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# Calculate entropy with n values

I trying to solve a quiz that asks the following.

The variable $X$ can be the values $1,2,3,...,n$ with the probabilities $\frac{1}{2^1}, \frac{1}{2^2},\frac{1}{2^3},...,\frac{1}{2^n}$

How can I calculate the entropy of $X$? Don't I have to know all the probability values of X?

-
Shouldn't the probabilities add up to $1$? Those probabilities add up to $1-\frac{1}{2^n}$ –  Thomas Andrews Jan 15 '13 at 16:56
Without knowing what happens the remaining $1/2^n$, you can only come up with a minimum value. –  Thomas Andrews Jan 15 '13 at 17:00
Hi Thomas. So that minimum value would be the $-0.5log_2(0.5)$? –  Favolas Jan 15 '13 at 17:03
Perhaps the $n$ term and the $1/2^n$ probability is meant to denote the general term, over all the positive integers, rather than a range? –  leonbloy Jan 15 '13 at 17:38
Ah, perhaps that is the case @leonbloy –  Thomas Andrews Jan 15 '13 at 17:40

Your probability space is incomplete, so the entroy is not well-defined.

The minimum entropy is achieved if the remaining $\frac{1}{2^n}$ of the time, it takes a single value, which we'll just assume is $X=0$. In that case (using $\log$ base $2$:)

\begin{align} H(X) &= - \sum_{k=0}^n P(X=k)\log P(X=k) \\ &=-\frac{1}{2^n}\log \frac{1}{2^n} - \sum_{k=1}^n \frac{1}{2^k}\log \frac{1}{2^k}\\ &=\frac{n}{2^n} + \sum_{k=1}^n \frac{k}{2^k} \end{align}

This is the absolute minimum for the entropy. There is no upper bound for the entropy.

Using $$\sum_{k=1}^n kx^k = x\frac {nx^{n+1}-(n+1)x^n +1}{(x-1)^2}$$ with $x=\frac 1 2$, we can compute $\sum_{k=1}^n \frac{k}{2^k}$ which I think yields a minimum entropy of:

$$2-\frac{1}{2^{n-1}}$$

But I don't entirely trust my calculation here.

Perhaps the commenter above is right, and the real intent of the problem is that the probability is over all natural numbers, $1,...,n,...$ with $P(X=n)=\frac{1}{2^n}$. Then the entropy is the limit as $n\to\infty$ of the above lower bound, which is $2$ (assuming my calculation is correct.)

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Entropy is simply

$$H = -\sum_{k=1}^{n} p_k \log_2{p_k} = \sum_{k=1}^{n} k \: 2^{-k} = \frac{1/2}{(1 - 1/2)^2} - \frac{n (1/2)^{n+1}}{1-1/2} = 2 \left [ 1 - n 2^{-(n+1)} \right ]$$

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What about the missing $2^{-n}$? –  MJD Jan 15 '13 at 17:24
Hi. Assume $n = 2$ and $H(X)=1$, $n = 3$ and $H(X)=1.375$, $n = 4$ and $H(X)=1.625$. So if $n$ keeps incrementing it will be impossible to calculate $H(X)$. Right? –  Favolas Jan 15 '13 at 17:25
In this case, $H>2$ and approaches 2 from above as $n \rightarrow \infty$. I do not know what you mean about the missing $2^{-n}$. And $H$ is certainly not impossible to calculate for increasing values of $n$; rather, it is very simple, as you stated it. –  Ron Gordon Jan 15 '13 at 18:38


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# Ex.9.3 Q2 Algebraic Expressions and Identities - NCERT Maths Class 8

Go back to  'Ex.9.3'

## Question

Complete the table

 --- First expression Second Expression Product (i) $$a$$ $$b + c + d$$ --- (ii) $$x + y - 5$$ $$5xy$$ --- (iii) $$p$$ $$6{p^2} - 7p + 5$$ --- (iv) $$4{p^2}{q^2}$$ $${p^2} - {q^2}$$ --- (v) $$a + b + c$$ $$abc$$ ---

## Text Solution

What is known?

Expressions

What is unknown?

Product

Steps:

The table can be completed as follows.

 --- First expression Second Expression Product (i) $$a$$ $$b + c + d$$ $$ab + ac + ad$$ (ii) $$x + y - 5$$ $$5xy$$ $$5{x^2}y + 5x{y^2} - 25xy$$ (iii) $$p$$ $$6{p^2} - 7p + 5$$ $$6{p^3} - 7{p^2} + 5p$$ (iv) $$4{p^2}{q^2}$$ $${p^2} - {q^2}$$ $$4{p^4}{q^2} - 4{p^2}{q^4}$$ (v) $$a + b + c$$ $$abc$$ $${a^2}bc + a{b^2}c + ab{c^2}$$

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# Delarmlnt wtothar tho " folowing serks 253 3+2Lct @k > reprotont Ihe magniudu ihvo lorms tho qkvan euthre Ionidy Soloct

###### Question:

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# LaTeX support in WIRIS plugins: Insights.

The WIRIS plugins integrates the WIRIS editor into the HTML editors. From release 3.0, it is possible to write LaTeX inside the HTML editor using the $$notation. In any case, formulas will be converted and stored as MathML. LaTeX is disabled in the WIRIS plugin for Moodle if the filter of TeX is enabled in order to avoid conflicts. ## How to edit using LaTeX For example, if you type$$\frac12$$in your HTML editor and you save the page, you will get a page with the fraction is never clicked for this purpose but means that the WIRIS plugin is installed and works behind the scenes to cope with LaTeX. Editing LaTeX in an HTML editor The browser displays the formula When you reedit the page, you will see again . Thus, during edition you get the LaTeX and you see the image of the formula in display time. A nice feature provided by the WIRIS plugin is that you can still use the WIRIS editor with LaTeX formulas. If the caret is inside a formula, you can click the WIRIS editor icon to edit it. After submitting the WIRIS editor form, the old LaTeX is replaced by the new LaTeX formula. Using the WIRIS editor to modify a LaTeX formula ## LaTeX is manipulated as MathML Above is described the user experience, but what is important to note is that formulas are actually stored and displayed using MathML. Upon submit the HTML editor, the WIRIS plugins searches all$$ and converts the LaTeX into MathML. During this process, the plugin stores the MathML and the original LaTeX text written by the author. Thus, when displaying formulas, the MathML mechanism takes place.

MathML is used internally whatever you do with LaTeX

LaTeX is stored inside a MathML using a standard tag called <semantics>. Following the previous example:

<math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mfrac><mn>1</mn><mn>2</mn></mfrac>
<annotation encoding="LaTeX">\frac12</annotation>
</semantics>
[/itex]

When the HTML editor is opened again, the stored LaTeX is recovered from the <semantics> tag and is displayed in the HTML editor area for further edition. What it is important to note is that it works even when the LaTeX cannot be converted into MathML while it is possible still to edit the erroneous LaTeX to fix any possible error.

## Which LaTeX is supported?

The supported LaTeX depends on the effectiveness when translating from/to MathML. The limits are due to the following facts:

1. Some formulas have no equivalence with MathML.
2. LaTeX is an extensible language with a lot of non-official packages. To do things worse, LaTeX has its own programming language.

Some examples about the current LaTeX supported can be found here.

## Can I disable this feature?

You might want to disable the interpret of LaTeX while you still desire to use WIRIS editor. In this case, you will need to edit the JavaScript configuration file and change the line

var _wrs_conf_parseModes = ['latex'];

by

var _wrs_conf_parseModes = [];


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# Transfer Function

In control theory the transfer function describes the relationship between the input and the output of a linear time-invariant (autonomous) system. If a system represented by the equation \begin{align} a_0 \, y^{(n)} + a_1 \, y^{(n-1)} + \dots + a_{(n-1)} \, y + a_n = b_0 \, x^{(m)} + b_1 \, x^{(m-1)} + \dots + b_{m-1} \, x + b_m \end{align} has the input also called driving function $\sum_i b_i \, x^{(m-i)}$ (where powers of $x$ denote derivatives when written in brackets), then the output is given by the solution $y$ to the ODE. In compact notation a general systems can be describes by \begin{align} \sum_{i=0}^n a_i \, y^{(n-i)} = \sum_{i=0}^m b_i \, x^{(m-i)} \end{align}

The transfer function $G(s)$ of a linear time-invariant system of differential equations is defined as the ratio between the laplace transform of the ouput (response function) and the laplace transform of the input (driving function), when all initial conditions are zero. \begin{align} G(s)=\left[ \frac{\mathcal{L}[output]}{\mathcal{L}[input]} \right]_{\text{zero i.c.}}=\frac{Y(s)}{X(s)}=\frac{\sum_i b_i \, y^{(m-i)}}{\sum_i a_i \, x^{(n-i)}} \end{align} By the use of Laplace transform it is possible to represent the system dynamics by algebraic equations.

## Transfer function for an Impulse input

For an impulse input the transfer function takes a very simple form. Acording to the property $\delta *x =x$ of the convolution we simply get \begin{align} Y(s)=\int_0^{\infty} \left\{ \delta*g \right\} \, e^{-st} dt=\int_0^{\infty} \left\{g \right\} \, e^{-st} dt=G(s) \\ \end{align} \begin{align} Y(s)=G(s) \\ \end{align} The inverse Laplace transform of $g(t)=\mathcal{F}^{-1}G(s)$ is called the impulse-response function or weighting function of the system. Notice, that we can determine the weighting function by measuring the response of a system to a unit impulse. In this case the weighting function is the inverse Laplace transform of the system response \begin{align} g(t)=\mathcal{F}^{-1}Y(s) \end{align}


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Why is gravitational potential energy negative, and what does that mean?

I usually think of gravitational potential energy as representing just what it sounds like: the energy that we could potentially gain, using gravity. However, the equation for it (derived by integrating Newton's law of gravitational force)...

$$PE_1 = -\frac{GMm}{r}$$

..has me thrown for a loop, especially after this answer.

• If potential energy really meant what I thought it did, then it would always have to be non-negative... but this equation is always negative. So what does "negative potential energy" mean!?
• If $KE + PE$ is always a constant, but PE is not only negative but becomes more negative as the particles attract, doesn't that mean the kinetic energy will become arbitrarily large? Shouldn't this mean all particles increase to infinite KE before a collision?
• If we are near the surface of the earth, we can estimate PE as $$PE_2 = mgh$$ by treating Earth as a flat gravitational plane. However, $h$ in this equation plays exactly the same role as $r$ in the first equation, doesn't it?
• So why is $PE_1$ negative while $PE_2$ is positive? Why does one increase with $h$ while the other increases inversely with $r$?
• Do they both represent the same "form" of energy? Since $PE_2$ is just an approximation of $PE_1$, we should get nearly the same answer using either equation, if we were near Earth's surface and knew our distance to its center-of-mass. However, the two equations give completely different answers! What gives!?

Can anyone help clear up my confusion?

• Energy is spent doing work. – Jossie Calderon Aug 13 '18 at 4:24

About negative energies: they set no problem:

On this context, only energy differences have significance. Negative energy appears because when you've made the integration, you've set one point where you set your energy to 0. In this case, you have chosen that $PE_1 = 0$ for $r = \infty$. If you've set $PE_1 = 1000$ at $r = \infty$, the energy was positive for some r.

However, the minus sign is important, as it is telling you that the test particle is losing potential energy when moving to $r = 0$, this is true because it is accelerating, causing an increase in $KE$:

let's calculate the $\Delta PE_1$ for a particle moving in direction of $r = 0$: $r_i = 10$ and $r_f = 1$:

$\Delta PE_1 = PE_f - PE_i = Gm(-1 - (-0.1)) = -Gm\times0.9 < 0$

as expected: we lose $PE$ and win $KE$.

Second bullet: yes, you are right. However, it is only true IF they are point particles: has they normally have a definite radius, they collide when $r = r_1 + r_2$, causing an elastic or inelastic collision.

Third bullet: you are right with $PE_2 = mgh$, however, again, you are choosing a given referential: you are assuming $PE_2 = 0$ for $y = 0$, which, on the previous notation, means that you were setting $PE_1 = 0$ for $r = r_{earth}$.

The most important difference now is that you are saying that an increase in h is moving farther in r (if you are higher, you are farther from the Earth center).

By making the analogy to the previous problem, imagine you want to obtain the $\Delta PE_2$. In this case, you begin at $h_i = 10$ and you want move to $h_f = 1$ (moving in direction to Earth center, like $\Delta PE_1$:

$\Delta PE_2 = PE_{f} - PE_{i} = 1mg - 10mg = -9mg < 0$.

As expected, because we are falling, we are losing $PE$ and winning $KE$, the same result has $PE_1$

Fourth bullet: they both represent the same thing. The difference is that $gh$ is the first term in the Taylor series of the expansion of $PE_1$ near $r = r_{Earth}$. As exercise, try to expand $PE_1(r)$ in a taylor series, and show that the linear term is:

$PE_1 = a + \frac{Gm(r-r_{earth})}{r_{earth}^2}$.

Them numerically calculate $Gm/r_{earth}^2$ (remember that $m=m_{earth}$). If you haven't made this already, I guess you will be surprised.

So, from what I understood, your logic is totally correct, apart from two key points:

• energy is defined apart of a constant value.

• in the $PE_1$, increase r means decrease $1/r$, which means increase $PE_2 = -Gm/r$. In $PE_2$, increase h means increase $PE_2=mgh$.

• Ah, I see, the trick is that it's a relative value - I keep thinking of energy as something absolute (though I guess even kinetic energy changes, depending on your frame of reference). I suppose we'd like to set PE=0 when r=0, but unfortunately, according to the equation it would take infinite energy to pull the particles apart! So I guess PE=0 when r=∞ is the only other reasonable choice. It all makes sense now - thanks! – BlueRaja - Danny Pflughoeft Nov 17 '11 at 22:18
• Also, the formula changes inside a non-point mass, so the $r\to 0$ limit is finite. – J.G. May 10 '18 at 19:57

I will first (1) summarize the differences between the definitions of PE1 and PE2 and then I will (2) equate the two.

(1) First, as this answer to "Why is gravitational energy negative?" says, PE1 defines the potential energy of a body of mass m in the gravitational field of a mass M as the energy (work) required to take it from its current position $r$ to infinity. PE1 assumes $r=\infty$ is $PE=0$ $$PE1=\frac{−GMm}{r}$$

PE2, on the other hand, is defined as the negative of the work done by gravity to lift a body of mass m from the surface of a planet to a height h above the planet.

$$PE2=-W=-Fdcos\theta =mgh$$

PE2 has a different frame of reference than PE1, as it assumes $PE=0$ at $r=R$, or on the surface of the planet. Also, and very importantly, PE2 is only used when an object is close to the surface of a planet, when $h<<<R$ (R is radius of planet), and g can be assumed to be constant:

$$g=\frac{GM}{(R+h)^2} \approx \frac {GM}{R^2}$$

(2) OK, now on to equating the two. Though the frames of references for PE1 and PE2 are different, $|\Delta PE|$ between two points should surely be the same. For the sake of example, let's say the two points are the surface of the planet and height h above the planet.

PE1 says $|\Delta PE|=mgh-mg(0)=mgh$

PE2 says $|\Delta PE|=\frac {-GMm}{R+h}- \frac {-GMm}{R}=GMm\left(\frac{1}{R}-\frac{1}{R+h}\right)=GMm\left(\frac{h+R-R}{(R)(h+R)}\right)=\frac{GMmh}{(R)(R+h)}$

and because $h<<<R$, $\frac{GMmh}{(R)(R+h)} \approx \frac{GMmh}{R^2} =mgh$

And thus, PE1 and PE2 both represent the same form of energy, but we must keep in mind the frames of reference and the conditions of use when we use them.

Hope this helps!! Peace.

It is because gravitational force is attractive and work is done by gravitational force itself. When system does work itself energy is taken as negative and when work is done by external agency on system energy is take as positive.

Gravity is an acceleration. No negative involved.

However, when you use acceleration to find a velocity, since velocity is a vector quantity, you must describe a direction. It is convention that anything that accelerates up, is described as a positive(+) like "The ball accelerates at 20m/s^2", whereas gravity describing a downward acceleration is described as (-) "-9.8m/s^2".

This applies to anything accelerating on the X axis as well. "The car accelerates at 10m/s^s when you apply the gas" or "The car accelerates at -4m/s^2 when you apply the brakes."

I believe this is done to make things easier when making graphs.

However, if you were to just say "I have a ball. It will be displaced, how far will it be displaced? (Notice how its not 'displaced north, or to the left ')" In a situation like that, you would use the acceleration of gravity without the negative. "It will be displaced by 9.8m every second^2".

I hope this helps. Then again, I might have completely misread your question. Either way, have a good day!

• This question is about potential energy, not acceleration vectors... – BlueRaja - Danny Pflughoeft Aug 23 '14 at 21:30

I think it's just a preference.

We could see gravitational potential energy as being positive, representing the energy "invested" in our position relative to a massive object. We can "regain" that energy (increase kinetic energy) by moving closer to the object, at which point we have lowered the amount of energy we could gain by moving further. So potential energy decreases as we move closer (approaching zero energy at zero distance), increases as we move further away, and the sum of PE and KE is constant.

But what value is the constant? When we are very very far away from the massive object we should have very very large potential energy. But even when we're quite close to the massive object, we're very very far away from every other massive object in the universe, and thus should have very very large gravitational potential energies relative to all those objects. We can approximately calculate a value for KE + PE by considering only the most relevant objects (the closest and/or largest ones), but our approximate value just grows and grows and grows as we try to get more accurate approximations by including smaller and more-distant objects in our category of "relevant" objects. So our KE + PE constant is some impossibly large value we can never really calculate or estimate as some specific value. In some ways it doesn't matter that we can't ever claim a value, since differences of energies are all we really need to work with, and we can still calculate those (by assuming that our PE relative to everything else in the universe has only changed negligably when we move around near the massive object we're considering). But it seems unsatisfying.

On the other hand, instead of considering PE as a positive quantity of energy "invested" in our position (energy we've already "spent" if we were moving away from the massive object, which we could gain by moving closer), we can instead consider it a negative quantity of energy we "owe" because of our position (energy we've gained "for free" if we moved closer to the object from infinity, which we would have to "spend" to escape to infinity again).

All the calculations of energy differences work out the same anyway. But now our PE relative to an object goes to zero as we get very very far away from the object. This mean that as we can calculate an approximation of our KE + PE constant by only considering the most relevant objects, and as we try to get better approximations by including smaller and more-distant objects in our calculation the effects of those additional objects gets closer and closer to zero. So we come up with an actual number we can justifiably say is the value for our KE + PE constant.

The fact that the gravitational potential energy as with all potential energies of attarctive forces are negative is based on the fact that we want to assume that when the particles are at infinity with respect to each other and at rest the system have zero total energy. Imagine if this was not the case and a system of two particles at infinite separation at rest would be taken to have a net energy then there would arise some confusion as to energy associated with the rest mass. The total energy of the system would then not be $E=Mc.c$ where $M$ is the sum of two masses. From where then would this extra energy come?

It is wrong to consider gravitational potential energy to be negative--tho common.

The big mistake is in assigning the P.E. at infinity = 0. This is clearly wrong--P.E. is clearly 0 at 0 separation, and large at great separations. The P.E. of objects far away from each other would have to be the summation of the P.E. for the first say 100' of separation plus the P.E. for the second 100' of separation plus---the P.E. for every 100' until the entire separation was accounted for. (I will express this as an integral after I brush up my calculus.) Viz, P.E. INCEASES as separation increases--starting at 0 at no separation.

Many people are making a big mistake in considering gravitational potential energy to be negative!

• With the field from a point source obeying the inverse-square law, the force is proportional to $r^{-2}$ and the potential (and the potential energy) is therefore proportional to $r^{-1}$. The linear $P=mgh$ is just an approximation for small changes in distance. – HDE 226868 May 10 '18 at 20:00
• @HDE226868 Did you mean to comment on a different answer? – diracula May 10 '18 at 20:48
• @diracula No - I should have made myself clearer. I was showing mathematically why the potential energy vanishes at infinity rather than growing to infinity; as $r\to\infty$, $r^{-1}$ goes to $0$. – HDE 226868 May 10 '18 at 20:49


text:
## NA-sampler

This page describes the package NA-sampler. It consists of a set of user callable subroutines which implement theNA multi-dimensional parameter space search algorithm. The package contains source code for the NA routines and an example driver program. The latter applies the NA algorithm to the estimation of crustal seismic velocity profiles from inversion ofseismic receiver functions (RFI). Several independent utility' programs are alsoprovided which manipulate the output files of the main program and plot some results. Some of these programs are specific to the RFI problem, and others may be more generally useful.

These pages are meant only as a guide. For more information look at the various README files in the src sub-directories, the comments in the source code for the RFI example, and the input files for the utility plot programs. Finally if you are that adventurous you can look at the code itself.

This package performs a derivative free parameter space search using the Neighourhood Algorithm. See references below for details. The latest details of NA-sampler and the related ensemble inference algorithm NA-Bayes can be found on the iEarth NA page

### Installation

When the tar file is unpacked it produces four directories:

src - contains all Fortran and C source code for NA subroutines and RFI example program and utility/plot programs.

data - contains data input files and example output files created by RFI example problem. Also contains data files for running splot graphics program, and RFI plot programs. (See utility programs for more details.)

bin - contains executable for RFI example program and all utility programs.

docs - contains these HTML pages.

To remove all object and source files prior to a re-installation type make clean'. Examine the Makefile if you do not want to compile all of the utility programs. Note that the file makena.macros' contains the location of the Fortran and C compilers and must be edited to suit your system prior to installation.Typical configuration files are provided for compaq alpha, sun, GNU, and Linux. These can be used by typing,

> cd src
> cp macros/makena.macros_sun makena.macros

To compile the complete set of programs from the src directory

> make all

If compilation is successful, all executables are placed in the bin directory and all examples demonstrating the utility programs can be run from the data directory (See below). The reverse of make all is,

> make clean

which will remove all .o files and all executables.

Note that the Demo directory, the Makefile and the file rfi_na.f can serve as a template for new user applications. If you are not interested in running, or looking at output from the seismic example program then you can remove all RFI specific source code and programs by,

> cd src
> \rm -fr rfi_subs
> cd ../data
> \rm -fr demo_results rfi_files

This will save disk space but please note that the RFI programs will be gone and the command make all' will no longer work. A re-installation from a tar file will be necessary to retrieve the RFI example files.

#### Parallelization with MPI

In May 2002 message passing interface (MPI) calls were added to the NA source code. This allows the possibility of running the search algorithm across a cluster of machines simultaneously (e.g. a linux Beowulf cluster). The advantage of this is that the forward modelling for each model in an ensemble can be performed on separate machines, giving much faster overall run times. Note that this does not mean that the user's own forward modelling code needs to be parallelized with MPI. All MPI calls appear in the NA source code and are used to farm the forward modelling out to different processors.

The MPI code is activated by a switch in the installation file makena.macros'. If you have say a linux cluster running MPI then you need to set the TYPE=mpi and the MF77 variable to the location of the MPI fortran 77 compiler on your system. See the example makena.macros files for examples. (If you want the usual single processor version then set TYPE=serial.) If installation occurs with these variables then you must execute the program using the mpirun command provided by your MPI installation in the usual way. For example one might run the seismic example executable with a command like,

> mpirun -np 20 rfi_na

which will execute the program rfi_na with the forward modelling distributed across the first 20 machines that your MPI is set up to run on.

Note that when the number of processors (e.g. 20 in this example) divides equally into the number of models generated at each iteration (see parameter Ns2 below) then one gets perfect parallelization in the forward modelling part of the code, and this usually means a significant overall speed up, e.g. a factor of Ns1/20 in the forward modelling part of the code. Since for many applications most time will be spent on forward modelling then this can represent a significant speed up of the overall code. When the number of processes is not a factor of Ns2 then one gets a smaller speed up factor. Note that there is no point in asking for more processors than Ns2 because there will be nothing for the extra processors to do.

#### Compiling problems and platform dependency

So far I have succesfully compiled and run all codes in this package on a SUN workstation (running Solaris 2.6 or later) using both the native compiler and the GNU compilers g77 and gcc; on a Compaq Alpha (running OSF1 V4.0) using the Digital Fortran and C compilers; on a linux box with g77, or intel fortran/C compilers ifc and icc. In all cases exactly the same results were obtained. Using the intel fortran and C compilers on Linux I obtained different answers. Possibly due to rounding error changing the Monte Carlo sequence. The default makena.macros file is for a linux box running the intel compiler. When installing on other platforms SUN you will need to edit the file src/makena.macros appropriately.

If you are using a Dec/Compaq alpha/compaq platform then you will need to use the compiling option -assume byterecl which makes the Fortran open statement for direct access files the same as on a sun, i.e. with record length specified in bytes rather than words. This can be done by editing the file src/makena.macros, which is used to define compilers and options during the make. An example file makena.macros_alpha is provided as an example.

Note that for the example problem direct access output (nad) files are provided with the package (in directory data/demo_results). To allow platform independence both little endian and big endian binary versions are included. The make file uses your choice of platform (specified in file makena.macros) to decide which type of binary file to link to the examples. This is a change over earlier distributions and means that you will now be able to run all of the utility programs, e.g. splot (showing the models generated by the NA ) on the example output files provided. The downside of this is that the distribution tar file is larger.

Some programs (from T. Shibutani, RCEP, DPRI, Kyoto Univ., Japan) are provided in the directory src/utl/byteconv for converting SAC files between a sun and a Compaq Alpha machine. This may be useful for seismologists with many sac files. Note that SAC files are the format used to store the input and output seismic receiver functions used in the RFI example. They are specific to this example and are read in by the user routines. They are not a generic NA input file.

#### Fortran 90

In January 2004 the source codes of the NA_sampler package were changed to incorporate dynamic memory allocation using Fortran 90. This means that array bounds in file na_param.inc only need to be set if using Fortran 77 compatibility mode, which is itself set by -DNA_F77_COMPATIBILITY=1 in the file makena.macros. Try the following

> cd src/macros
> diff makena.macros_alpha_f90 makena.macros_alpha_f77

Fortran 90 will be the default compiler from January 2004.

### Code structure and interfacing

The figure above shows the structure of the NA code and how it is interfaced with the user code. The green routines must be provided by the user. The main routine passes control to the NA code by calling the subroutine na' with no arguments. This then calls three user written subroutines.

user_init - provides the definition of the parameter space to the NA routines and performs any user problem specific tasks and I/O.

forward - calculates the misfit/objective function for a given input model.

writemodels - allows the user to write out, in their own format, all of the models, misfit values etc, from the ensemble produced by the NA.

Only forward is called more than once. Each of these subroutines are fully described in the header of the example source files for the RFI problem.

The file rfi_na.f contains the main driver program and the three subroutines which implement the RFI example problem. In this case the driver routine merely calls na and exits. All data I/O and initialization is performed inside the routine user_init. The three RFI interface routines make use of many other problem specific subroutines and libraries. Source files for all of these are in the directory src/rfi_files. They are compiled and linked into the main program by the Makefile.

To use the NA subroutines you need to do the following,

• Create the three subroutines for your problem. (Use the RFI examples as a template)

• Create the main routine that calls na (This can be trivially simple as in the RFI example.)

• Edit the file na_param.inc to adjust array sizes if necessary. Since January 2004 this is not needed if using Dynamic memory allocation under the fortran 90 compiler (set NA_F77_COMPATIBILITY=0 in makena.macros)

• Edit the Makefile and makena.macros to compile the users subroutines

• Compile the new program with

> make rfi_na (or the equivalent)
and run it from the data directory.

Note that the only essential information that the NA routines need from the user routines is the definition of the parameter space (from user_init) and the misfit for a given model (from forward). The routine writemodels is called once after the NA has finished and can be null.

Output files

The two generic output files are na.sum and na.nad The first is an ascii file with summary information on the performance, the second is a direct access file binary file containing all of the models generated by the NA during the search. The format of this file is described in Appendix C. The NAD file is used as a compact storage format which allows users to add their own information in a header. NAD files are read in by certain utility programs like the plot program splot (See appendix A). NAD files are also read in by the appraisal algorithm NA-Bayes, which is a separate package that can be downloaded. See the NA homepage for details.

Code development

The development of this code is an ongoing research effort and I expect that the NA subroutines will be updated from time to time, however, it is intended that the interface routines will remain unchanged. (The input file na.in will probably change) In this way the incorporation of updated NA routines into an existing application should be quite straightforward, i.e. compiling the new na.f to obtain na.o and linking with the existing driver code. Note that the NA source code in sub-directory NA_src should not normally need editing ! (Edit at your own risk !)

The code was developed on a sun network under Solaris 2.6. Note that there are some issues to consider when transporting the code across to new platforms and compilers systems. The ones that I am aware of are described in the section on Compiling problems.

### Controlling the Neighbourhood algorithm

The NA control parameters and other options are read in from the file na.in.
Each line is described below.

#
# Neighbourhood Algorithm input options file
#
ialg          : Algorithm type (1=Uniform search, 2=Neighbourhood algorithm)
maxit         : Maximum number of iterations
Ns1           : Number of models for initial sample
Ns2           : Number of models generated at each iteration
Nr            : Number of Neighbourhoods re-sampled at each iteration
YorN, iseed   : Use quasi random number generator ? (y/n); random seed
olevel        : Output info level (0=silent;1=summary info;2=1+write models)
YorN          : Turn timing mode on ? (y/n)
YorN          : Turn debug mode on ? (y/n)

Most of these are self-explanatory. The examples in the data directory contain typical' values for the parameters, although the main tuning parameters Ns2 and Nr should be tuned for the particular problem. (See tuning the control parameters)

The parameter istype controls the type of initial ensemble of models used. Several options are available.

• istype=0 - then the initial sample is generated using a uniform random distribution and has size Ns1.

• istype=1 - then the initial sample is read in from a direct access file in nad format called na.nad' and the initial sample size Ns1 is reset to the number of models in the nad file.

• istype=-1 - same as istype=1 except that the forward problem is not solved for the initial ensemble and instead the misfit values are taken from the file.

• istype=2 - then the initial sample is read in from an ascii file models.in. For each model the format is a triplet of a blank line followed by the misfit value on a line by itself followed by the nd values of the model. This is repeated for each model. See the data directory for an example. If the number of models in the file is less that Ns1 then the remaining models are generated randomly. If greater than Ns1, then Ns1 is reset to that value.

• istype=-2 - same as istype=2 except that the forward problem is not solved for the initial ensemble and instead the misfit values are taken from the file.

With these options it is possible to restart the NA using the output ensemble from a previous run of NA (istype=1). This might be useful if one wanted to change the tuning parameters (Ns2 and Nr), or perhaps change the misfit function its self (say to add regularization or incorporate some new type of data), and continue on. If istype=+-2 then its possible to seed the initial distribution with one or more previously known good' models with low misfit. This will naturally attract the sampling towards them and allow a gradually focusing search in their neighbourhood. Examples of programs which read and write NAD files are contained in the utility programs section.

If a quasi random number generator is not chosen then a pseudo-random number generator is used. (See the papers cited below or Press et al 1992 Numerical Recipes, C.U.P., for a discussion of the difference.)

### Tuning the control parameters

The two control parameters usually need to be tuned for the problem in hand and the style of sampling that you want. There are no well defined rules but experience has proven to be a useful guide. Most of my understanding is based on empirical evidence from applications at RSES. The usual objective with a direct sampling algorithm is to try and achieve a balance between exploration of the parameter space and exploitation of the previous samples. The more explorative an algorithm is, the more likely it will be robust against local minima, and the starting random seeds, the more exploitative it is the faster it will reduce the objective function. The amount of exploration and exploitation performed by the NA is largely controlled by the two control parameters ns (ns2) and nr.

ns = Number of models generated at each iteration.

nr = Number of Voronoi cells re-sampled at each iteration.

and so ns/nr models are placed in each cell.

See the papers below for more details. As ns and nr are increased it will be more explorative and as they are decreased it will be more exploitative.

Our experience has shown that reasonable results can be obtained after enough iterations when ns is of the order of twice the number of dimensions, and nr is in the range 2 - ns/2. The number of iterations is more difficult to judge. I have used used 10 - 100 times ns. It is clear that the larger values of ns and nr (i.e. more explorative) require more iterations to concentrate sampling (especially in high dimensions). In our experiments we have found in all cases that the sampling does eventually concentrate quite well, even for larger nr values, although you might not want to wait that long!

My strategy is often that it is better to get good sampling slowly than poor sampling quickly. However, small values of nr do seem to concentrate sampling quickly, especially in small dimensional (e.g. d ~ 4) problems. These issues are also discussed in Sambridge (1999a). I would be interested to hear of other peoples experiences (see contacting the author ).

### Utility programs

When the package is compiled the following general utility programs are put in the bin directory (source in src/utl).

Execute with

asc2nad - goes the other way

Execute with

splot - reads in a nad file and plots a scatter diagram of all models (PS or X)

projected onto a pair of axes. See the splot manual for more details.

The following are utility programs specific to the receiver function example:

plot_rf - plots the observed and predicted receiver function

for the best fitting velocity profile found by the NA.

plot_model - a density plot of all velocity profiles produced by the NA search.

synrf - calculates a synthetic receiver function from an input velocity profile.

synrf_noise - Same as synrf but adds noise to the result.

nad2rfi - converts direct access NAD file to an ascii file containing

velocity profiles, in a format specific to RFI problem.

Executables from all utility programs are put in the bin directory. Further details can be found at the RFI utility programs page.

### Running the receiver function inversion example

The example driver program provided with the NA-sampler package is for the inversion of receiver functions for crustal seismic structure. This is described further in the two NA papers cited below. Note that the driver program is not intended to be a general purpose receiver function code, but merely an example of using the NA routines on a realistic seismic problem. Changes to the code may be required if it is applied to other data sets (For example, the maximum number of layers, and waveforms may need adjusting etc.)

To run the RFI examples after compilation of the package (see above),

> cd data
> cp na.in_examples/na.in_9_2 na.in
> rfi_na

This will run a short (3 iteration) test run with Ns=9 and Nr=2. The output should be similar to that in demo_results/na.sum_3_9_2. A similar set of steps is required to run one of the longer example inputs in directory na.in_examples. You could try editing na.in to adjust input parameters and repeating the test run.

The RFI subroutines use several input and output files. All of which are in sub-directories of data/rfi_files. The main one is rfi.in, which contains parameters associated with the input data and the names of the other I/O files.

Two other input files are used. In the default setup these are:

rfi_param - in sub-directory NA_MDL, which contains the bounds of the model parameters (Thicknesses, velocities and Vs/Vp ratio's in each of 6 layers).

rec_func - in sub-directory ORF, which contains the one or more input (stacked) receiver functions.

Two output files are produced. In the default setup these are:

rec_models - in sub-directory NA_MDL, which contains all models generated at each iteration of the NA in a convenient format.

rec_func - in sub-directory NA_SRF, which contains the predicted receiver function from the best fit model.

The output files produced by a 500 iteration example are included in the demo_results directory. They can be used to demonstrate the use of splot and the RFI graphics programs. See the splot manual for examples of using splot on the 500 iteration results. See RFI utility programs for examples of plotting receiver functions and the ensemble of velocity profiles produced by the NA.

### Conditions of use

• The NAB routines may not be distributed to third parties.
Interested parties should contact the author directly.

• Due acknowledgment must be made of the use of the NA routines in research reports or publications. Whenever such reports are released for public access, a copy should be forwarded to the author.

• The NA routines may only be used for not-for-profit research and development, unless it has been agreed in advance with the author in writing.

Use of any part of this package will be taken as acceptance of these conditions.

### Contacting the author

The purpose of this documentation and the associated README files, is to reduce the likelihood of running into problems. I would be grateful if you do not contact me with queries on the installation, or running, of the code unless you have thoroughly looked for the answer yourself. (Note that either these HTML pages, the README files, example files and the source code itself may provide an answer). Note that for detailed queries on the RFI example it may be best to contact Dr. T. Shibutani, (shibutan@rcep.dpri.kyoto-u.ac.jp) who wrote the original versions of the RFI code and plot programs. His contribution is much appreciated.

I am interested in obtaining feedback on people's experiences with this code. Especially to hear what type of problems it has been applied to. I expect that the code will be updated from time to time. Information on new developments will be posted on the NA homepage. If you develop your own web pages showing work where NA has been used, or publish papers then I would be happy to add a reference on the NA homepage.

### Related sites:

• NA homepage - where the latest versions of NA-Bayes and NA-sampler can be accessed.

• splot - graphics program for display of multi-dimensional ensembles.

• RFI - Receiver function utility programs and graphics (included with NA-Sampler package.)

• NA-Bayes - the manual for the neighbourhood algorithm Bayesian inference package.

• naplot - graphics program for displaying Bayesian estimators (included with NA-Bayes package).

• summary.pdf - A short summary of NA-sampler written for the IASPEI handbook (PDF file)

References:

The Neighbourhood algorithm for searching a multi-dimensional parameter space for models (points) with `acceptable data fit' is implemented in the program NA-sampler, and is described in the paper:

Geophysical Inversion with a Neighbourhood Algorithm -I. Searching a parameter space, Sambridge, M., Geophys. J. Int., 138 , 479-494, 1999. doi:10.1046/j.1365-246X.1999.00876.x

A related paper describes the Neighbourhood re-sampling algorithm (implemented in program NA-bayes) for calculating Bayesian integrals from a finite set of samples produced by the NA algorithm or any other search method (e.g. GA or SA etc.):

Geophysical Inversion with a Neighbourhood Algorithm -II. Appraising the ensemble,Sambridge, M., Geophys. J. Int., 138 ,727-746, 1999. doi:10.1046/j.1365-246x.1999.00900.x


text:
## a general framework for updating belief functions

Posted in Books, Statistics, University life with tags , , , , , , , , , on July 15, 2013 by xi'an

Pier Giovanni Bissiri, Chris Holmes and Stephen Walker have recently arXived the paper related to Sephen’s talk in London for Bayes 250. When I heard the talk (of which some slides are included below), my interest was aroused by the facts that (a) the approach they investigated could start from a statistics, rather than from a full model, with obvious implications for ABC, & (b) the starting point could be the dual to the prior x likelihood pair, namely the loss function. I thus read the paper with this in mind. (And rather quickly, which may mean I skipped important aspects. For instance, I did not get into Section 4 to any depth. Disclaimer: I wasn’t nor is a referee for this paper!)

The core idea is to stick to a Bayesian (hardcore?) line when missing the full model, i.e. the likelihood of the data, but wishing to infer about a well-defined parameter like the median of the observations. This parameter is model-free in that some degree of prior information is available in the form of a prior distribution. (This is thus the dual of frequentist inference: instead of a likelihood w/o a prior, they have a prior w/o a likelihood!) The approach in the paper is to define a “posterior” by using a functional type of loss function that balances fidelity to prior and fidelity to data. The prior part (of the loss) ends up with a Kullback-Leibler loss, while the data part (of the loss) is an expected loss wrt to l(THETASoEUR,x), ending up with the definition of a “posterior” that is

$\exp\{ -l(\theta,x)\} \pi(\theta)$

the loss thus playing the role of the log-likelihood.

I like very much the problematic developed in the paper, as I think it is connected with the real world and the complex modelling issues we face nowadays. I also like the insistence on coherence like the updating principle when switching former posterior for new prior (a point sorely missed in this book!) The distinction between M-closed M-open, and M-free scenarios is worth mentioning, if only as an entry to the Bayesian processing of pseudo-likelihood and proxy models. I am however not entirely convinced by the solution presented therein, in that it involves a rather large degree of arbitrariness. In other words, while I agree on using the loss function as a pivot for defining the pseudo-posterior, I am reluctant to put the same faith in the loss as in the log-likelihood (maybe a frequentist atavistic gene somewhere…) In particular, I think some of the choices are either hard or impossible to make and remain unprincipled (despite a call to the LP on page 7).  I also consider the M-open case as remaining unsolved as finding a convergent assessment about the pseudo-true parameter brings little information about the real parameter and the lack of fit of the superimposed model. Given my great expectations, I ended up being disappointed by the M-free case: there is no optimal choice for the substitute to the loss function that sounds very much like a pseudo-likelihood (or log thereof). (I thought the talk was more conclusive about this, I presumably missed a slide there!) Another great expectation was to read about the proper scaling of the loss function (since L and wL are difficult to separate, except for monetary losses). The authors propose a “correct” scaling based on balancing both faithfulness for a single observation, but this is not a completely tight argument (dependence on parametrisation and prior, notion of a single observation, &tc.)

The illustration section contains two examples, one of which is a full-size or at least challenging  genetic data analysis. The loss function is based on a logistic  pseudo-likelihood and it provides results where the Bayes factor is in agreement with a likelihood ratio test using Cox’ proportional hazard model. The issue about keeping the baseline function as unkown reminded me of the Robbins-Wasserman paradox Jamie discussed in Varanasi. The second example offers a nice feature of putting uncertainties onto box-plots, although I cannot trust very much the 95%  of the credibles sets. (And I do not understand why a unique loss would come to be associated with the median parameter, see p.25.)

Watch out: Tomorrow’s post contains a reply from the authors!

Posted in Statistics with tags , , , , , , , , , on January 28, 2013 by xi'an

Last Monday, my student Li Chenlu presented the foundational 1962 JASA paper by Allan Birnbaum, On the Foundations of Statistical Inference. The very paper that derives the Likelihood Principle from the cumulated Conditional and Sufficiency principles and that had been discussed [maybe ad nauseam] on this ‘Og!!! Alas, thrice alas!, I was still stuck in the plane flying back from Atlanta as she was presenting her understanding of the paper, as the flight had been delayed four hours thanks to (or rather woe to!) the weather conditions in Paris the day before (chain reaction…):

I am sorry I could not attend this lecture and this for many reasons: first and  foremost, I wanted to attend every talk from my students both out of respect for them and to draw a comparison between their performances. My PhD student Sofia ran the seminar that day in my stead, for which I am quite grateful, but I do do wish I had been there… Second, this a.s. has been the most philosophical paper in the series.and I would have appreciated giving the proper light on the reasons for and the consequences of this paper as Li Chenlu stuck very much on the paper itself. (She provided additional references in the conclusion but they did not seem to impact the slides.)  Discussing for instance Berger’s and Wolpert’s (1988) new lights on the topic, as well as Deborah Mayo‘s (2010) attacks, and even Chang‘s (2012) misunderstandings, would have clearly helped the students.

## empirical Bayes (CHANCE)

Posted in Books, Statistics, University life with tags , , , , , , on April 23, 2012 by xi'an

As I decided to add a vignette on empirical Bayes methods to my review of Brad Efron’s Large-scale Inference in the next issue of CHANCE [25(3)], here it is.

Empirical Bayes methods can crudely be seen as the poor man’s Bayesian analysis. They start from a Bayesian modelling, for instance the parameterised prior

$x\sim f(x|\theta)\,,\quad \theta\sim\pi(\theta|\alpha)$

and then, instead of setting α to a specific value or of assigning an hyperprior to this hyperparameter α, as in a regular or a hierarchical Bayes approach, the empirical Bayes paradigm consists in estimating α from the data. Hence the “empirical” label. The reference model used for the estimation is the integrated likelihood (or conditional marginal)

$m(x|\alpha) = \int f(x|\theta) \pi(\theta|\alpha)\,\text{d}\theta$

which defines a distribution density indexed by α and thus allows for the use of any statistical estimation method (moments, maximum likelihood or even Bayesian!). A classical example is provided by the normal exchangeable sample: if

$x_i\sim \mathcal{N}(\theta_i,\sigma^2)\qquad \theta_i\sim \mathcal{N}(\mu,\tau^2)\quad i=1,\ldots,p$

then, marginally,

$x_i \sim \mathcal{N}(\mu,\tau^2+\sigma^2)$

and μ can be estimated by the empirical average of the observations. The next step in an empirical Bayes analysis is to act as if α had not been estimated from the data and to conduct a regular Bayesian processing of the data with this estimated prior distribution. In the above normal example, this means estimating the θi‘s by

$\dfrac{\sigma^2 \bar{x} + \tau^2 x_i}{\sigma^2+\tau^2}$

with the characteristic shrinkage (to the average) property of the resulting estimator (Efron and Morris, 1973).

…empirical Bayes isn’t Bayes.” B. Efron (p.90)

While using Bayesian tools, this technique is outside of the Bayesian paradigm for several reasons: (a) the prior depends on the data, hence it lacks foundational justifications; (b) the prior varies with the data, hence it lacks theoretical validations like Walk’s complete class theorem; (c) the prior uses the data once, hence the posterior uses the data twice (see the vignette about this sin in the previous issue); (d) the prior relies of an estimator, whose variability is not accounted for in the subsequent analysis (Morris, 1983). The original motivation for the approach (Robbins, 1955) was more non-parametric, however it gained popularity in the 70’s and 80’s both in conjunction with the Stein effect and as a practical mean of bypassing complex Bayesian computations. As illustrated by Efron’s book, it recently met with renewed interest in connection with multiple testing.

## Bayes posterior just quick and dirty on X’idated

Posted in Statistics, Travel, University life with tags , , , , on February 22, 2012 by xi'an

As a coincidence, I noticed that Don Fraser’s recent discussion paper `Is Bayes posterior just quick and dirty confidence?’ will be discussed this Friday (18:00 UTC) on the Cross Validated Journal Club. I do not know whether or not to interpret the information “The author confirmed his presence at the event” as meaning Don Fraser will be on line to discuss his paper with X’ed members Feel free to join anyway if you have 20 reputation points or plan to get those by Friday! (I will be in the train coming back from Oxford. Oxford, England, not Mississippi!)

## Principles of Applied Statistics

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on February 13, 2012 by xi'an

This book by David Cox and Christl Donnelly, Principles of Applied Statistics, is an extensive coverage of all the necessary steps and precautions one must go through when contemplating applied (i.e. real!) statistics. As the authors write in the very first sentence of the book, “applied statistics is more than data analysis” (p.i); the title could indeed have been “Principled Data Analysis”! Indeed, Principles of Applied Statistics reminded me of how much we (at least I) take “the model” and “the data” for granted when doing statistical analyses, by going through all the pre-data and post-data steps that lead to the “idealized” (p.188) data analysis. The contents of the book are intentionally simple, with hardly any mathematical aspect, but with a clinical attention to exhaustivity and clarity. For instance, even though I would have enjoyed more stress on probabilistic models as the basis for statistical inference, they only appear in the fourth chapter (out of ten) with error in variable models. The painstakingly careful coverage of the myriad of tiny but essential steps involved in a statistical analysis and the highlight of the numerous corresponding pitfalls was certainly illuminating to me.  Just as the book refrains from mathematical digressions (“our emphasis is on the subject-matter, not on the statistical techniques as such p.12), it falls short from engaging into detail and complex data stories. Instead, it uses little grey boxes to convey the pertinent aspects of a given data analysis, referring to a paper for the full story. (I acknowledge this may be frustrating at times, as one would like to read more…) The book reads very nicely and smoothly, and I must acknowledge I read most of it in trains, métros, and planes over the past week. (This remark is not  intended as a criticism against a lack of depth or interest, by all means [and medians]!)

A general principle, sounding superficial but difficult to implement, is that analyses should be as simple as possible, but not simpler.” (p.9)

To get into more details, Principles of Applied Statistics covers the (most!) purposes of statistical analyses (Chap. 1), design with some special emphasis (Chap. 2-3), which is not surprising given the record of the authors (and “not a moribund art form”, p.51), measurement (Chap. 4), including the special case of latent variables and their role in model formulation, preliminary analysis (Chap. 5) by which the authors mean data screening and graphical pre-analysis, [at last!] models (Chap. 6-7), separated in model formulation [debating the nature of probability] and model choice, the later being  somehow separated from the standard meaning of the term (done in §8.4.5 and §8.4.6), formal [mathematical] inference (Chap. 8), covering in particular testing and multiple testing, interpretation (Chap. 9), i.e. post-processing, and a final epilogue (Chap. 10). The readership of the book is rather broad, from practitioners to students, although both categories do require a good dose of maturity, to teachers, to scientists designing experiments with a statistical mind. It may be deemed too philosophical by some, too allusive by others, but I think it constitutes a magnificent testimony to the depth and to the spectrum of our field.

Of course, all choices are to some extent provisional.“(p.130)

As a personal aside,  I appreciated the illustration through capture-recapture models (p.36) with a remark of the impact of toe-clipping on frogs, as it reminded me of a similar way of marking lizards when my (then) student Jérôme Dupuis was working on a corresponding capture-recapture dataset in the 90’s. On the opposite, while John Snow‘s story [of using maps to explain the cause of cholera] is alluring, and his map makes for a great cover, I am less convinced it is particularly relevant within this book.

The word Bayesian, however, became more widely used, sometimes representing a regression to the older usage of flat prior distributions supposedly representing initial ignorance, sometimes meaning models in which the parameters of interest are regarded as random variables and occasionaly meaning little more than that the laws of probability are somewhere invoked.” (p.144)

My main quibble with the book goes, most unsurprisingly!, with the processing of Bayesian analysis found in Principles of Applied Statistics (pp.143-144). Indeed, on the one hand, the method is mostly criticised over those two pages. On the other hand, it is the only method presented with this level of details, including historical background, which seems a bit superfluous for a treatise on applied statistics. The drawbacks mentioned are (p.144)

• the weight of prior information or modelling as “evidence”;
• the impact of “indifference or ignorance or reference priors”;
• whether or not empirical Bayes modelling has been used to construct the prior;
• whether or not the Bayesian approach is anything more than a “computationally convenient way of obtaining confidence intervals”

The empirical Bayes perspective is the original one found in Robbins (1956) and seems to find grace in the authors’ eyes (“the most satisfactory formulation”, p.156). Contrary to MCMC methods, “a black box in that typically it is unclear which features of the data are driving the conclusions” (p.149)…

If an issue can be addressed nonparametrically then it will often be better to tackle it parametrically; however, if it cannot be resolved nonparametrically then it is usually dangerous to resolve it parametrically.” (p.96)

Apart from a more philosophical paragraph on the distinction between machine learning and statistical analysis in the final chapter, with the drawback of using neural nets and such as black-box methods (p.185), there is relatively little coverage of non-parametric models, the choice of “parametric formulations” (p.96) being openly chosen. I can somehow understand this perspective for simpler settings, namely that nonparametric models offer little explanation of the production of the data. However, in more complex models, nonparametric components often are a convenient way to evacuate burdensome nuisance parameters…. Again, technical aspects are not the focus of Principles of Applied Statistics so this also explains why it does not dwell intently on nonparametric models.

A test of meaningfulness of a possible model for a data-generating process is whether it can be used directly to simulate data.” (p.104)

The above remark is quite interesting, especially when accounting for David Cox’ current appreciation of ABC techniques. The impossibility to generate from a posited model as some found in econometrics precludes using ABC, but this does not necessarily mean the model should be excluded as unrealistic…

The overriding general principle is that there should be a seamless flow between statistical and subject-matter considerations.” (p.188)

As mentioned earlier, the last chapter brings a philosophical conclusion on what is (applied) statistics. It is stresses the need for a careful and principled use of black-box methods so that they preserve a general framework and lead to explicit interpretations.

## May I believe I am a Bayesian?!

Posted in Books, Statistics, University life with tags , , , , , , , , , on January 21, 2012 by xi'an

…the argument is false that because some ideal form of this approach to reasoning seems excellent n theory it therefore follows that in practice using this and only this approach to reasoning is the right thing to do.” Stephen Senn, 2011

Deborah Mayo, Aris Spanos, and Kent Staley have edited a special issue of Rationality, Markets and Morals (RMM) (a rather weird combination, esp. for a journal name!) on “Statistical Science and Philosophy of Science: Where Do (Should) They Meet in 2011 and Beyond?” for which comments are open. Stephen Senn has a paper therein entitled You May Believe You Are a Bayesian But You Are Probably Wrong in his usual witty, entertaining, and… Bayesian-bashing style! I find it very kind of him to allow us to remain in the wrong, very kind indeed…

Now, the paper somehow intersects with the comments Stephen made on our review of Harold Jeffreys’ Theory of Probability a while ago. It contains a nice introduction to the four great systems of statistical inference, embodied by de Finetti, Fisher, Jeffreys, and Neyman plus Pearson. The main criticism of Bayesianism à la de Finetti is that it is so perfect as to be outworldish. And, since this perfection is lost in the practical implementation, there is no compelling reason to be a Bayesian. Worse, that all practical Bayesian implementations conflict with Bayesian principles. Hence a Bayesian author “in practice is wrong”. Stephen concludes with a call for eclecticism, quite in line with his usual style since this is likely to antagonise everyone. (I wonder whether or not having no final dot to the paper has a philosophical meaning. Since I have been caught in over-interpreting book covers, I will not say more!) As I will try to explain below, I believe Stephen has paradoxically himself fallen victim of over-theorising/philosophising! (Referring the interested reader to the above post as well as to my comments on Don Fraser’s “Is Bayes posterior quick and dirty confidence?” for more related points. Esp. about Senn’s criticisms of objective Bayes on page 52 that are not so central to this discussion… Same thing for the different notions of probability [p.49] and the relative difficulties of the terms in (2) [p.50]. Deborah Mayo has a ‘deconstructed” version of Stephen’s paper on her blog, with a much deeper if deBayesian philosophical discussion. And then Andrew Jaffe wrote a post in reply to Stephen’s paper. Whose points I cannot discuss for lack of time, but with an interesting mention of Jaynes as missing in Senn’s pantheon.)

The Bayesian theory is a theory on how to remain perfect but it does not explain how to become good.” Stephen Senn, 2011

While associating theories with characters is a reasonable rethoretical device, especially with large scale characters as the one above!, I think it deters the reader from a philosophical questioning on the theory behind the (big) man. (In fact, it is a form of bullying or, more politely (?), of having big names shoved down your throat as a form of argument.)  In particular, Stephen freezes the (Bayesian reasoning about the) Bayesian paradigm in its de Finetti phase-state, arguing about what de Finetti thought and believed. While this is historically interesting, I do not see why we should care at the praxis level. (I have made similar comments on this blog about the unpleasant aspects of being associated with one character, esp. the mysterious Reverent Bayes!) But this is not my main point.

…in practice things are not so simple.” Stephen Senn, 2011

The core argument in Senn’s diatribe is that reality is always more complex than the theory allows for and thus that a Bayesian has to compromise on her/his perfect theory with reality/practice in order to reach decisions. A kind of philosophical equivalent to Achille and the tortoise. However, it seems to me that the very fact that the Bayesian paradigm is a learning principle implies that imprecisions and imperfections are naturally endowed into the decision process. Thus avoiding the apparent infinite regress (Regress ins Unendliche) of having to run a Bayesian analysis to derive the prior for the Bayesian analysis at the level below (which is how I interpret Stephen’s first paragraph in Section 3). By refusing the transformation of a perfect albeit ideal Bayesian into a practical if imperfect bayesian (or coherent learner or whatever name that does not sound like being a member of a sect!), Stephen falls short of incorporating the contrainte de réalité into his own paradigm. The further criticisms found about prior justification, construction, evaluation (pp.59-60) are also of that kind, namely preventing the statistician to incorporate a degree of (probabilistic) uncertainty into her/his analysis.

In conclusion, reading Stephen’s piece was a pleasant and thought-provoking moment. I am glad to be allowed to believe I am a Bayesian, even though I do not believe it is a belief! The praxis of thousands of scientists using Bayesian tools with their personal degree of subjective involvement is an evolutive organism that reaches much further than the highly stylised construct of de Finetti (or of de Finetti restaged by Stephen!). And appropriately getting away from claims to being perfect or right. Or even being more philosophical.

## principles of uncertainty

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , , , on October 14, 2011 by xi'an

Bayes Theorem is a simple consequence of the axioms of probability, and is therefore accepted by all as valid. However, some who challenge the use of personal probability reject certain applications of Bayes Theorem.”  J. Kadane, p.44

Principles of uncertainty by Joseph (“Jay”) Kadane (Carnegie Mellon University, Pittsburgh) is a profound and mesmerising book on the foundations and principles of subjectivist or behaviouristic Bayesian analysis. Jay Kadane wrote Principles of uncertainty over a period of several years and, more or less in his own words, it represents the legacy he wants to leave for the future. The book starts with a large section on Jay’s definition of a probability model, with rigorous mathematical derivations all the way to Lebesgue measure (or more exactly the McShane-Stieltjes measure). This section contains many side derivations that pertain to mathematical analysis, in order to explain the subtleties of infinite countable and uncountable sets, and the distinction between finitely additive and countably additive (probability) measures. Unsurprisingly, the role of utility is emphasized in this book that keeps stressing the personalistic entry to Bayesian statistics. Principles of uncertainty also contains a formal development on the validity of Markov chain Monte Carlo methods that is superb and missing in most equivalent textbooks. Overall, the book is a pleasure to read. And highly recommended for teaching as it can be used at many different levels. Continue reading


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# Fractional quantum Hall effect

The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of $e^2/h$. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations have a fractional elementary charge and possibly also fractional statistics. The 1998 Nobel Prize in Physics was awarded to Robert Laughlin, Horst Störmer, and Daniel Tsui for the discovery and explanation of the fractional Hall effect.[1] However, Laughlin's explanation was a phenomenological guess and only applies to fillings $\nu = 1/m$ where $m$ is an odd integer. The microscopic origin of the FQHE remains unknown and is a major research topic in condensed matter physics.

## Introduction

 What mechanism explains the existence of the $\nu=5/2$ state in the fractional quantum Hall effect?

The fractional quantum Hall effect (FQHE) is a collective behaviour in a two-dimensional system of electrons. In particular magnetic fields, the electron gas condenses into a remarkable liquid state, which is very delicate, requiring high quality material with a low carrier concentration, and extremely low temperatures. As in the integer quantum Hall effect, the Hall resistance undergoes certain quantum Hall transitions to form a series of plateaus. Each particular value of the magnetic field corresponds to a filling factor (the ratio of electrons to magnetic flux quanta)

$\nu = p/q,\$

where p and q are integers with no common factors. Here q turns out to be an odd number with the exception of two filling factors 5/2 and 7/2. The principal series of such fractions are

${1\over 3}, {2\over 5}, {3\over 7}, \mbox{etc.,}$

and

${2\over3}, {3\over 5}, {4\over 7}, \mbox{etc.}$

There were several major steps in the theory of the FQHE.

• Laughlin states and fractionally-charged quasiparticles: this theory, proposed by Laughlin, is based on accurate trial wave functions for the ground state at fraction $1/q$ as well as its quasiparticle and quasihole excitations. The excitations have fractional charge of magnitude $e^*={e\over q}$.
• Fractional exchange statistics of quasiparticles: Bertrand Halperin conjectured, and Daniel Arovas, J. R. Schrieffer, and Frank Wilczek demonstrated, that the fractionally charged quasiparticle excitations of the Laughlin states are anyons with fractional statistical angle $\theta = {\pi \over q}$; the wave function acquires phase factor of $e^{i \theta}$ (together with an Aharonov-Bohm phase factor) when identical quasiparticles are exchanged in a counterclockwise sense. A recent experiment seems to give a clear demonstration of this effect.[2]
• Hierarchy states: this theory was proposed by Duncan Haldane, and further clarified by Halperin, to explain the observed filling fractions not occurring at the Laughlin states' $\nu = 1/q$. Starting with the Laughlin states, new states at different fillings can be formed by condensing quasiparticles into their own Laughlin states. The new states and their fillings are constrained by the fractional statistics of the quasiparticles, producing e.g. $\nu = 2/5$ and $2/7$ states from the Laughlin $\nu = 1/3$ state. Similarly constructing another set of new states by condensing quasiparticles of the first set of new states, and so on, produces a hierarchy of states covering all the odd-denominator filling fractions. This idea has been validated quantitatively,[3] and brings out the observed fractions in a natural order. Laughlin's original plasma model was extended to the hierarchy states by MacDonald and others.[4]
• Composite fermions: this theory was proposed by Jain, and further extended by Halperin, Lee and Read. The basic idea of this theory is that as a result of the repulsive interactions, two (or, in general, an even number of) vortices are captured by each electron, forming integer-charged quasiparticles called composite fermions. The fractional states of the electrons are understood as the integer QHE of composite fermions. For example, this makes electrons at filling factors 1/3, 2/5, 3/7, etc. behave in the same way as at filling factor 1, 2, 3, etc. Composite fermions have been observed, and the theory has been partially verified by experiment and computer calculations. Composite fermions are valid even beyond the fractional quantum Hall effect; for example, the filling factor 1/2 corresponds to zero magnetic field for composite fermions, resulting in their Fermi sea. Composite fermion theory provides a complementary description of the Laughlin and hierarchy states. It gives trial wave functions which, though not identical to those produced from the hierarchy picture (the wave functions for the Laughlin states are identical), are in the same universality class, as shown by Read. There are no experimental tests for fractional quantum Hall states that, even in principle, allow one to confirm the composite fermion description while excluding the hierarchy description.

The FQHE was experimentally discovered in 1982 by Daniel Tsui and Horst Störmer, in experiments performed on gallium arsenide heterostructures developed by Arthur Gossard. Tsui, Störmer, and Laughlin were awarded the 1998 Nobel Prize for their work.

Fractionally charged quasiparticles are neither bosons nor fermions and exhibit anyonic statistics. The fractional quantum Hall effect continues to be influential in theories about topological order. Certain fractional quantum Hall phases appear to have the right properties for building a topological quantum computer.

## Evidence for fractionally-charged quasiparticles

Experiments have reported results that specifically support the understanding that there are fractionally-charged quasiparticles in an electron gas under FQHE conditions.

In 1995, the fractional charge of Laughlin quasiparticles was measured directly in a quantum antidot electrometer at Stony Brook University, New York.[5] In 1997, two groups of physicists at the Weizmann Institute of Science in Rehovot, Israel, and at the Commissariat à l'énergie atomique laboratory near Paris, detected such quasiparticles carrying an electric current, through measuring quantum shot noise.[6][7] Both of these experiments have been confirmed with certainty.

A more recent experiment,[8] which measures the quasiparticle charge extremely directly, appears beyond reproach.

## Impact of fractional quantum Hall effect

The FQH effect shows the limits of Landau's symmetry breaking theory. Previously it was long believed that the symmetry breaking theory could explain all the important concepts and essential properties of all forms of matter. According to this view the only thing to be done is to apply the symmetry breaking theory to all different kinds of phases and phase transitions. From this perspective, we can understand the importance of the FQHE discovered by Tsui, Stormer, and Gossard.

Different FQH states all have the same symmetry and cannot be described by symmetry breaking theory. Thus FQH states represent new states of matter that contain a completely new kind of order—topological order. The existence of FQH liquids indicates that there is a whole new world beyond the paradigm of symmetry breaking, waiting to be explored. The FQH effect opened up a new chapter in condensed matter physics. The new type of orders represented by FQH states greatly enrich our understanding of quantum phases and quantum phase transitions. The associated fractional charge, fractional statistics, non-Abelian statistics, chiral edge states, etc. demonstrate the power and the fascination of emergence in many-body systems.


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Manifolds with two coordinate charts - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T12:38:56Z http://mathoverflow.net/feeds/question/117457 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117457/manifolds-with-two-coordinate-charts Manifolds with two coordinate charts Lee Mosher 2012-12-28T23:19:26Z 2013-01-11T12:56:04Z
What is an early reference for the fact that if a compact, connected $n$-manifold $M$ is covered by two open sets homeomorphic to $\mathbb{R}^n$ then $M$ is homeomorphic to $S^n$?

And is it true that if $M$ is a compact, connected $n$-manifold with boundary, and if $M$ is covered by two open sets homeomorphic to $\lbrace(x_1,\ldots,x_n) \in \mathbb{R}^n | x_n \ge 0\rbrace$, then $M$ is a closed ball?
 http://mathoverflow.net/questions/117457/manifolds-with-two-coordinate-charts/117475#117475 Answer by Andy Putman for Manifolds with two coordinate charts Andy Putman 2012-12-29T04:55:50Z 2012-12-29T16:58:54Z
I'll only discuss the first question (EDIT : Actually, I address the second question at the end). As Agol pointed out in the comments, for $n \geq 5$ this is an easy consequence of Newman's 1966 proof of the Poincare conjecture in the topological category.

I don't know if it was explicitly stated earlier than this. However, it can easily be derived from the main result of the paper

MR0126835 (23 #A4129) Brown, Morton The monotone union of open n-cells is an open n-cell. Proc. Amer. Math. Soc. 12 1961 812–814.

In fact, this works in all dimensions (including $3$ and $4$).

Brown's theorem is as follows. Assume that $M$ is a topological $n$-manifold and that for all compact $K \subset M$, there exists some open set $U \subset M$ with $K \subset U$ and $U \cong \mathbb{R}^n$. Then $M \cong \mathbb{R}^n$. Brown's proof is clever, but completely elementary.

To get the desired result from this, assume that $X = U_1 \cup U_2$ with $U_i \cong \mathbb{R}^n$ and that $X$ is compact. Let $\phi : \mathbb{R}^n \rightarrow U_1$ be a homeomorphism. It is enough to prove that $X \setminus \{\phi(0)\} \cong \mathbb{R}^n$. We will do this with Brown's theorem. Consider a compact set $K \subset X \setminus \{\phi(0)\}$. To verify Brown's criteria, it is enough to construct a homeomorphism $\psi : X \setminus \{\phi(0)\} \rightarrow X \setminus \{\phi(0)\}$ such that $\psi(K) \subset U_2$.

For $r>0$, let $B(r) \subset \mathbb{R}^n$ be the ball of radius $r$. The set $U_1 \setminus U_2$ is compact, so there exists some $R>0$ such that $U_1 \setminus \phi(B(R)) \subset U_2$. Also, there exists some $\epsilon > 0$ such that $K \cap \phi(B(\epsilon)) = \emptyset$. It is easy to construct a homeomorphism $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $f(B(\epsilon)) = B(2R)$ and $f(0)=0$ and $f|_{\mathbb{R}^n \setminus B(3R)} = \text{id}$. We can therefore define a homeomorphism $\psi : X \setminus \{\phi(0)\} \rightarrow X \setminus \{\phi(0)\}$ by $\psi(p) = \phi \circ f \circ \phi^{-1}(p)$ for $p \in U_1 \setminus \{\phi(0)\}$ and $\psi(p) = p$ for $p \notin U_1$. Clearly $\psi(K) \subset U_2$.

EDIT : Lee suggested that this might be able to address his second question too. I thought a bit about it, and I believe that it can. The key is the following "relative" version of Brown's theorem, which can be proven exactly like Brown's theorem.

Theorem : Let $(M,N)$ be a pair consisting of a topological $n$-manifold $M$ and a closed submanifold $N \subset M$. Assume that for all compact $K \subset M$, there exists some open set $U \subset M$ such that $K \subset U$ and such that the pair $(U,U \cap N)$ is homeomorphic to the pair $(\mathbb{R}^n,\mathbb{R}^{n-1})$ (the second embedded in the standard way). Then $(M,N) \cong (\mathbb{R}^n,\mathbb{R}^{n-1})$.

To apply this, assume that $X$ is a compact manifold with boundary and that $X = U_1 \cup U_2$ with $(U_i,\partial U_i) \cong (\mathbb{R}^n_{\geq 0},\mathbb{R}^{n-1})$. Double $X$ to get a closed manifold $Y$, and let $Y' \subset Y$ be the image of the boundary of $X$. The open sets $U_i$ double to give an open cover $Y = V_1 \cup V_2$. Letting $V_i' = V_i \cap Y'$, we have $(V_i,V_i') \cong (\mathbb{R}^n,\mathbb{R}^{n-1})$. Let $(M,M')$ be the result of deleting the image of $0$ in $(V_1,V_1')$. It is enough to prove that $(M,M') \cong (\mathbb{R}^n,\mathbb{R}^{n-1})$, and this can be proven just like above.

Of course, Agol answered the second question first -- it follows from the topological Schonfleiss theorem applied to the double, which was proven by Brown in

MR0117695 (22 #8470b) Reviewed Brown, Morton A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc. 66 1960 74–76. 54.00 (57.00)

Mazur had earlier proven a weaker result. This requires the sphere to be bicollared, but this holds. Indeed, from the assumptions the sphere is locally bicollared, and Brown proved in

MR0133812 (24 #A3637) Brown, Morton Locally flat imbeddings of topological manifolds. Ann. of Math. (2) 75 1962 331–341.

that this implies that the sphere is bicollared. See

MR0267588 (42 #2490) Connelly, Robert A new proof of Brown's collaring theorem. Proc. Amer. Math. Soc. 27 1971 180–182.

for a super-easy proof of Brown's collaring theorem.