min-U}
	\mathcal{U}_{\min} = \left\{
		u: B(u) \leq B_{\min}
		\right\}
\end{equation}
where
\begin{equation}\label{eq:min-g}
	B_{\min} = \min_\gamma\mbox{$
	B\left(
	 \sum_{k=1}^K \gamma_k \Phi(X_{(1)}-\vartheta_k) ,...,
	 \sum_{k=1}^K \gamma_k \Phi(X_{(n)}-\vartheta_k)\right)
	 $}
\end{equation}
This induces the maximum plausibility set for $\gamma$
\begin{equation}\label{eq:mpe-gamma}
	\Gamma_{\min} =
	\left\{(\gamma_1, ..., \gamma_K):\;
	\mbox{$
	B\left(
	 \sum_{k=1}^K \gamma_k \Phi(X_{(1)}-\vartheta_k) ,...,
	 \sum_{k=1}^K \gamma_k \Phi(X_{(n)}-\vartheta_k)\right)
	 \leq B_{\min}
	 $}
		\right\}
\end{equation}
It follows that the maximum plausibility
is necessarily one and is achieved for all $\gamma\in \Gamma_{\min}$.
Since $\Gamma_{\min}$ given in \eqref{eq:mpe-gamma} is a set,
we have a set of MPEs of $\theta_i$, one for each $\gamma\in \Gamma_{\min}$.

For simple practical applications, we can summarize the above set of MPEs by
two extremes, the lower and upper posterior expectations
\begin{equation}\label{eq:mpe-extremes}
	\hat{\theta}_i^{(\mbox{lower})} = \min_{
		\gamma \in \Gamma_{\min}
		} \hat{\theta}_i(\gamma)
		\qquad\mbox{and}\qquad
	\hat{\theta}_i^{(\mbox{upper})} = \max_{
		\gamma \in \Gamma_{\min}
		} \hat{\theta}_i(\gamma)
\end{equation}
for $i=1,...,n.$
When a single value estimate is desired,
the mean of the two extremes may be used, that is,
\begin{equation}\label{eq:mpe-middile}
\hat{\theta}_i = \frac{
	\hat{\theta}_i^{(\mbox{lower})} +
	\hat{\theta}_i^{(\mbox{upper})}}{2}
\end{equation}
This one is used to evaluate PRS performance in the simulation
study in Section \ref{s:simu}.

\subsection{Plausibility intervals}
\label{ss:CIs}

Interval estimation can be made to
have desirable frequency-calibration property.
Suppose that one is interested in the
simultaneous interval estimation, one for each $\theta_i$.
Denote by $C_i^{(1-\alpha)}(X)$
the interval estimate of $\theta_i$ with coefficient
$1-\alpha$.
A frequency-calibration property commonly used in practice
is that
\begin{equation}\label{eq:cis-property}
	E\left(
	\frac{1}{n}\sum_{i=1}^n
	I\left\{\theta_i\in C_i^{(1-\alpha)}(X)\right\}
	\right) \geq 1 - \alpha.
\end{equation}
In words, it requires on average at least $1-\alpha$ percent of
the true values of $\theta_i$'s will be covered,
in a frequentist sense within and cross experiments.
This, more precisely, the case with only the equality constraint in \eqref{eq:cis-property}
has been a popular standard in practice
\citep{wahba1983bayesian, brown2011confidence}.

The validity of IMs that guarantees  frequency-calibration
has been established in \cite{martin2013inferential}.
Formally, we have the following result.

\begin{theorem}[IMs Validity]
Let $B_{1-\pi}$ be the $(1-\alpha)$-th quantile of $B(U)$
and denote by $\Gamma_{1-\pi}$ the $\alpha$-plausibility region
\begin{equation}\label{eq:cis-property}
	\Gamma_{1-\pi} =
	\left\{(\gamma_1, ..., \gamma_K):\;
	\mbox{$
	B\left(
	 \sum_{k=1}^K \gamma_k \Phi(X_{(1)}-\vartheta_k) ,...,
	 \sum_{k=1}^K \gamma_k \Phi(X_{(n)}-\vartheta_k)\right)
	 \leq B_{1-\pi}
	 $}
		\right\}.
\end{equation}
Then,
	$E\left((\gamma_1,...,\gamma_n) \in \Gamma_{1-\gamma}\right)
	\ge 1-\pi$.
\end{theorem}

\noindent
Suppose that for each $\gamma \in \Gamma_{1-\pi}$,
we construct the two-sided ``confidence interval''
with the left-end point
\begin{equation}\label{eq:cis-left-end}
 \theta_{\alpha/2}^{(\mbox{lower})}(\gamma) =
	\max_\ell\left\{
	\theta_\ell:
	\hat{\theta}_i(\gamma) =
	\frac{\sum_{k=1}^\ell \gamma_k\phi(X_i-\vartheta_k)}
	{\sum_{k=1}^K \gamma_k\phi(X_i-\vartheta_k)}
	\leq  \frac{\alpha}{2}
		\right\}
\end{equation}
	and the right-end point
\begin{equation}\label{eq:cis-right-end}
 \theta_{\alpha/2}^{(\mbox{upper})}(\gamma)
	\min_r\left\{
	\theta_r:
	\hat{\theta}_i(\gamma) =
	\frac{\sum_{k=r}^n \gamma_k\phi(X_i-\vartheta_k)}
	{\sum_{k=1}^K \gamma_k\phi(X_i-\vartheta_k)}
	\leq \frac{\alpha}{2}
		\right\}.
\end{equation}
This leads to the following plausibility interval
\begin{equation}\label{eq:cis-def}
	C_i(1-\pi, 1-\alpha) \equiv
\left[
\min_{\gamma\in \Gamma_{1-\pi}} \theta_{\alpha/2}^{(\mbox{lower})}(\gamma),\;
\max_{\gamma\in \Gamma_{1-\pi}} \theta_{\alpha/2}^{(\mbox{upper})}(\gamma)
\right]
\end{equation}
The following theorem establishes a lower bound for
the ``confidence'' coefficient of \eqref{eq:cis-def}.

\begin{theorem}[Coverage Property]
\label{thm:coverage-01}
	The coverage probability of the confidence interval \eqref{eq:cis-def}
	is at least $(1-\gamma)(1-\pi)$, {\it i.e.},
\[
	\mbox{\rm Pr}\left(
C_i(1-\pi, 1-\alpha) \ni \theta_i
	\right)
	\ge (1-\pi)(1-\alpha)
\]
for all $\pi \in (0,1)$ and $\alpha\in(0,1)$.
\end{theorem}
\begin{proof}
The IMs validity says that $\gamma$ is covered by $\Gamma_{1-\pi}$
$1-\pi$ percent. In such a case,
the interval defined by the two end points
\eqref{eq:cis-left-end} and \eqref{eq:cis-right-end}
covers the true value, generated from $G$, with probability
	$1-\alpha$. The claimed result follows from the
	multiplication rule of probability.
\end{proof}

Theorem \ref{thm:coverage-01}
provides a way of constructing exact confidence intervals.
Because the set for $\mathbb{G}$ or $\gamma$ can be large,
rendering conservative intervals,
we provide some relevant comments on this issue, aiming to
a method of adaptive construction of shorter intervals
with approximate coverage via Monte Carlo simulations.

\begin{remark}[Within-experiment Validation]
Within-experiment frequency evaluation of confidence interval
is possible when $n$ is large.
The fractional number of $\theta_i$'s falling outside of
their $1-\alpha$ confidence intervals is expected to be
$n\alpha$ and in such cases, the corresponding observed $X_i$'s
have probability 0.5 to fall outside of the intervals as well.
On the basis of this argument,
we expect to see that the number of $X_i$ falling outside of
corresponding individual $1-\alpha$ confidence intervals for $\theta_i$
is about $n\alpha/2$.
\end{remark}

\begin{remark}[Selection of $\pi$ as a Way of Modeling]
When confidence intervals are excessively large,
modeling can be useful. Efron's $g$-modeling,
which assumes $\gamma \propto e^{Q\zeta}$
for some fixed $(K\times p)$ matrix $Q$ with
unknown $\zeta\in\mathbb{R}^p$, serves a good example.
Alternatively, one can also
choose $\pi$ and attribute the ignored relevant error to
the modeling error.
\end{remark}

\begin{remark}[Adaptive Adjustment]
	\label{remark:adaptive-adjustment}
Efficient confidence intervals can be constructed via Monte Carlo
as follows. Without loss of generality, suppose that
$95\%$ CIs are of interest. So, take $1-\alpha = \sqrt{0.95}$,
evaluate coverage via Monte Carlo simulation
with $\gamma \in ...$ $1-\pi = \sqrt{0.95}$
for a sequence of $m$, say, $m=10$, quantiles $B(U)$ with probabilities
$(s/m)\sqrt{0.95}$, $s=1,...,m$.
The MPE estimate of $\mathbb{G}$, for example, can be used to
generate data for this Monte Carlo approach.
The approximate intervals for the real data
are the $(s/m)\sqrt{0.95}$ intervals, where $s$ is the index of
	the simulation intervals with their actual coverage
	in the simulated data matching the targeted coverage.
This approach is used in the real-data example in Section \ref{ss:real-data-example}.
\end{remark}

\section{ IMs for MNMs of the classic kind}
\label{s:classic}

\subsection{Constructing an association via augmenting latent auxiliary variables}

The Stein phenomenon implies that
when performance assessment of inferential methods
involves simultaneous inference about $\theta_i$'s,
such as mean square errors for point estimation,
efficient methods can be developed by allowing
the individual estimators of $\theta_i$'s to share
risk with each other.
To see this intuitively from an IMs perspective,
we write $Z_1,..., Z_{n}$ in \eqref{eq:association-01}
in terms of their sorted values $Z_{(1)},..., Z_{(n)}$ with
an auxiliary random permutation of $(1,...,n)$, denoted by $(\tau_1,...,\tau_n)$:
\begin{equation}\label{eq:association-02}
        X_i = \theta_i + Z_{(\tau_i)}
\end{equation}
Since everyone of the $n!$ permutations is equally likely,
the sampling models \eqref{eq:association-01} and \eqref{eq:association-02}
are equivalent to each other.
Clearly, efficient simultaneous inference can be obtained, conditional on
$(\tau_1,...,\tau_n)$.
Here we develop efficient methods by estimating
the unknown permutation $\tau$ as a way of
controlling the level of risk sharing.
Before proceeding we record a result that is useful in what follows,
where optimization over $(Z_{(1)},...,Z_{(n)})$ and $\tau=(\tau_1,...,\tau_n)$
can be carried out over $\theta \in \mathbb{R}^n$.
\begin{prop}
The observed data $X_1$,..., $X_n$,
	the auxiliary variables $(Z_{(1)},...,Z_{(n)})$ and $\tau=(\tau_1,...,\tau_n)$
determine and are determined by $\theta_i$'s.
\end{prop}

What is interesting here for IMs is that the number of
auxiliary variables is larger than that of the observed data.
For this reason, we call the auxiliary permutation
$\tau$ a {\it latent} auxiliary variable.
For technical clarity, we
denote by RP the abbreviation of ``random permutation'' and
summarize the above discussion into the following lemma concerning
the equivalence of three sampling associations,
including \eqref{eq:association-01}
and \eqref{eq:association-02}.

\begin{lemma}[Equivalence of Three Sampling Associations]
	\label{thm:association-equivalence}
The sampling associations \eqref{eq:association-01},
\eqref{eq:association-02}, and
\begin{equation}\label{eq:association-03}
	X_{\tau_i} = \theta_{\tau_i} + Z_i
	\qquad(\tau \sim \mbox{RP}(1,...,n);\;
	Z_i\stackrel{iid}{\sim}N(0, 1))
\end{equation}
are all equivalent in the sense that the induced
sampling distributions of $X$'s, given
	$\theta$'s, are the same, i.e.,
	$X_i\stackrel{iid}{\sim}N(\theta_i, 1)$ for all $i=1,...,n$.
\end{lemma}

Lemma \ref{thm:association-equivalence} can be easily
established with standard technical arguments in terms of sampling distributions.
This result allows for flexibility in formulating
IMs for prior-free probabilistic inference.
For example, Equation \eqref{eq:association-03} implies that
the sampling scheme in the classic kind can be viewed
as follows. There is a collection of $\theta$'s, denoted by
$\{\theta_{(k)}: k=1,...,n\}$.
One takes $\theta$'s from $\{\theta_{(k)}: k=1,...,n\}$ without
replacement,
draws a sample of size of $n$ from $N(0,1)$, and
creates $X$'s via either \eqref{eq:association-03} or
\(
	X_{\tau_i} = \theta_{\tau_i} + Z_{(i)}
	\) for \(i=1,...,n\).

Enforcing risk sharing clearly prevents $\tau$ from being
integrated out. Note that different permutations
produce observed data samples as a collection of
data points looking like each other or,
more precisely, having the same distribution.
Given the observed data, however, the permutation
that was chosen to generate the data would be the best to
be conditioned on for inference via predicting
$Z_{(\tau_1)}, ..., Z_{(\tau_n)}$.
This suggests that the collection of data points $X_i$'s
be used to establish an association, which we call
an {\it augmented association}.
In other words, we can make use of the  sampling
model of the order and sufficient statistics  $X_{(1)}\leq ...\le X_{(n)}$
to create such an augmented association.

Note that the marginal distribution for a random draw from
$\{X_{(1)},...,X_{(n)}\}$ is given by
\begin{equation}\label{eq:resampling-dist-01}
        F_{\theta}(x)
        =  \sum_{k=1}^n \frac{1}{n}\Phi(x - \theta_k)
\end{equation}
In terms of the true realizations of the auxiliary variables
we can also write \eqref{eq:resampling-dist-01} as
\begin{equation}\label{eq:resampling-dist-02}
        F_{\theta}(x)
	=  \sum_{k=1}^n \frac{1}{n}\Phi(x - [X_k-Z_{(\tau_k)}])
\end{equation}
With Equation \eqref{eq:resampling-dist-02}
we have the following association
\begin{equation}\label{eq:resampling-association-01}
	F_{\theta}(X_{(i)})
	=  \sum_{k=1}^n \frac{1}{n}\Phi(X_{(i)} - [X_k-Z_{(\tau_k)}])
\end{equation}
for $i=1,...,n$.
Before proceeding we make the following two remarks, which motivates
the PRS construction in the next section.
\begin{remark}[Dependence of Latent Auxiliary Variables]
\label{remark:latent-auxiliary}
Comparison of \eqref{eq:association-02} and
\eqref{eq:resampling-association-01} reveals that
\eqref{eq:resampling-association-01}
unsurprisingly depends on $\Phi(Z_i)$'s. This can be
seen clearly, for example, by the extreme case where all the $\theta_i$'s are the same and, thereby,
	in this extreme case \eqref{eq:association-02} and
\eqref{eq:resampling-association-01} are identical.
\end{remark}
\begin{remark}[Sampling W/O Replacement]
\label{remark:without-replacement}
Although $F_{\theta}(.)$ defined in \eqref{eq:resampling-dist-01}
is the distribution function for all $X_i$'s,
$\{X_{(1)},...,X_{(n)}\}$ is not a sample from $F_{\theta}(.)$.
This is because that $\{X_{(1)},...,X_{(n)}\}$ is
equivalent to the ordered sample of observations $X_i$'s taken
	from the $n$ subpopulations $N(\theta_i, 1)$
	without replacement (of subpopulations),
	whereas a sample from $F_{\theta}(.)$, i.e., the $n$ subpopulations
	$N(\theta_i, 1)$, is necessarily drawn with replacement.
\end{remark}

\subsection{Point and interval estimations}

The PRS defined in Section \ref{s:partial-bayes} for the sorted uniform samples
can be used for sorted sample of $\Phi(Z_i)$'s.
The remaining challenges concern a specification of a PRS
for the latent permutation $\tau$, which is to be utilized for inference  through
\eqref{eq:resampling-association-01}.
Intuitively, for largest possible risk sharing,
it is important to control the accuracy of predicting $\tau$ using PRS.
For the extreme case in this way of reasoning, we would ignore predicting
$Z_{(i)}$'s, at least not predicting directly, {\it i.e.},
formally predicting $Z_{(i)}$'s with their entire probability sample space.
In other words, predicting $\tau$ alone effectively
estimates the collection of $\theta$'s.
What is needed is to create a random bound $\mathcal{B}$ on
$B(F_\theta(X_{(1)}),...,F_\theta(X_{(n)}))$,
working in the same way as the basic IMs to induce
a random set for $\theta$'s.
Theoretically, valid inference can be made
as long as we use a random bound $\mathcal{B}$ that is stochastically
not smaller than $B(F_\theta(X_{(1)}),...,F_\theta(X_{(n)}))$.
This is supported rigorously by the following theorem.

\begin{theorem}[Validity of conservative PRSs]
	\label{thm:validity-of-conservative-PRSs}
	Suppose that $B$ is a random variable
	that is stochastically larger than
	$B(F_\theta(X_{(1)}),...,F_\theta(X_{(n)}))$
	with $X$ and $\theta$ pairs given in \eqref{eq:association-02}.
	Then inference on $\theta$ in the Inferential Models framework
	through the induced
	random set on the parameter space
\begin{equation}\label{eq:prs-classic-01}
	\Theta(\mathcal{B}) =
	\left\{\theta:\,
	\theta\in\mathbb{R}^n;\;
	B(F_\theta(X_{(1)}),...,F_\theta(X_{(n)}))
	\leq \mathcal{B}
		\right\}
\end{equation}
is valid.
\end{theorem}

\begin{proof}
	This result can be proved by using
	arguments similar to those in the proof of
	Theorem 2 of \cite{martin2013inferential};
	see also \cite{martin2015plausibility}.
\end{proof}

In the simulation study of Section \ref{s:simu}, we use
$\mathcal{B} = B(U_{(1)},...,U_{(n)})$, where
$U_{(1)},...,U_{(n)}$ is the sorted $n$ uniforms.
A heuristic argument for this is that
the sorted sample of $X$'s without replacement (of $\theta$'s)
spreads more evenly than that with replacement from the
same marginal distribution \eqref{eq:resampling-dist-01}.
Together with \eqref{eq:resampling-dist-01},
this implies that the sorted $X$s have about the same component-wise locations
but smaller variability.
As a result, $\mathcal{B} = B(U_{(1)},...,U_{(n)})$
serves as a random bound in Theorem \ref{thm:validity-of-conservative-PRSs}
for producing valid inference on $\theta$, at least for
values of $B(F_\theta(X_{(1)}),...,F_\theta(X_{(n)}))$ that are not large;
see Figure \ref{fig:conservatie-prs} for a pictorial illustration that
supports this argument.
Although it guarantees validity, the use of
$\mathcal{B} = B(U_{(1)},...,U_{(n)})$
can lead to conservative inference.
Since the distribution of
$B(F_\theta(X_{(1)}),...,F_\theta(X_{(n)}))$ depends on $\theta$'s,
efficient inference can be made by using
the so-called assertion-specific PRS
\citep{martin2015conditional}.
More precisely, we summarize this approach into the following theorem,
followed by remark on its potential applications and computational challenges.

\begin{theorem}[Validity of Assertion-specific PRSs for MPE and Constructing CIs]
	\label{thm:assertion-specific-prs}
	Suppose that the assertion
	$\{\vartheta'_{(k)}:\; k=1,...,n\} \in \mathbb{R}^n$
	is of interest.  Define
	the induced random set on $\theta$
\begin{equation}\label{eq:prs-classic-01}
	\Theta(X') =
	\left\{\theta:\,
	\theta\in\mathbb{R}^n;\;
	B(F_\theta(X_{(1)}),...,F_\theta(X_{(n)}))
	\leq
	B(F_{\vartheta'}(X'_{(1)}),...,F_{\vartheta'}(X'_{(n)}))
		\right\}
\end{equation}
where $X'_i \stackrel{iid}{\sim}N(\vartheta'_i, 1)$ for
	$i=1,...,n$.
	Then inference based on $\Theta(X')$ in the IMs framework
	is valid.
\end{theorem}
\begin{proof}
	This result can also be proved by using
	arguments similar to those in the proof of
	Theorem 2 of \cite{martin2013inferential}.
\end{proof}

\begin{remark}
	According to
	Theorems \ref{thm:validity-of-conservative-PRSs}
	and \ref{thm:assertion-specific-prs},
a way of constructing efficient confidence regions
is to find the minimum plausibility for any
	given $\theta \in \Theta(\mathcal{B})$.
	This minimum plausibility is the plausibility
	of $\Theta(\mathcal{B})$.
	Evaluation of this plausibility appears intensive.
	It would be interesting to develop efficient computational
	methods for this approach.
\end{remark}

The theory established so far provides inference about
the $\theta$'s as a collection.
Recall that given the $\theta$ collection, $X$'s were generated
by drawing $\theta$'s without replacement.
Standard algebraic operations of applying the Bayes theorem lead to
the following results.

\begin{theorem}[Conditional Distribution of $\theta$]
	\label{thm:posterior-dist-of-theta}
	Suppose $n\geq 2$ and assume that the value of $\theta$, $\{\vartheta_k: \vartheta_k=\theta_{(k)}, k=1,...,n\}$, are known.
	Then the partial conditional distribution of $\theta_i$ given $X_i$
	has the probability mass function (pmf)
\begin{equation}\label{eq:partial-cond}
		\mbox{Pr}(\theta_i = \vartheta_j|X_i)
	=\frac{\phi(X_i-\vartheta_j)}
	      {\sum_{k=1}^n\phi(X_i-\vartheta_k)}
\end{equation}
	and the full conditional distribution of $\theta$ given $X$ has the pmf
\begin{equation}\label{eq:full-cond}
	\mbox{Pr}(\theta_1 = \vartheta_{\tau_1},
	..., \theta_n = \vartheta_{\tau_n}
	|X_1,...,X_n)
	=\frac{\frac{1}{n!}\prod_{j=1}^n\phi(X_j-\vartheta_{\tau_j})
	}{
		\sum_{\tau'} \frac{1}{n!}\prod_{j=1}^n\phi(X_j-\vartheta_{\tau'_j})
		}
\end{equation}
\end{theorem}

Theorem \ref{thm:posterior-dist-of-theta} makes it possible
to estimate $\theta$'s via $E(\theta|X)$, fulfilling
the required operation to overcome the fundamental difficulty
of making inference for the classic kind ({\it c.f.}, Section
\ref{s:intro} and \cite{stigler19901988}).
While it is appealing to use the full conditional distribution
\eqref{eq:full-cond} for MPE, for computational simplicity
here we use the following simple estimator, obtained from
\eqref{eq:partial-cond},
\begin{equation}\label{eq:mpe-classic-01}
	\hat{\theta}_i
	=\frac{\sum_{k=1}^n\vartheta_k\phi(X_i-\vartheta_k)}
	      {\sum_{k=1}^n\phi(X_i-\vartheta_k)}
\end{equation}
for $i=1,...,n$,
with $\{\vartheta_{(k)}: k=1,...,n\}$ obtained from
the approach in either Theorem \ref{thm:validity-of-conservative-PRSs}
or Theorem \ref{thm:assertion-specific-prs}.
The computational strategies for the empirical-Bayes kind
provided in Appendices can be applied here for the classic kind.
More discussion is given in Section \ref{s:discussion}
on partial conditioning versus full conditioning
in Theorem \ref{thm:posterior-dist-of-theta} for
estimation $\theta$.

\ifthenelse{1=1}{}{
	\color{red}\tiny
Note that the same PRS could be used if $X_i$ had been taken
from $F_\theta(.)$ with replacement. Approximate inference obtained by
taking the sample as obtained with replacement from $F_\theta(.)$.
This is a special case of the empirical-Bayes kind
where the mixing distribution $G(.)$ is uniform
on $n$ locations $\theta_i$.
\[
	\Theta_U \equiv
	\left\{\vartheta:
	\vartheta \in \mathbb{R}^n,
	\vartheta_1 \leq \vartheta_2 \le ... \le \vartheta_n,
	B\left(F_\theta(X_{(1)}),
	..., F_\theta(X_{(n)}) \right)
	\leq ??? B\left(U\right)
	\right\}
	\]

Here we focus on an alternative approximation by resampling
or bootstrapping $\{X_i: i=1,...,n\}$.
More precisely, let $\tilde{X}_{(1)}^{(m)},...,\tilde{X}_{(n)}^{(m)}$
denote the ordered $m$-th sample in a total number of $M$ bootstrap samples.
We consider the following induced random set
for sorted $\theta_i$'s:
\begin{equation}\label{eq:resampling-prs-tau}
	\Theta_U \equiv
	\left\{\vartheta:
	\vartheta \in \mathbb{R}^n,
	\vartheta_1 \leq \vartheta_2 \le ... \le \vartheta_n,
	\lim_{M\rightarrow \infty}
	\frac{1}{M}\sum_{m=1}^{M} B\left(F_\theta(\tilde{X}_{(1)}^{(b)}),
	...,
	F_\theta(\tilde{X}_{(n)}^{(b)}) \right)
	\leq ??? B\left(U\right)
	\right\}
\end{equation}
where $U$ is the vector of sorted uniform sample of size of $n$.
{\color{red} Assertion-specific PRS???}
\footnote{\color{red}
	Averaging over bootstrap $\theta$-estimates
	seems to work well...
	Computational strategies in Appendix C
	}
	\color{black}
}

\section{Numerical illustration}
\label{s:simu}

\subsection{A simulation study}

In this simulation study,
we consider the three scenarios used in the simulation study
of \cite{jiang2022estimation} for their many-normal-means example:
\begin{quote}
\begin{description}
	\item[\it Single-Mode.] $\theta_i \stackrel{iid}{\sim}N(0, 0.01)$,
	 the case in favor of James-Stein estimator;
	\item[\it Two-Mode.] $\theta_i \stackrel{iid}{\sim}
		\frac{1}{2}	N(-2, 0.01) +\frac{1}{2}	N(2, 0.01)$;
		and
	\item[\it Single-Mode with Outliers.] $\theta_i \stackrel{iid}{\sim}
		0.9\delta_{0} +	0.1N(-3, 1)$, where
		$\delta_0$ denotes the degenerated distribution at $0$.

\end{description}
\end{quote}
For all simulation studies, three different sample sizes, $n = 10,20, $ and $50$ are considered.
The numerical deconvolution uses
$K=1000$ points equally spaced in the range
from $X_{(1)}+\Phi^{-1}(0.0001)$
to from $X_{(n)}+\Phi^{-1}(0.9999)$,
where $\Phi^{-1}(.)$ stands for the inverse
mapping of the standard normal cdf $\Phi(.)$.
For each case, $M = 200$ data sets are generated
to compute the Monte Carlo approximations to, for example,
mean square error (MSE).

\setlength{\tabcolsep}{5pt}
\begin{table}[!htp]
\centering
\footnotesize
\caption
	MSE results in three simulation studies with different methods.
	}
	\label{tb:1}
\begin{tabular}{ccccccccccccc}
	&&&&&&&&&&&&\\
\hline
\hline
	&&&&&&&&&&&&\\
	\multirow{3}{*}
		\begin{tabular}[c]{@{}c@{}}
			\\
			\textit{Problem Kind}\\
			\textit{/Method}
		\end{tabular}
	}
	&& \multicolumn{3}{c}{{\scriptsize$N(0,0.01)$}}        && \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}{\scriptsize$\frac{1}{2}N(-2,0.01)+\frac{1}{2}N(2,0.01)$}\end{tabular}} && \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}{\scriptsize$0.9\delta_{0}+0.1N(-3,1)$}\end{tabular}} \\
		\cline{3-5}
		\cline{7-9}
		\cline{11-13}
		&&&&&&&&&&&&\\

   && \underline{$n = 10$} & \underline{$n = 20$} & \underline{$n = 50$}
   && \underline{$n = 10$} & \underline{$n = 20$} & \underline{$n = 50$}
   && \underline{$n = 10$} & \underline{$n = 20$} & \underline{$n = 50$}
			   \\
		\cline{1-1}
	&&&&&&&&&&&&\\
	\underline{\it The EB Kind} &&&&&&&&&&&&\\
	$g$-modeling                     && 0.419          & 0.395          & 0.168
	&& 0.748      & 0.724     & 0.737
	&& 0.554      & 0.552     & 0.364                         \\
	Auto-modeling                    && 0.199 & 0.110 & 0.054
	&& 0.600                      & 0.437          & 0.356
	&& 0.420                 & 0.418                 & 0.312                \\

	IMs     &&0.219&0.134&{\color{black} 0.062}
	&& 0.678&0.581&{\color{black} 0.500}
	&& 0.462&0.370& {\color{black} 0.361}\\
	&&&&&&&&&&&&\\
	\underline{\it Classic Kind}&&&&&&&&&&&&\\
	MLE                              && 1.022          & 0.972          & 0.985          && 0.990                               & 1.009                              & 1.003                              && 0.971                          & 1.021                          & 0.983                         \\
	James-Stein                      && 0.300          & 0.167          & 0.066          && 0.876                               & 0.850                              & 0.826                              && 0.521                          & 0.516                          & 0.482                         \\
	IMs
	&&0.312&0.166&0.080
	&& 0.719&0.557&0.422
	&& 0.484&0.376&0.317\\

	&&&&&&&&&&&&\\
	\hline
\end{tabular}
\end{table}

\begin{figure}[!htb]
\centering
\includegraphics[width=4in]{fig-two-modal.pdf}
	\caption{A typical example of IMs for the empirical-Bayes kind in the two-modal case with $n=20$, where
	X --- $X$'s, curves --- lower and upper $E(\theta_i|X)$,
	circles --- true values of $\theta$'s, and
	vertical segments --- confidence intervals computed at the MPE (set).
	The coefficients of the confidence intervals is about 97\%, according
	to the simulation results given in Table \ref{tb:2}.
	}
\label{fig:cis-two-mode}
\end{figure}

\ifthenelse{1=1}{

\setlength{\tabcolsep}{4pt}
\begin{table}[!htp]
\centering
\footnotesize
	\caption{Actual coverage and average interval length (in parentheses)
	obtained by the IMs method for the empirical-Bayes kind
	in three simulation studies.
	}
	\label{tb:2}
\begin{tabular}{cccccccccc}
	&&&&&&&&&\\
\hline
\hline
	&&&&&&&&&\\
	\multirow{3}{*}
		\begin{tabular}[c]{@{}c@{}}
			\\
			\textit{}\\
			\textit{Targeted Coverage}
		\end{tabular}
	}
	&& \multicolumn{2}{c}{{\scriptsize$N(0,0.01)$}}        && \multicolumn{2}{c}{\begin{tabular}[c]{@{}c@{}}{\scriptsize$\frac{1}{2}N(-2,0.01)+\frac{1}{2}N(2,0.01)$}\end{tabular}} && \multicolumn{2}{c}{\begin{tabular}[c]{@{}c@{}}{\scriptsize$0.9\delta_{0}+0.1N(-3,1)$}\end{tabular}} \\
		\cline{3-4}
		\cline{6-7}
		\cline{9-10}
		&&&&&&&&&\\

   && \underline{\;\;\;\;$n = 10$\;\;\;\;} & \underline{\;\;\;\;$n = 20$\;\;\;\;}
   && \underline{\;\;\;\;$n = 10$\;\;\;\;} & \underline{\;\;\;\;$n = 20$\;\;\;\;}
   && \underline{\;\;\;\;$n = 10$\;\;\;\;} & \underline{\;\;\;\;$n = 20$\;\;\;\;}
			   \\
		\cline{1-1}
	&&&&&&&&&\\
	MPE	&&83.77 (1.57)&94.14 (1.63)   &&91.58 (2.95) &96.94 (2.96)   &&86.09 (2.00)&93.84 (1.93)\\
	50\%	&&93.43 (2.14)&96.79 (1.99)  &&97.91 (4.34)&99.08 (3.42) &&94.72 (2.67)&95.73 (2.36) \\
	75\%	&&97.56 (2.47)&97.93 (2.18) &&98.56 (4.68)&99.23 (3.97) &&96.06 (3.03)&97.01 (2.56) \\
	90\%	&&99.18 (2.84)&98.93 (2.42) &&98.42 (4.89)&99.44 (4.50) &&97.97 (3.43)&98.41 (2.79) \\
	95\%	&&99.50 (3.19)&99.14 (2.59)  &&98.71 (5.09)&99.29 (4.64) &&98.28 (3.76)&98.54 (2.97)\\
	&&&&&&&&&\\
	\hline
\end{tabular}
\end{table}

	}{
\begin{table}[!htp]
\centering
\footnotesize
	\caption{Actual coverage and average interval length (in parentheses)
	obtained by the IMs method for the empirical-Bayes kind
	in three simulation studies. {\color{red}[FIXME]}
	}
	\label{tb:2}
\begin{tabular}{ccccccccccccc}
\hline
\hline
	&&&&&&&&&&&&\\
	\multirow{3}{*}
		\begin{tabular}[c]{@{}c@{}}
			\\
			\textit{}\\
			\textit{Targeted Coverage}
		\end{tabular}
	}
	&& \multicolumn{3}{c}{{\scriptsize$N(0,0.01)$}}        && \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}{\scriptsize$\frac{1}{2}N(-2,0.01)+\frac{1}{2}N(2,0.01)$}\end{tabular}} && \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}{\scriptsize$0.9\delta_{0}+0.1N(-3,1)$}\end{tabular}} \\
		\cline{3-5}
		\cline{7-9}
		\cline{11-13}
		&&&&&&&&&&&&\\

   && \underline{$n = 10$} & \underline{$n = 20$}
   && \underline{$n = 10$} & \underline{$n = 20$}
   && \underline{$n = 10$} & \underline{$n = 20$}
			   \\
		\cline{1-1}
	&&&&&&&&&&&&\\
	MPE	&&83.77&94.14&99.97   &&91.58 &96.94&98.99   &&86.09&93.84&98.00\\
	&&(1.57)&(1.63)&(1.77) &&(2.95)&(2.96)&(3.22)  &&(2.00)&(1.93)&(2.52)\\
	&&&&&&&&&&&&\\
	50\%	&&93.43&96.79&--- &&97.91&99.08&--- &&94.72&95.73&--- \\
		&&(2.14)&(1.99)&() &&(4.34)&(3.42)&() &&(2.67)&(2.36)&()\\
	&&&&&&&&&&&&\\
	75\%	&&97.56&97.93&--- &&98.56&99.23&--- &&96.06&97.01&--- \\
	&&(2.47)&(2.18)&()  &&(4.68)&(3.97)&()  &&(3.03)&(2.56)&()\\
	&&&&&&&&&&&&\\
	90\%	&&99.18&98.93&--- &&98.42&99.44&--- &&97.97&98.41&--- \\
	&&(2.84)&(2.42)&() &&(4.89)&(4.50)&()  &&(3.43)&(2.79)&()\\
	&&&&&&&&&&&&\\
	95\%	&&99.50&99.14&---&&98.71&99.29&--- &&98.28&98.54&---\\
	&&(3.19)&(2.59)&() &&(5.09)&(4.64)&()  &&(3.76)&(2.97)&()\\
	&&&&&&&&&&&&\\
	\hline
	\hline
\end{tabular}
\end{table}

		}

\begin{figure}[!htb]
\centering
\includegraphics[width=5.5in]{conservative-prs.pdf}
	\caption{Illustration of conservative PRSs in four different
	collections of $n=20$ $\theta$'s:
	(1) one simulated values from the single mode model,
	(2) one simulated values from the two-mode model, and
	(3) one simulated values from the single mode model
	    with outliers.
	}
\label{fig:conservatie-prs}
\end{figure}

The MSE performance results of different methods for point estimation
are summarized into Table \ref{tb:1}.
The results for Efron's $g$-modeling,
	the auto-modeling method of \cite{jiang2022estimation},
	the method of maximum likelihood estimation (MLE), and
	the James-Stein shrinkage estimator are
	taken from \cite{jiang2022estimation}.
	From these simulation-based numerical results,
	we see that IMs for the classic kind outperforms
	the James-Stein estimator.
	This is not surprising because of the difficulty
	recognized, for example, by \cite{stigler19901988}
	and the sound reasoning with uncertainty
	introduced in this paper for the classic kind.
	Figure \ref{fig:cis-two-mode}
	displays the IMs results for a typical data set
	generated from the two-mode model.
	It is interesting to notice that
	IMs  ``shrinks'' toward local modes, whereas
	the James-Stein estimates are of course linear function of $X$'s.

	The results also show that our IMs for the empirical-Bayes
	kind performs better than Efron's $g$-modeling method.
	Possible explanations may be that
	Efron's $g$-modeling method is subject to
	a larger model error due to the $g$-modeling.
	The use of the method of (penalized) MLE for estimation of
	the hyper-parameters introduced in the $g$-modeling
	could also be a contributing factor.

	In addition, it is encouraging to observe that the IMs
	are comparable with the auto-modeling method of
	\cite{jiang2022estimation}.
	Good performance is expected for the IMs, although
	they may be further improved a bit when better PRSs are used.
	An intuitive explanation for the similar good performance for both the IMs and
	the auto-modeling method of \cite{jiang2022estimation}
	is that both methods take it into consideration
	that generated data from the estimated model should look like
	the observed.
	The IMs formulated this paper does it through a PRS on
	$F_\theta(X_{(i)})$'s, whereas auto-modeling explicitly
	makes this clear from a modeling perspective.

	The results for interval estimation
	produced by the IMs for the empirical-Bayes kind
	are summarized into Table \ref{tb:2} for $n=10$ and $n=20$.
	The ``Targeted Coverage'' is the nominal coefficients,
	except for the row of $MPE$. The values
	in the row of $MPE$ are computed with respect to all $\gamma$'s in the
	maximum plausibility region.
	The confidence coefficient for $\theta$'s conditional on
	$\gamma$'s is set to be $1-\alpha = 0.95^{1/2} = 0.9747.$
	The plausibility coefficient $1-\pi$ is taken to fulfill the
	nominal level, by making use of Theorem \ref{thm:coverage-01}.
	We see that these intervals have more coverage
	than their nominal interval coefficients.
	This is somewhat expected because of the implicit use of
	``confidence'' region for the high 1,000-dimensional
	$\gamma$.
	More efficient intervals can be obtained via
	adaptive construction of shorter intervals; see Remark
	\ref{remark:adaptive-adjustment}.
	This can be in fact seen clearly from able \ref{tb:2}.
	This is also used in Section \ref{ss:real-data-example} for the real-data example.

\ifthenelse{1=1}{}{
\color{blue}
\begin{verbatim}
> source("table1.R")
load results-1-10-01.RData ...
load results-1-10-02.RData ...
Example: 1 n: 10 MSE: 0.189
Example: 1 n: 10 coverage:
          [,1]      [,2]      [,3]
[1,] 1229.0000 1603.0000 1608.0000
[2,] 1620.0000 1620.0000 1620.0000
[3,]   75.8600   98.9500   99.2600
[4,]  401.0000   55.0000   39.0000
[5,]    0.4951    0.0679    0.0481
Example: 1 n: 10 interval.len: 1.211804 2.739934 3.055028
load results-2-10-01.RData ...
load results-2-10-02.RData ...
Example: 2 n: 10 MSE: 0.727
Example: 2 n: 10 coverage:
          [,1]      [,2]     [,3]
[1,] 1593.0000 1764.0000 1772.000
[2,] 1810.0000 1810.0000 1810.000
[3,]   88.0100   97.4600   97.900
[4,]  107.0000   23.0000   19.000
[5,]    0.1182    0.0254    0.021
Example: 2 n: 10 interval.len: 2.553787 4.932374 5.22502
load results-3-10-01.RData ...
load results-3-10-02.RData ...
Example: 3 n: 10 MSE: 0.413
Example: 3 n: 10 coverage:
          [,1]      [,2]      [,3]
[1,] 1283.0000 1609.0000 1615.0000
[2,] 1650.0000 1650.0000 1650.0000
[3,]   77.7600   97.5200   97.8800
[4,]  354.0000   43.0000   32.0000
[5,]    0.4291    0.0521    0.0388
Example: 3 n: 10 interval.len: 1.650383 3.359621 3.692479
\end{verbatim}
\color{black}
}

\ifthenelse{1=0}{}{
\subsection{A real-data example}
\label{ss:real-data-example}

\ifthenelse{1=1}{}{
We refer to Zhou et al. (2003)
for a comprehensive review of meta-analysis. Here
we focus on a popular statistical model for meta-analysis, namely,
the many-normal-means model
\begin{equation}\label{eq:ma-sampling-model}
y_i \stackrel{ind}{\sim} N(\theta_i, s_i^2)
\qquad(i=1,...,n),
\end{equation}
where $y_i$ is the estimated/observed treatment effect
in the $i$-th study with $\theta_i$ the unknown
treatment effect to be inferred and $s_i^2$ the
associated standard deviation that is taken to be known,
assuming that the sample sizes in all the studies are not too small.
}

\setlength{\tabcolsep}{4pt}
\begin{table}[!htp]
\footnotesize
\begin{center}
	\caption{The data of \cite{rubin1981estimation} on SAT coaching experiments
 \label{tbl:coaching-experiments} }
\begin{tabular}{rcrcr}
\multicolumn{5}{c}{}\\
\hline
\hline
\multicolumn{5}{c}{}\\
       &&  Estimated &&    Associated \\
 School $(i)$ &&   Treatment-Effects $(y_i)$ &&    Standard-Error $(s_i)$\\
\cline{1-1} \cline{3-3} \cline{5-5}
\multicolumn{5}{c}{}\\
 1&&  28.39&&   14.9 \\
 2&&   7.94&&   10.2 \\
 3&&  -2.75&&   16.3 \\
 4&&   6.82&&   11.0 \\
 5&&  -0.64&&    9.4 \\
 6&&   0.63&&   11.4 \\
 7&&  18.01&&   10.4 \\
 8&&  12.16&&   17.6 \\
\multicolumn{5}{c}{}\\
\hline
\end{tabular}
\end{center}
\end{table}

\cite{rubin1981estimation} considered assessing effects of SAT coaching programs
based on  $n=8$ parallel randomized experiments in 8 schools.
The estimated individual effects and the associated
standard deviations are tabulated in
Table \ref{tbl:coaching-experiments}.
It is arguable to use the following sampling model
\begin{equation}\label{eq:ma-sampling-model}
y_i \stackrel{ind}{\sim} N(\mu_i, s_i^2)
\qquad(i=1,...,n),
\end{equation}
because each $y_i$ was obtained from
a data set that is large enough for the acceptance of the normality
assumption and the assumption that $s_i^2$'s are known.

Let $X_i=y_i/s_i$ and let $\theta_i=\mu_i/s_i$ for $i=1,...,8$.
Inference about $\mu_i$ from $y$'s via \eqref{eq:ma-sampling-model}
can be obtained from inference about $\theta$'s from $X$'s based on
the sampling model \eqref{eq:association-01}.
In this case, the Efron-Morris shrinkage estimator over-shrinks
$\theta$'s due to the fact that the sample variance of $X$'s is small,
resulting a strong negative correlation between $X$'s and the
shrinkage estimates of their corresponding $\theta$s.
As suggested in the literature,
this can be of course easily modified
in the same way of creating the James-Stein positive-part estimator
\citep{js1961}.

\begin{figure}[!htb]
\centering
\includegraphics[width=4in]{real-example-02.pdf}
	\caption{The results for the real data example by the
	IMs for the empirical-Bayes kind, where
	X --- $X$s, curves --- lower and upper $E(\theta_i|X)$,
	circles --- Efron-Morris shrinkage estimates, and
	vertical segments --- confidence intervals computed at the MPE (set).
	The coefficients of the confidence intervals is about 97\%, suggested
	by an application of the method of adaptive adjustment;
	see Remark \ref{remark:adaptive-adjustment}.
	}
\label{fig:cis-real-data-example}
\end{figure}

IMs results based on the empirical-Bayes kind are shown in
Figure \ref{fig:cis-real-data-example}.
Similar results can be extended to $\mu$'s because of
the simple scaling transformation $\mu_i = s_i\theta_i$.
The coefficients of the confidence intervals is about 97\%, suggested
        by an application of the method of adaptive adjustment;
        see Remark \ref{remark:adaptive-adjustment}.
It is interesting to see that the MPEs of $\theta$'s are
about the same. To verify this directly,
consider the mean-removed statistic
\[
S^2 = 	X'(I-\frac{1}{n}J)X
 = 4.2899
	\]
where $I$ is the $n\times n$ identity matrix
and $J$ is the $n\times n$ matrix of ones.
Note that the sampling distribution of
$S^2$ is the non-central chi-square with
7 degrees of freedom and non-centrality parameter
$\delta^2=\theta'(I-\frac{1}{n}J)\theta$.
The cdf value of $S^2$ under the central chi-square distribution
is 0.2542.
This doesn't provide much strong evidence against
the assertion that $\delta^2=0$ or, equivalently,
all the $\theta$s are the same.
So, it is not surprising to see that the MPEs
shown in Figure \ref{fig:cis-real-data-example} are about the same;
see \cite{leaf2015valid} for more discussion
from an IMs perspective.

}

\ifthenelse{1=1}{}{
\begin{figure}[!htb]
\centering
	\vspace{2in}
	\caption{\color{red} CI plot... with CI values given in the data table?}
\label{fig:conservatie-prs}
\end{figure}

{\color{blue} Results based on the classic kind ...}

}

\section{Discussion}
\label{s:discussion}

Taking an IMs approach,
we have investigated on making exact and efficient inference for the two
kinds of many-normal-means.
For the empirical Bayes kind, with IMs one can
aim to producing efficient inference by balancing validity and flexibility.
Since the IMs framework offers prior-free probabilistic inference, which
says more than confidence, IMs plausibility can be used for
point and interval estimations.
In addition to the desired exactness, efficiency of the resulting
methods is satisfactory, as shown in the simulation study.
Furthermore, in the case further $g$-modeling is preferred,
the IMs results could be useful for model building
inasmuch as valid non-parametric methods are useful for
model checking;
see \cite{tukey1962future} for more discussion along this line.

It should be noted that IMs works well for the
over-parameterized $\mathbb{G}$.
This can be viewed as a good property of IMs, unlike frequentist and Bayesian schools
of thought.
Frequentist statisticians often do inference by first constructing point estimators and then develop confidence methods based on point estimators.
For over-parameterized models, it is often that
some kinds of regularization via penalty methods are used.
Bayesians have to fully specify a probability model for everything, making it identifiable in a probabilistic sense with the usual theory of probability.
With IMs, our way of reasoning is done with respect to the auxiliary variables. Non-identifiability is no longer an issue, because we work with set-valued functions/mappings.

There are also other kinds of the many-normal-means problem.
A popular kind is obviously the so-called
normal-normal empirical Bayes, where the unobserved $\theta$'s
are assumed to have followed a normal distribution
with unknown hyper-parameters.
This model has been widely used in applied statistics;
see, for example,
\cite{rubin1980using}, \cite{morris1983parametric},
\cite{gelman1995bayesian},
\cite{qiu2018exact}, and references therein.

Our understanding of the classic kind from the IMs perspective
motivated the new idea of introducing latent permutation auxiliary
variable. While the proposed methods are satisfactory in the context
of our numerical results for illustration and comparison,
there are remaining theoretical challenges
as far as optimal solutions are of interest.
This is mainly due to the technical difficulties in
dealing with data collected via sampling without replacement, whereas
investigations of the empirical Bayes kind enjoys the
relative simplicity offered by  data collection through
sampling with replacement.
In particular, it would be interesting
to develop efficient computational algorithms for
evaluation of the full conditional expectation
in Theorem \ref{thm:posterior-dist-of-theta}.
Nevertheless, it is expected that
there is a large difference between
partial and full conditional expectations for small $n$
and not so for large $n$.

The importance of the many-normal-means problem suggests
immediately that our new methods
can be developed
for the many-normal-means type of problems, such as linear and
non-parametric regression. These new methods can also be potentially
extended to situations beyond normal models,
including Poisson and Binomial models.
Incidentally, the idea of considering latent associations
would inspire new research on developments of the IMs theory
itself as well as methods of other schools of thought. For example,
an alternative to popular penalty methods
for maximum likelihood estimation is to make use of
the idea of using latent associations as a data-adaptive way of
regularization, which has been more or less a standard method
in contemporary high-dimensional statistics.
We plan to work on these topics and report results elsewhere.

redpajama_set_name: RedPajamaArXiv


Ezone is back with mega sale (not actually mega) but there are couple of good grabs.
If you have Standard Chartered card, get additional 5% off.
Might be when the OP posted it all were in stock then!! Removed the RED face!!
It is out of stock for many days………..
listen…its out of stock from past many weeks or may be months..!!
hey!!!!!!!!!chill………those who went to this site will know that it is out of stock for many days……….


text:
Alkaline Trio Release New EP
Surprise! Alkaline Trio have released a new three-track EP
Alkaline Trio have released a surprise new three-track EP, simply titled E.P.
The new record is available to stream now (scroll down for Spotify) and features the tracks Minds Like Minefields, Radio Violence and Smokestack.
"With the tour postponed and the current state of the world, we feel and hope that some new music will help people transcend the uncertainty and possible fear they might be feeling, if even for a few minutes," says frontman Matt Skiba.
READ THIS: Alkaline Trio albums ranked by Matt Skiba
"Music has always had a way of helping us all cope with unpleasant times. We can only hope that ours can do the same for our fans on some level. We hope you enjoy the songs."
The EP will also be released physically on a limited-edition two-song 7". There will be 1000 blue variants available in the US and 500 red vinyl in Europe. The two songs included will be Minds Like Minefields, Radio Violence.
"[Radio Violence is] a song about communication breakdowns," says bassist Dan Andriano. "It's about the lies we tell ourselves and others and it's about shutting out those closest to us when we need them most."
This is the first new music we've heard from Alkaline Trio since 2018's Is This Thing Cursed? studio album. Since then, Matt has been busy in his 'other job' with blink-182, releasing album Nine last year.
Matt Skiba says he's "truly grateful" for time with blink-182 in new statement
Alkaline Trio's Matt Skiba has thanked blink-182 and all their fans for having him as frontman for the last seven years: "You were delicious!"
Tom DeLonge thanks Matt Skiba for keeping blink-182 "alive and thriving in my absence"
With Tom DeLonge rejoining blink-182 once more, the frontman has taken to social media to publicly thank Alkaline Trio's Matt Skiba for his "ability to jump in and save the day".
From Dahmer to Bundy: 12 gruesome songs about serial killers
From Jeffrey Dahmer to Richard Ramirez, some of history's most notorious serial killers have been immortalised in rock and metal songs…
Riot Fest announce 2022 line-up: My Chem, Misfits and Nine Inch Nails to headline
My Chemical Romance, the original Misfits (performing Walk Among Us in full) and Nine Inch Nails will top the bill at this year's long-awaited Riot Fest.
In pictures: Taking Back Sunday and Alkaline Trio's sold-out show at London's O2 Academy Brixton
A gallery of Taking Back Sunday and Alkaline Trio absolutely ruling in the capital on Sunday night…
Album review: Dan Andriano & The Bygones – Dear Darkness
Alkaline Trio man Dan Andriano turns down the volume on his latest solo effort…
25 Years Of Vagrant Records: The label that accidentally took emo to the mainstream
From a family-funded Get Up Kids album to becoming a lynchpin of 2000s emo, we look back on Vagrant Records' reluctant rise…
11 great albums disliked by the people who made them
Here are 11 awesome rock and metal albums whose creators don't care for them as much as most fans do…


text:
from django import forms

from satchless.product.forms import BaseVariantForm, variant_form_for_product

from . import models

has_side_effects = True

def _get_existing_variants_choices(queryset, field_names):
    existing_choices = {}
    existing_variants = queryset.values_list(*field_names)

    if existing_variants:
        for index, existing_field_choices in enumerate(zip(*existing_variants)):
            field_name = field_names[index]
            original_choices = queryset.model._meta.get_field(field_name).choices
            flt = lambda choice: choice[0] in existing_field_choices
            existing_choices[field_names[index]] = filter(flt,
                                                          original_choices)
    else:
        for field_name in field_names:
            existing_choices[field_name] = []
    return existing_choices

class VariantWithSizeAndColorForm(BaseVariantForm):
    color = forms.CharField(max_length=10,
            widget=forms.Select(choices=models.ColoredVariant.COLOR_CHOICES))

    def __init__(self, *args, **kwargs):
        super(VariantWithSizeAndColorForm, self).__init__(*args, **kwargs)
        all_variants = self.product.variants.all()
        existing_choices = _get_existing_variants_choices(all_variants,
                                                          ('color', 'size'))
        for field_name, choices in existing_choices.items():
            self.fields[field_name].widget.choices = choices

    def _get_variant_queryset(self):
        color = self.cleaned_data.get('color')
        size = self.cleaned_data.get('size')
        return self.product.variants.filter(color=color, size=size)

    def clean(self):
        if not self._get_variant_queryset().exists():
            raise forms.ValidationError("Variant does not exist")
        return self.cleaned_data

    def get_variant(self):
        return self._get_variant_queryset().get()

@variant_form_for_product(models.Cardigan)
class CardiganVariantForm(VariantWithSizeAndColorForm):
    size = forms.CharField(max_length=10,
            widget=forms.Select(choices=models.CardiganVariant.SIZE_CHOICES))

@variant_form_for_product(models.Dress)
class DressVariantForm(VariantWithSizeAndColorForm):
    size = forms.CharField(max_length=10,
            widget=forms.Select(choices=models.DressVariant.SIZE_CHOICES))

@variant_form_for_product(models.Hat)
class HatVariantForm(BaseVariantForm):
    def get_variant(self):
        return self.product.variants.get()

@variant_form_for_product(models.Jacket)
class JacketVariantForm(VariantWithSizeAndColorForm):
    size = forms.CharField(max_length=10,
            widget=forms.Select(choices=models.JacketVariant.SIZE_CHOICES))

@variant_form_for_product(models.Shirt)
class ShirtVariantForm(VariantWithSizeAndColorForm):
    size = forms.CharField(max_length=10,
            widget=forms.Select(choices=models.ShirtVariant.SIZE_CHOICES))

@variant_form_for_product(models.TShirt)
class TShirtVariantForm(VariantWithSizeAndColorForm):
    size = forms.CharField(max_length=10,
            widget=forms.Select(choices=models.TShirtVariant.SIZE_CHOICES))

@variant_form_for_product(models.Trousers)
class TrousersVariantForm(VariantWithSizeAndColorForm):
    size = forms.CharField(max_length=10,
            widget=forms.Select(choices=models.TrousersVariant.SIZE_CHOICES))

redpajama_set_name: RedPajamaGithub


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text:
Shiites win dominating role in Iraq
Interim Prime Minister Allawi's list came in third in the polls
Iraq – The Shi'a United Iraqi Alliance (UIA) candidates have won at least 132 of 275 seats in the Iraq National Assembly, according to the results announced Sunday.
Capturing 48.1% of the national vote, the UIA is the largest single block in the Assembly. Yet even so large a block must create alliances with other parties to form a unity government. There has been ongoing speculation the UIA is in negotiations with the Democratic Patriotic Alliance of Kurdistan in advance of the poll results.
The union of the Khurdish parties garnered 25.4% of the national vote, equivalent to 71 seats, giving Khurds a vital voice in the National Assembly. The parties have reiterated their call for a Khurdish president or prime minister.
al-Qayimaal Iraqia, dominated by Iyad Allawi's Iraqi National Accord party, polled at 13.8%, and secured 38 seats.
Sunni voters stayed away from the election in droves, leaving them with very little representation in the Assembly. The Iraqis, the largest Sunni party in Iraq, received only 150,000 votes, less than two percent of the votes reported. The next largest, the Independent Democrats Movement, gained less than 0.1 percent.
The National Assembly vote on January 30 also included elections for governors, and the Khurdistan Assembly.
There are approximately 14.2 million eligible voters and 7,761 candidates in the Transitional National Assembly race. Approximately 58% of eligible voters cast their votes in the election, in line with pre-election projections.
Iraqis vote in massive numbers
First free Iraqi election begins; insurgents strike
Iraqi elections kept low-key, but secure, in Paris
"Shiites win resounding victory in Iraq's landmark election" — Agence France-Presse, 02-13-2005, 17h55
Jason Keyser. "Shiites Win Nearly Half of Iraqi Votes" — American Broadcasting Company News, Feb 13, 2005
"Shiites win most votes in Iraqi elections" — Associated Press, Sunday, February 13, 2005 Updated at 11:54 AM EST
"Shiite Alliance Wins Most Ballots in Iraqi Election" — Cihan News Agency, 02.13.2005 Sunday - ISTANBUL 21:16
"United Iraqi Alliance wins Iraq polls" — Rediff.com, February 13, 2005 22:08 IST
"Shia parties triumph in Iraq poll" — BBC News, February 13, 2005
"The Iraqi vote: outcome known, future uncertain" — Radio Netherlands, February 14, 2005
Independent Electoral Commission of Iraq
Retrieved from "https://en.wikinews.org/w/index.php?title=Shiites_win_dominating_role_in_Iraq&oldid=1985045"
Politics and conflicts


text:
UNESCO » Culture » World Heritage Centre » About World Heritage » Reporting & Monitoring » State of Conservation
State of Conservation
Marine & coastal
Earthen Architecture
Religious & Sacred
only with
Cultural Criteria:
Natural Criteria:
Property on the List in Danger
With DSOCR
Properties and threats
List of Threats
Other Threats
With Threats
2019 state of conservation reports (upload in process)
Mont-Saint-Michel and its Bay
Factors affecting the property in 2000*
Erosion and siltation/ deposition
Management systems/ management plan
Factors* affecting the property identified in previous reports
Siltation of the Bay
Re-definion of the limits of the inscribed area needed
International Assistance: requests for the property until 2000
Requests approved: 0
Total amount approved : 0 USD
Missions to the property until 2000**
Information presented to the Bureau of the World Heritage Committee in 2000
- Palace and Park of Fontainebleau Inscribed on the World Heritage List in1981
- Chateau and Estate of Chambord Inscribed on the World Heritage List in 1981
- Amiens Cathedral Inscribed on the World Heritage List in 1981
- Chartres Cathedral Inscribed on the World Heritage List in 1979
- Strasbourg- Grande Ile Inscribed on the World Heritage List in1988
- Cathedral of Notre-Dame, Former Abbey of Saint-Rémi and Palace of Tau Reims Inscribed on the World Heritage List in1991
- Paris, Banks of the Seine Inscribed on the World Heritage List in1991
- Mont-Saint-Michel and its Bay Inscribed on the World Heritage List in1979
- Palace and Park of Versailles Inscribed on the World Heritage List in1979
New Information: Following the December 1999 storm, the above-mentioned French sites inscribed on the World Heritage List were seriously damaged. The French authorities provided the Secretariat with information about the major damage and estimated costs of repair. The damage was particularly severe at the following World Heritage sites:
- Fontainebleau: material damages to the roofing, 800 trees uprooted in the English garden. Damage was assessed at an amount of FF 4 Million.
- Notre Dame de Paris: collapse of a number of outside sculptures. Damage was assessed at an amount of FF 22.2 Million.
-Palace and Park of Versailles: material damage to the roof of the Palace and severe damage in the Park. Damage was assessed at an amount of FF 250 Million.
The report was transmitted to ICOMOS for further advice.
The Bureau may wish to examine information that will be provided at the time of its session and may wish to recommend the French authorities to submit a request for international assistance in this regard.
Decisions adopted by the Committee in 2000
Next 2003 Previous 1999
Date of Inscription: 1979
Nomination records (Year): 1979, 2007, 2018
Category: Cultural
Criteria: (i)(iii)(vi)
Other information on site
Read the SOUV
Documents examined by the Committee in 2000
WHC-2000/CONF.202/5
* : The threats indicated are listed in alphabetical order; their order does not constitute a classification according to the importance of their impact on the property.
Furthermore, they are presented irrespective of the type of threat faced by the property, i.e. with specific and proven imminent danger ("ascertained danger") or with threats which could have deleterious effects on the property's Outstanding Universal Value ("potential danger").
** : All mission reports are not always available electronically.
The Convention
Convention Text
Policy Compendium
Committee Decisions
World Heritage Fund
List of factors affecting the properties
Reactive Monitoring


text:
package org.fusesource.fabric.internal;

import java.io.BufferedReader;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.util.Properties;

public class FabricConstants {

    public static final String FABRIC_VERSION;
    public static final String FRAMEWORK_VERSION;

    static {
        String fabricVersion="unknown";
        String frameworkVersion="unknown";
        InputStream is = FabricConstants.class.getResourceAsStream("version.properties");
        InputStreamReader reader = null;
        Properties properties = new Properties();

        try {
            reader = new InputStreamReader(is, "UTF-8");
            properties.load(reader);
            fabricVersion = (String) properties.get("FABRIC_VERSION");
            frameworkVersion = (String) properties.get("FRAMEWORK_VERSION");
        } catch (Throwable e) {
        } finally {
            try {
                reader.close();
            } catch (Throwable e) {
            }
          try {
            is.close();
          } catch (Throwable e) {
          }

        }
        FABRIC_VERSION = fabricVersion;
        FRAMEWORK_VERSION = frameworkVersion;
    }
}


text:
Le traité de Berlin est signé, à la suite de la conférence de Berlin, par la France, le Royaume-Uni, l'Italie, les États-Unis, la Belgique, les Pays-Bas, la Suède et l'Allemagne. Il détermine les règles de comportement auxquelles les puissances acceptent de se soumettre dans le cadre de leurs politiques africaines respectives. C'est aussi le moment de la création de l'État Indépendant du Congo.

 Les puissances s'engagent à interdire l'esclavage dans leurs possessions actuelles et acquisitions ultérieures. Ces engagements restent de principes et ne seront pas appliqués.

 Établissement d'une vaste zone de libre-échange dans le bassin conventionnel du Congo ainsi que la liberté de commerce sur le fleuve Niger.

Obligation d'avoir effectivement occupé une côte pour en être déclaré possesseur et obligation de prévenir les signataires des acquisitions en cours afin de permettre d'éventuels arbitrages (rien n'est dit pour l'intérieur des terres).

L'État Indépendant du Congo est reconnu, avec comme souverain Léopold II de Belgique. Ses frontières sont sensiblement les mêmes que celles de l'actuelle République Démocratique du Congo. Du fait du "droit de préférence" accordé par l'Association internationale du Congo (à l'origine du nouvel état) à la France, le nouvel état ne peut être cédé qu'à celle-ci. Ce droit sera encore souligné en 1960 par le général de Gaulle au moment de l'indépendance de ce qui était alors le Congo belge. Pour satisfaire les Britanniques et les Allemands, l'État Indépendant du Congo se déclare libre-échangiste.

Les nombreuses questions laissées en suspens par cette conférence (Égypte, Maroc, sources du Nil, souveraineté sur le Niger, République boer, Éthiopie) qui seront à l'origine de conflits diplomatiques violents prouve que rien ou presque ne fut réglé à Berlin en 1885.

Articles connexes
 Conférence de Berlin
 Colonialisme

Liens externes
  Découpage colonial

Berlin
Berlin 1885
Berlin 1885
Berlin 1885
Berlin 1885
Histoire de l'Afrique
Berlin sous l'Empire allemand
1885
Traité signé par les États-Unis
Traité signé par la Belgique
Traité signé par les Pays-Bas
Traité signé par la Suède

en:Treaty of Berlin, 1885

redpajama_set_name: RedPajamaWikipedia