text:
Sweat for success. Exercise could be key to boosting your productivity
Step into your local gym after 6.30 in the evening and it�s highly unlikely that you�ll find an empty locker.
Britons have long been seen by their continental counterparts as an unhealthy bunch who would rather spend their evenings drinking in the pub than doing anything physically stimulating. But with the rise of corporate gyms and companies offering membership as employment perks, the gym is no longer just the preserve of aspirant Mr Universe candidates, but the humble office worker too.
And many top business people reckon a regular workout is crucial to career success. Catherine Watkin, a consultant at Brighton recruiter GRS Risk, says her life has been transformed by her decision to move away from recruiting in the City and opt for a more active lifestyle.
Originally from Snowdonia and the daughter of an outdoor pursuits instructor, exercise had always been a regular part of her life, and working long hours was draining her mentally and physically. �I was tired all the time and would drag myself into work and back home. But now I have time to exercise it energises me,� she observes.
After quitting the job, Watkin suffered from a serious case of burn-out. �I never wanted to work in recruitment again and needed to take a sabbatical,� she recalls.
Watkin believes that the exercise, coupled with a more balanced style of living, has made her much more productive. She claims to achieve the same results at GRS Risk by working sensible hours and enjoying outdoor hobbies as she did by working long, unsociable hours in London. Her interests include mountain biking, long-distance running � she ran this year�s London Marathon � and Lindy Hop, an energetic partner dance.
Julia Robertson, managing director of Carlisle Staffing Services, started training four years ago to keep healthy. Unlike Watkin, she sees exercise as a daily chore that has to be done. �When I go into the gym early in the morning, I am sure the people in there are doing similar jobs to me,� she observes.
Meanwhile, Tony Reeves, chairman and chief executive of AIM-listed recruiter Hot Group, actively encourages his staff to take part in sport events. Reeves, who says he has been sports-mad all his life, has run eight marathons and encourages his employees to take part with him. He said: �It helps build camaraderie in the office. It also helps create an equal playing field outside the office because no one is really the boss any more.�
Sport also helps boost his own performance, he claims: �I am no saint and sometimes I drink too much wine. But if I have not had a workout for more than two days I start to feel lethargic.�
GRS Risk�s Watkin says the recruitment industry is missing a trick by not allowing people the time off to do exercise. Agencies expecting consultants to work over-long hours will find productivity is damaged in the long run.
She warns: �A lot of recruitment companies would do well to realise that we work in a pressured industry. Recruitment bosses should allow employees the time to go out and do things, otherwise they lose productivity and their job just becomes a drag.�
Richard Staines


RIYADH: Saudi Arabian billionaire Prince Alwaleed bin Talal, one of the kingdom's top international businessmen, was released from detention on Saturday, more than two months after he was taken into custody in a sweeping crackdown on corruption.
His release came hours after he told Reuters in an interview at Riyadh's opulent Ritz-Carlton hotel that he expected to be cleared of any wrongdoing and be freed within days.
A senior Saudi official said Prince Alwaleed was freed after he reached a financial settlement with the attorney general.
"The attorney general has approved this morning the settlement that was reached with Prince Alwaleed bin Talal, and the prince returned home at 1100 a.m. (0800 GMT)," the official told Reuters, without giving details of the terms.
The decision to free him, and the release of several other well-known tycoons on Friday, suggested the main part of the corruption probe was winding down after it sent shockwaves through Saudi Arabia's business and political establishment.
In his first interview since being detained, conducted hours before his release, Prince Alwaleed told Reuters he maintained his innocence of any corruption in talks with the authorities.
He said he expected to keep full control of his global investment firm Kingdom Holding Co 4280.SE without being required to hand assets to the state. He said he had been able to communicate with executives at his business while detained.
Prince Alwaleed, who is in his early 60s, described his confinement as a "misunderstanding" and said he supported reform efforts by Crown Prince Mohammed bin Salman.
"There are no charges. There are just some discussions between me and the government," he said.
Prince Alwaleed had been confined at the Ritz-Carlton since early November, along with dozens of other senior officials and businessmen, part of the crown prince's plan to reform oil superpower Saudi Arabia and consolidate his position.
The source declined to give further details, but confirmed that Prince Alwaleed would remain head of Kingdom Holding.
A Gulf banker who deals with Saudi Arabia said the authorities appeared keen to conclude the probe partly because foreign investors were concerned their assets or local business partners could be targeted in the wide-ranging crackdown.
Prince Alwaleed's detention was particularly worrying for foreigners because of his international prominence as an investor in top Western companies such as Twitter (TWTR.N) and Citigroup (C.N), and in top hotels including the George V in Paris and the Plaza in New York, the banker said.
"The government is signaling that it wants to move to a new phase now, away from the crackdown and into other economic reforms," the banker said.
Outside Prince Alwaleed's Riyadh palace, dozens of cars lined the entrances as a huge Saudi flag flapped above. Guards cracked jokes and drank coffee. His office said the prince was out visiting family, but declined to give any details.
The attorney general said earlier this week that 90 detainees had been released after their were charges dropped, while others traded cash, real estate and other assets for their freedom. The authorities were still holding 95 people, he said. Some are expected to be put on trial.
An official Saudi source said on Friday that several prominent businessmen had reached financial settlements with the authorities, including Waleed al-Ibrahim, owner of regional television network MBC, who was released. Terms of his settlement were not revealed.
Saudi authorities have said they expect to raise some $100 billion for the government through such settlements – a huge windfall for the state, which has seen its finances squeezed by low oil prices. Some private analysts think that target will be hard to hit, given how many suspects have seen charges dropped.
Allegations against Prince Alwaleed included money laundering, bribery and extorting officials, a Saudi official told Reuters in November. The prince's net worth has been estimated by Forbes magazine at $17 billion.
While declaring his innocence in the Reuters interview, the prince appeared greyer and thinner than in his last public appearance, a television interview in October, and had grown a beard while in detention.
"I have nothing to hide at all. I'm so comfortable, I'm so relaxed. I shave here, like at home. My barber comes here. I'm like at home, frankly speaking," he said, adding he had been able to communicate with family members while in detention.
He granted the 30-minute interview to Reuters, including a tour of his suite, partly to disprove rumors about mistreatment and that he had been moved from the hotel to a prison.
After being released, Prince Alwaleed said, he would stay in Saudi Arabia and would return to the challenge of juggling his global business interests.


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Ontwikkeld door de University of Cambridge is dit spel speciaal voor mensen die naast hun moedertaal de Engels taal erbij willen leren.
Players of Academy Island have to progress through different difficulty levels by helping the alien use English in a range of situations such as shopping in a bakery and visiting places such as an art gallery and library. Our game is designed to help learners improve their English language ability, in a fun way.


text:
Q: How can I merge or search an object? Here's my issue:
I have an object filled with arrays that look like this.
[376339] => Array
    (
        [0] => 1f422730-f54b-4e4d-9289-10258ce74446
        [1] => 60dc4646-06ce-44d0-abe9-ee371847f4df
    )

I need to search another object to find objects with the matching IDs, like below. Is there a way of doing this without a foreach? There are SEVERAL and I would like to not have to loop over the entire object every time.
stdClass Object
(
    [id] => 1f422730-f54b-4e4d-9289-10258ce74446
    [percentage] => 32
    [destinations] => Array
        (
            [0] => stdClass Object
                (
                    [id] => 59826
                    [destination_id] => 59826
                    [type] => Destination
                    [dequeue] =>
                    [value] => xxxxxxxxxxx
                )

        )

)
stdClass Object
(
    [id] => 60dc4646-06ce-44d0-abe9-ee371847f4df
    [percentage] => 68
    [destinations] => Array
        (
            [0] => stdClass Object
                (
                    [id] => 60046
                    [destination_id] => 60046
                    [type] => Destination
                    [dequeue] =>
                    [value] => xxxxxxxxxxxx
                )

        )

)

I need it to end up looking like this.
[376339] => Array
    (
        [0] => Array
            (
                [id] => 1f422730-f54b-4e4d-9289-10258ce74446
                [percentage] => 32
                [destinations] => Array
                    (
                        [0] => stdClass Object
                            (
                                [id] => 59826
                                [destination_id] => 59826
                                [type] => Destination
                                [dequeue] =>
                                [value] => xxxxxxxxxxx
                            )
                    )
            )
        [1] => Array
            (
                [id] => 60dc4646-06ce-44d0-abe9-ee371847f4df
                [percentage] => 68
                [destinations] => Array
                    (
                        [0] => stdClass Object
                            (
                                [id] => 60046
                                [destination_id] => 60046
                                [type] => Destination
                                [dequeue] =>
                                [value] => xxxxxxxxxxxx
                            )

                    )
            )
    )

I'm not sure if this makes any sense, that's why I had my two inital outputs I need to have merged into one somehow. This is all coming from one huge json object and I'm just using json_decode($jsonStuff) to decode it.
Would this be easier if I added true in the decode function? If I could just search for it like I could in python, that would be neat. But as it is, I'm at a loss as to how to get the output I need.
Note: Input json CANNOT be changed, I have no affiliation with the people that created it.

A: First loop over your input array and create an array with the key as the id
$input = json_decode($json_input);
$output = array();
foreach($input as $obj){
    $output[$obj->id] = $obj;
}

then you can build your other array by searching the id on the array key
$massive_search_array = array(376339 => array
    (
        0 => 1f422730-f54b-4e4d-9289-10258ce74446,
        1 => 60dc4646-06ce-44d0-abe9-ee371847f4df
    )
);

$final_output = array();
foreach($massive_search_array as $index => $searches){
    foreach($searches as $search){
        if(isset($output[$search])){
            $final_output[$index][] = $output[$search];
        }
    }
}

redpajama_set_name: RedPajamaStackExchange


text:
74 arrive on patera
4th Dec 2022 @ 10:14 am
Maritime Rescue services yesterday intercepted an inflatable craft carrying 74 people, 16 miles south-east of Lanzarote.
58 men, 12 women and four minors, all of African origin, were on the boat and apparently in good health. The boat was escorted to the main pier in Arrecife where passengers were attended by the Cruz Roja. Three of the migrants were taken to the Dr José Molina Orosa Hospital for checks.
Our Online Advent Calendar is back for another year! Be sure to follow our Facebook page "Gazette Life Lanzarote" for upcoming posts.
https://gazettelife.com/wp-content/uploads/2022/12/pateras-main.jpg 482 579 Shay https://gazettelife.com/wp-content/uploads/2019/08/logo-clean5.png Shay2022-12-04 10:14:332022-12-04 10:14:3474 arrive on patera
This week's weather in Lanzarote
Lanzarote webcams guide
Lanzarote Live Music Guide
What's On in Lanzarote in February
What can Flower Power do for my garden?
A guide to Lanzarote's buses
Airport updates
Burglar charged
Who will be mayor? Car flips in Puerto del Carmen

redpajama_set_name: RedPajamaCommonCrawl


And sometimes surgery leaves you worse for wear.
Here at Southern Pain and Neurological of Greater New Orleans, we often treat our patients suffering from chronic neck pain with nerve blocks. This procedure is far less invasive than surgery, and offers incredible results.
During any procedure, we locate the exact source of the patient's pain. That way we can provide treatment to that affected nerve area. When we do locate the affected region, we apply nerve blocks (an anesthetic) that stop the brain from every receiving pain messages. In essence, pain lessens or even ceases because the brain, and the affected nerves, are no longer communicating the way they had.
And the best part is, this procedure requires little to no special preparation or postoperative healing. This is an outpatient procedure that delivers results. Fast. Unlike surgery, there are no permanent changes made to your body, and you can stop treatment anytime you wish.
Following your procedure, our team will follow up with you to see how you are doing and determine if further action (such as more injections) is required.
Compared to surgery, nerve blocks are far more pleasant to experience and don't come with the same health risks.
If you suffer from chronic neck pain, surgery isn't your only option. It's often not even your best option. Schedule a consultation with our team to learn more about nerve blocks for chronic neck pain. We have offices in Marrero, Metairie, and Covington, LA.


The experience and expertise which we have gained since the company was first established has allowed us to offer a first class, professional service. We recognise that all sites and clients are different with their own individual requirements, therefore Tisdale Demolition, using our specialist knowledge will tailor the demolition techniques accordingly.
In a worldwide industry, where Health and Safety standards vary greatly, Tisdale Demolition Ltd endeavour to offer an unwavering commitment to H&S excellence. Where projects and environmental restrictions demand innovative demolition solutions, we are on hand to offer a comprehensive and adaptable service from; planning and resident liaison, through to completion. We strive to achieve the ultimate standard of service to not only secure the satisfaction of each and every client, but to ensure the utmost safety of our workforce and the local community. Our stringent management systems reflects the company's commitment to Health, Safety and the Environmental protection and our main objective is to recognise and implement sound working practices. A comprehensive operational plan is produced for every project that we undertake which includes project timescale, Environmental and Health and Safety risk assessments, hazardous materials survey, neighbourhood impact and security assessment s. Our aim is to minimise the impact of the project on the environment and adjacent neighbours, who are routinely visited prior to work commencement to address any potential concerns.
in accordance with the Construction (Design and Management) Regulations 2007, and presented to both Client and CDM Co-ordinator for discussion. A preliminary programme, outlining the sequence of intended operations will also be prepared. Both can subsequently be altered to incorporate agreed amendments and/or client requirements.
Working with numerous Social Housing Landlords has allowed us to develop and hone our in house customer care processes with the objective of ensuring that the needs of the client and customers are not only met, but are exceeded.
A dedicated site supervisor will be appointed for each project not only to monitor the day to day work on an on-going basis but to act as a focal point to address any of the local resident's problems or concerns.
Demolition works to domestic housing and industrial properties can sometimes be carried out in built up areas, with occupied premises in the immediate vicinity. In this respect, the site(s) will be fully secured with hoarding to prevent un-authorised access and only authorised personnel will be allowed access to the site by reporting to the site supervisor and then only when they were issued with the appropriate PPE. Demolition works by its nature will have an impact on the surrounding environment due to the levels of dust which can be created, and as such we have recently purchased a mobile dust suppression bowser to minimise the impact on the surrounding environment.
and to advise of the steps which they should take to minimise disruption to themselves.
While working in the proximity of schools we always feel it to be advantageous to visit the head teacher to explain the extent of the works, traffic management requirements, site access and egress and to advise the pupils of the potential dangers regarding demolition sites.


A mature plantsman's garden of one acre with an interesting collection of plants sourced from Ulster Gardens Scheme plant stalls over the last 30 years. It has been developed around a traditional farmyard. There are formal and informal areas of trees, shrubs, roses, clematis and herbacaceous plants which have been established in artistic combinations to give seasonal interest. It also contains raised beds for vegetables and propagation. Mostly suitable for wheelchairs.
Location: On the A27 main Portadown to Tandragee Road approximately 3 miles from Portadown and 2 miles from Tandragee. Leave the A27 at Ballylisk roundabout signposted to Laurelvale. Garden is 0.25 miles on left hand side beside Mullavilly Parish Church.


The Errea Edmonton Windbreaker is one of the Italian sports giant's most popular items in their immensely successful training line. Featuring a waterproof outer with a quarter-zip opening at the side of the neck, the Edmonton is designed to keep the elements at bay and allow the wearer to train in comfort, even on stormy nights. Available in the 3 most popular colours, the Edmonton has something suitable for any team's training gear.
Errea are one of Europe's favourite suppliers of football team kits and training gear, kitting out clubs ranging from 5-A-Side teams right up to professional clubs in the Premiership, Serie A and La Liga, providing recreational and professional clubs with their high quality, excellent value performance sportswear. The Edmonton Windbreaker style pullover jacket is an essential for any team's training kit, and this is one of the best value on the market.


At the IA Summit in Phoenix, AZ Kejun Xu talked over a few insights for optimizing sign-up conversions in her presentation Conversion Rates: Small Design Tweaks That Make a Difference. Here's my notes from her talk.
An online mortgage foreclosure business based on conversions explored what gets people to sign up using usability testing, surveys, and a/b testing.
Take-aways from the research: professionalism, usability, answering people's questions up front, appropriate calls to action, content above the fold, and social validation increased conversion numbers.
Professionalism: had two versions of website. The more professional looking website positively impacted conversion rates (over 30 days).
Usability: the site asked for credit card numbers despite a promise of being free for seven days. People went back to an earlier page to see if it was really free. This wiped out the data people entered. Conversion rates went up when this issue was fixed. Surprise!
Answer people's questions: people wanted to know why a credit card number was required for a free site. The site added a link that explained why on mouseover. Drop off rate feel 2% when this information was added.
Social validation: people will do what others do. Showing content featured on TV raised conversion. Promote comments and reviews of your service.


text:
Highway funding meetings to begin; first at Monticello
Special to The Commercial
Jan 6, 2020 at 1:07 PM Jan 6, 2020 at 2:06 PM
The Arkansas Highway Commission has scheduled 12 regional public meetings. The purpose is to inform the public and solicit input on proposed projects that could be funded by a continuation of the temporary half-cent sales tax.
Area meetings include:
MONTICELLO — 5:30 p.m. Tuesday, Jan. 21, at the fine arts center at the University of Arkansas at Monticello;
TEXARKANA — 5:30 p.m. Thursday, Jan. 30, Trinity Baptist Church;
EL DORADO — 5:30 p.m. Thursday, April 2, El Dorado Municipal Auditorium.
A total of 12 meetings have been scheduled. Other meetings are planned in West Memphis, Harrison, Mena, Fort Smith, Hot Springs, Maumelle, Springdale, Batesville and Jonesboro, according to the news release.
These meetings will consist of an open house session where attendees can view displays and visit with Arkansas Department of Transportation (ARDOT) staff.
This will be followed by a brief formal presentation on the current funding and condition of the Arkansas Highway System as well as the impact that continuation of the half-cent sales tax would have in funding system preservation and new capital projects in the future, according to the release.
In early 2019, Gov. Asa Hutchinson announced his highway funding legislative package that resulted in Act 416 and a proposed Constitutional Amendment, House Joint Resolution 1018. Act 416 provides revenue from motor fuel taxes, electric and hybrid vehicle registration fees, and casinos resulting in an estimated $95 Million in new funding to ARDOT. If the constitutional amendment is approved by voters, it will provide an additional estimated $205 Million annually to ARDOT and an estimated $43 million each to cities and counties, according to the release.
The community is invited to attend the meetings. Attendees will have an opportunity to ask questions directly to the Commission and ARDOT Director Scott E. Bennett.


text:
It's Never Too Late to Shine
Not every collegiate journey follows a traditional path straight from high school to higher education, and even those that do may encounter struggles along the way. But no matter the path you take, there's always the chance to succeed in the end and achieve academic success. Our members remind us of that on a regular basis.
Tammy Lilly – Fontbonne University
Lilly started with a traditional route – entering college right after high school. Soon, though, she decided to leave school to focus on marriage and family. After 22 years, she returned with a new goal. "I was starting all over from the beginning, a different degree, no credit hours transferable, two kids, a husband, and a full-time career," she shared. "I re-entered university with a focus to finish with a 4.0 GPA." The added maturity, she feels, led to a richer experience.
"It's never too late to pursue something that is important to you. And, self-satisfaction should always be the primary reason why someone wants to better themselves. Not what others think or don't think. Not what others say or don't say," said Lilly. As advice to other students, she adds, "If it was easy everyone would have a degree and then the value of a degree would be minimized simply by virtue of is commonality. You are worth it so invest in yourself!"
In 2008, Lilly accomplished her goal and graduated with a B.A. and a membership in Phi Kappa Phi. "Done! One of my greatest achievements!"
Jacob Lisak – Western Michigan University
Lisak struggled in high school and grades were not a focus. "I think there's a lot of young people out there who feel that they're stuck on a path for life," he noted. Lisak has three people to thank for getting back on track. At the top of his list are his parents who always believed in his potential. He also is grateful for his guidance counselor. "He was always someone I could talk to and seemed to dedicate himself to the kids who were struggling. He gave me a big chance to join a law enforcement vocational tech program in high school that was new and difficult to get into. The class was amazing, and it was the first time I felt passionate about school and learning," said Lisak.
He hopes college students take advantage of the opportunities available to them. "Enjoy it," he advises. "It's so much more difficult to socialize when you're older. Take time to join the clubs and groups that you find interesting in school. There will be far less opportunities to do this later in life."
Now with a bachelor's degree in secondary education and a master's degree in social work, Lisak is proud of his accomplishments – including his invitation to join Phi Kappa Phi. "Finding out I was in the top 7.5 [percent] of my class felt great, and it was the first time I had acknowledgement that I was excelling at anything school related."
Skyler Pretto – Florida Institute of Technology
Twelve years after dropping out of high school, Pretto decided to work toward a G.E.D. His appetite for learning combined with the hopes of broadening his career possibilities were the driving force behind the decision. "If there is anything I would tell myself in high school, it would probably be that life does not get any easier when one chooses to not pursue a better education for themselves, a better life, a better career path," he shared.
Now as an electrical engineering student at FIT, the goal of graduation is right around the corner. To his fellow students, he urges them not to worry about their pace or compare themselves to others. "Their desire to learn—their willingness to challenge themselves—is what will drive their success, not just scholastically, but in life," said Pretto.
Earning an invitation to join Phi Kappa Phi has been a highlight of his return to school. He said, "This is the greatest scholastic achievement of my life, thus proving that it is never too late to get a better education."
Do you have your own inspirational story? We'd love to hear it! Email news@phikappaphi.org to share.


Oh how I wish this were available at the Cooperative Extension in Machias! I'm a little bit very jealous. I'd love to take these classes. If you attend any of them I'd love to hear about them. A description of each coarse is available at the University of Maine Cooperative Extension in Cumberland County Extension's website.
University of Maine Cooperative Extension in Cumberland County announces a new series called "From Scratch: Your Maine Kitchen" offered on Saturdays, 10:00 a.m. – 1:00 p.m. monthly between Nov. 2014 – Oct. 2015. A unique taste of Maine from a variety of local food sources to excite your appetite. The series will kick off with "From the Maine Wild" followed in December by "Gifts from the Maine Kitchen"; start the new year with "Making Sourdough bread" in January. Learn more about cheese making, seafood and seaweed, fermentables, foraging for greens and drinking the harvest in the months to come.
Sign up today for this new series of food workshops offered by University of Maine Cooperative Extension Cumberland County, 75 Clearwater Dr., Suite 104 in Falmouth. For special accommodations or for more details call 207.781.6099 or 1.800.287.1471 (in Maine) or register online.


Application. This unit describes the performance outcomes, skills and knowledge required to undertake introductory programming tasks using an object-oriented programming language, including tool usage, documentation, debugging, and testing techniques. how to clean a lingcod Application. This unit describes the performance outcomes, skills and knowledge required to undertake introductory programming tasks using an object-oriented programming language, including tool usage, documentation, debugging, and testing techniques.
I need some help figuring this out if it is possible. I am still very new to this so please forgive me. I have an array of an object that I want a method to manipulate for me.
14/02/2017�� One of the benefits of using Object Oriented Programming (OOP) is the reusability of code. The typical way to do so is to instantiate an object of an already defined class with the Set variable = New ClassName syntax.


The official athletics website for the Mercer University Bears. I went to Team Sports 642 Shurling Drive Macon, Georgia to purchase and confirm the order for (30) new basketball jerseys for our two christian basketball teams.
Cleats, jerseys, helmets, and more — come to Team Sports in Warner Robins to suit up for your favorite sport or refresh your current equipment.Mercer senior Jibri Bryan was shot and killed on Tuesday afternoon in the parking lot of a convenience store in Macon, Ga.Home to more than 3,000 artifacts, the Georgia Sports Hall of Fame is a must-see attractions for anyone visiting Macon in central GA.Georgia Sports Hall of Fame Explore the history of sports in the Peach State at the Georgia Sports Hall of Fame, one of the most distinct museums in the City of Macon.
With more than 43,000 square feet of space, it is the largest individual state sporting hall of fame in the country.NeSmith Field - 374 GA-87 BUS, Cochran, GA John Drew Smith Tennis Center - 3280 N Ingle Pl, Macon, GA Middle Georgia Equestrian Center - 128 MacThompson Road Cochran, Ga. Swim Macon is a year-round competitive swim team that was founded in 1981.There are sporting events happening throughout the year in Macon.
Coverage of Macon County high school sports including baseball, basketball, football, girls basketball, girls soccer, soccer, softball and wrestling.Our Macon, GA, Hibbett Sports is conveniently located in the Walnut Creek Plaza off Gray Highway (U.S. 129) between Shurling Drive and Woodlawn Drive, close to Dollar Tree and Walmart.
MACON, GA. — The Peoria Rivermen took a six-hour bus ride from Pensacola to Macon on Thursday.I am trying to help a couple former business associates with this fantastic piece of artwork.You can find the GSHoF near downtown along Martin Luther King Jr. Blvd.
Home Team Sports Collectibles is just the place you're looking to explore in Macon.
ROCKMART, Ga. — The Georgia Bureau of Investigation released the name of the suspect accused of shooting 4 people in Rockmart, Georgia last night.
The professional sports teams, the Macon Whoopee (minor league hockey) and the Macon Knights (arena football), as well as numerous parks, round out leisure options. Shortly thereafter, he accepted a position with the Macon-Bibb Parks and Recreation department in 2007. Features include news, discounts, retiree lists, policy updates, inclement weather alerts, service tips, and a connection to the Georgia state employee and manager self-service portal.Our organization is dedicated to the development of the competitive swimmer of all ages and abilities.
Nashville, Memphis, and Chattanooga, Tennessee, as well as Birmingham, Alabama, also had teams in the league.
If you need team pants, shirts, uniforms, they can take care of it.OurSports Central is your source for the latest news and scores from alternative and minor league sports in Macon, Georgia\.
About Team Sports Inc is located at the address 642 Shurling Dr Ste B in Macon, Georgia 31211. They can be contacted via phone at (478) 741-9677 for pricing, hours and directions.


text:
Home»Empower»Women In Numbers: Women in the Workplace — Troublemakers By Trade
Women In Numbers: Women in the Workplace — Troublemakers By Trade
March 1, 2022 Updated: March 1, 2022 2 Mins Read
Women often struggle to get respect in their industries, but now we're fighting back.
By Jess Bugg, Illustrations by Jessica Wetterer
According to The New York Times, in 2018 Google silently paid millions of dollars in exit packages to male executives accused of sexual misconduct. Seven female employees retaliated by organizing a walkout that turned into more than 20,000 people around the world. This led to immediate changes including the ability for employees to bring sexual misconduct claims to court.
After a record number of women ran for office in 2018, a record number of women won. While women make up 51% of the U.S. population, they make up just over a quarter of all members of the 117th Congress. However, this is by far the highest percentage in U.S. history and represents a 50% increase from only a decade ago.
All 28 members of the U.S. Women's National Soccer Team sued the U.S. Soccer Federation (USSF) for gender discrimination, alleging unequal pay and working conditions. According to The New York Times, some women players make just 38% of what their male counterparts make despite consistently outperforming them as well as having higher viewership. After court documents revealed that the USSF argued biological differences justified the pay discrepancy between the men and the women, federation president Carlos Cordeiro resigned.
Women in the U.S. have earned more degrees than men for the last four decades. However, it wasn't until 2019 that they surpassed college-educated men in the workforce. A Pew Research study found that as of 2019, 29.5 million women in the labor force had at least a bachelor's degree. Women now make up 50.2% of the college-educated work force, which is up about 11% since 2000.
Lucille Ball was 41 when the first episode of I Love Lucy aired, turning her into one of the most iconic female comedians in history. To this day, her success is still extremely rare as most famous women in the entertainment industry reach stardom at a much younger age. Ball went on to break other norms during her career such as being the first openly pregnant woman on television. The characters Lucy and Ricky (played by her real-life husband, Desi Arnaz) were also the first interracial couple portrayed on television.
READ MORE FROM THE MARCH ISSUE
Lucille Ball Women in the Workplace
Previous ArticleWoman to Watch: Eleni Ottalagana
Next Article LASIK Q&A with Zarmeena Vendal, M.D.


Looking for Social Media Expert to manage the social media activities of the company.
DealMarkaz.pk is a free classified ads website for marketing, advertising & pr in Multan. Post and search free ads on Multan's rapidly growing free advertisement platform. Buy or sell apartments, cars, bikes, mobile phones, find jobs, home services, repair electronic or kitchen appliances and many more. Sell your used stuff at a good price through DealMarkaz.pk in your nearby location.


Ironically enough, I recorded this episode long before I scrolled social media on this very day only to found what I knew would happen: a local "health coach" is encouraging women to go on a sugar detox from November 1 for 21 days. As she put it, the sugar detox was intentionally set to begin the day after Halloween.
Nothing makes me want to pull my eye brows off one hair at a time.
Restricting will lead to a binge.
These women will undoubtedly shove their face with candy on Halloween because they simply are anticipating the deprivation that lies ahead.
An eventual binge on sugar (and any other foods they enjoy) sometime in the future.
If stuff like this worked, why do I have a sneaking suspicion that they'll feel the need to go on another sugar detox next year? How's that workin' for ya? ;) Not so hot, right?
I'll leave the rest to the episode.
Share this with any woman who might be considering a detox on November 1. I just... I just can't.


Some of the country's most notorious street gangs have gotten Web-savvy, showcasing illegal exploits, making threats, and honoring killed and jailed members on digital turf. Crips, Bloods, MS-13, 18th Street and others have staked claims on various corners of cyberspace. "Web bangers" are posting potentially incriminating photos of members holding guns, messages taunting other gangs and boasts of illegal exploits on personal Web sites and social networking sites.


I had hoped to publish this blog post over two weeks ago, but you know how that goes – work travel, projects for clients, jury duty, and a head cold took priority. However late, I still want to clarify an answer I gave as part of the COVE conference legal panel on February 21. A conference attendee asked about the enforceability of clickthrough agreements. I am concerned that I may have provided a confusing explanation of browsewrap and clickthrough agreements and that I may have given the wrong impression that clickthrough agreements might not be enforceable. To be clear, clickthrough agreements are enforceable when properly implemented (which is not difficult).
What follows is a little more information about browsewrap and clickthrough agreements. I'll discuss one component of the legal formation of a contract (mutual assent), compare browsewrap and clickthrough agreements, explain why clickthrough agreements are generally enforceable, and then describe the problems with browsewrap agreements and ways in which browsewrap agreements may be enforceable.
Lame attorney disclaimer – for advice specific to your circumstances on how to implement an enforceable agreement on your website, please contact an attorney.
Although there are some aberrational cases to the contrary, for the most part courts do not treat browsewraps as a contract, and anyone relying on a so-called browsewrap does so at their extreme peril.
What I tried to describe (too quickly and unclearly) at the COVE conference, is that it is pretty easy to show that a website user assents to a clickthrough agreement: as described above, the user must accept an offer to use the website. In contrast, proving that a website visitor has assented to a browsewrap agreement is more difficult.
However, showing assent through a browsewrap agreement can be done. Enforceability of browsewrap agreements depends on a website user's actual or constructive knowledge of the agreement. Here are some factors courts have considered for determining whether actual or constructive knowledge of a browsewrap agreement exists.
Was the agreement prominently referenced many times throughout the website?
Was the agreement present during an order/purchase/signup process?
Would a reasonable prudent user be on notice of the agreement?
Did the link to the agreement stand out from the rest of the webpage? Does a website user have to scroll to find it?
Clickthrough agreements are by far the preferred method for implementing binding agreements through website use, and it might be possible to create binding browsewrap agreements. I was concerned that I may have given the impression that clickthrough agreements are not be enforceable at the COVE conference. That is not the case. Again, clickthrough agreements are enforceable when properly implemented.


text:
\section{Introduction}

In this article, we investigate the properties of some restricted type II maximum likelihood (type II ML) priors on regression coefficients in examples in model selection and uncertainty. Along the way, we establish connections with the Bayesian Information Criterion (BIC; \cite{schwarz1978estimating}) and proper priors. Operationally, parametric type II ML proceeds as follows: (1) start with a parametric model for the data $y$, specified with a sampling density $f(y \mid \theta)$ and a prior $\pi_\eta(\theta)$ that depends on a hyperparameter $\eta \in \mathcal{C}$ and (2) set $\eta$ by maximizing the marginal likelihood $m(y)$ of the model, that is
$$
\eta =  \arg  \max_{\eta \in \mathcal{C}} \int f(x \mid \theta) \pi_\eta (\theta) \, \mathrm{d} \theta  = \arg  \max_{\eta \in \mathcal{C}} m(y).
$$
Type II ML was named and extensively studied in \cite{good1965estimation}, and it can be seen as a particular instance of empirical Bayes which, in general, ``estimates'' the hyperparameter $\eta$ from the data (although not necessarily by maximizing the marginal likelihood: a popular alternative is the method of moments).

The motivation for this work was to seek a compromise between use of conventional priors for model uncertainty (e.g., Zellner's $g$-priors
or Zellner-Siow prior; \cite{zellner1980posterior, zellner1986assessing}) and BIC. Conventional
priors are typically centered at the smallest (or null) model, and can be quite far from the likelihood function arising from a larger model. Prior distributions centered at null models can be oriented in directions away from the likelihood function, which would seem to unduly favor the null model.

\cite{raftery1995bayesian} shows that
BIC is similar to the use of a normal prior for a model that is centered at the maximum likelihood estimate (MLE)
of the unknown parameter $\theta$. This is actually a type II ML prior, arising from estimating the prior mean by type II ML. However, it seems like an extreme use of type II ML, centering the prior completely around the model likelihood function and favoring the more complex model.

The compromise studied
herein is to keep the prior centered at the null model, as with current conventional priors, but allow the prior variance or covariance matrix
to be estimated by type II ML, thus orienting the prior towards the model likelihood function and allowing a reasonable amount of the prior
mass to ``reach'' the model likelihood function. It was hoped that this would strike a balance between the two extremes.
Indeed, we find that this ``variance-oriented" type II ML prior does yield compromise results that are intermediate between the results from conventional priors and BIC. Surprisingly, however, the compromise results were typically closer to results from BIC than results from
conventional priors, providing an unexpected additional justification for BIC.

A second surprise was that the ``variance-oriented" type II ML priors (and resulting Bayes factors) can be computed in closed form, even for the case of entire
unknown prior covariance matrices and even when restricting the hyper-parameters to be at least as vague as conventional priors (
this is a computational advantage over, e.g., the Zellner-Siow priors). The
importance of making  restrictions on the hyperparameters is also highlighted; without them, one can even have inconsistent model selection: for example, if the scale parameter $g$ of a Zellner $g$-prior \cite{zellner1980posterior} is estimated without restrictions, the resulting procedure is not consistent if the null model is true \citep{liang2008mixtures}.

The scenarios we consider in the paper involve regression coefficients in normal linear models (Section~\ref{sec:linmod}), high-dimensional ANOVA  (Section~\ref{sec:ANOVA}), and the nonparametric regression example in \cite{shibata1983asymptotic} (Section~\ref{sec:shibata}). In this latter
section we also highlight how type II ML can be fruitfully used when prior information is available. The article ends with conclusions. All the proofs are relegated to the supplementary material.

\section{Type II ML priors in normal linear models} \label{sec:linmod}

\subsection{Derivation of the type II ML prior}
Consider the normal linear model
$$
Y  = X_0 \beta_0 + X \beta + \epsilon, \, \, \epsilon \sim N_n(0_n, \sigma^2 I_n),
$$
where $Y \in \mathbb{R}^n$, $X_0 \in \mathbb{R}^{n \times p_0}$ contains common predictors, and $X \in \mathbb{R}^{n \times p}$ contains model-specific predictors. We assume that the predictors are linearly independent and the common and model-specific parameters are orthogonal, so that $X_0'X = 0_{p_0 \times p}$ (if $X_0 = 1_n$, this amounts to centering $X$). In this section, the prior on the common parameters is the right-Haar prior $\pi(\beta_0, \sigma^2) \propto 1/\sigma^2$, which is supported by group invariance arguments in \cite{berger1998invariant} and \cite{bayarri2012criteria}.

The prior distribution we consider for $\beta$, given $\sigma^2$, is the $N_p(\beta \mid 0_p, \sigma^2 W)$ normal prior, with mean $0_p$ and
unspecified positive definite covariance matrix $W$.
For a fixed $W$ and $n \ge p+p_0$, the marginal likelihood is
\begin{align*}
m_W(Y) &= \int_{\mathbb{R}} \int_{\mathbb{R}^p} \int_{\mathbb{R}_+} N_n(Y \mid X_0 \beta_0 + X \beta, \sigma^2 I_n) \, N_p(\beta \mid 0, \sigma^2 W) \, 1/\sigma^2 \, \mathrm{d}  \beta_0 \, \mathrm{d}  \beta \, \mathrm{d}  \sigma^2 \nonumber \\
&=  \frac{\Gamma\left(\frac{n-p_0}{2}\right) \pi^{-(n-p_0)/2}}{\left( |X' X| \, |X_0' X_0| \, |(X'X)^{-1}+W|  \right)^{1/2}} [ \mathsf{SSE} + \widehat{\beta}'[W + (X'X)^{-1}]^{-1} \widehat{\beta} ]^{-\frac{(n-p_0)}{2}}, \label{marginal}
\end{align*}
where $\widehat{\beta} = (X'X)^{-1} X' Y$,
$\mathsf{P}_{X} = X (X' X)^{-1} X'$, $\mathsf{P}_{X_0} = X_0(X_0'X_0)^{-1}X_0'$, and $\mathsf{SSE} = Y'(I_n - \mathsf{P}_{X_0}-\mathsf{P}_{X})Y$.
The type II ML approach to determination of $W$ is simply to maximize the marginal likelihood over $W$, using the result as the prior
covariance matrix. An earlier version of this \citep{george2000calibration, hansen2003minimum, liang2008mixtures} considered
$g$-priors arising from $W$ of the form $W= g \,  \sigma^2 (X'X)^{-1}$, and then maximizing the marginal likelihood over the choice of $g$.

While this maximization over $W$ can be done in closed form, the result is not satisfactory, in that the result is
a singular matrix. We thus must constrain $W$ under the maximization, and will do
so through the concept of a ``unit-information prior."
The expected Fisher information of the regression coefficient $\beta$ is $(X'X)/\sigma^2$, so one can argue that $(X'X)/(n \sigma^2)$ contains as much information as a ``typical'' observation in the sample \citep{kass1995reference, raftery1995bayesian,hoff2009first}. Thus
the $N_p(0,n \sigma^2(X'X)^{-1})$ prior is called the unit-information (normal) prior, and it is argued that this
is a reasonably vague (but necessarily proper) prior for dealing with model uncertainty; in particular,
smaller covariance matrices than this are deemed unreasonable. Motivated by this, we study  the restricted type II ML prior
\begin{align*}
\beta \mid \sigma^2 &\sim N_p(0_p, \sigma^2 \, \widehat{W}) \\
\widehat{W} &= \mathrm{arg \, max}_{W \succeq n(X'X)^{-1}} m_W(Y) \,,
\end{align*}
Where $A \succeq B$ means that $A-B$ is positive semidefinite. This will ensure that the restricted type II ML covariance will be
at least as disperse as the unit information prior covariance. The lower bound is also an instance of Zellner's $g$-prior with $g=n$.

 In the context of estimation instead of testing,  \citet{dasgupta1988frequentist}, \citet{leamer1978specification}, and \citet{polasek1985sensitivity} study priors that resemble our type II ML prior, bounding the prior covariance matrix both above and below.

Proposition~\ref{prop:wmatrix} below, which is proved in the supplementary material, shows that the covariance matrix that maximizes $m_W(Y)$ subject to $W \succeq n(X'X)^{-1}$ is a linear combination of the unrestricted maximum over all positive semidefinite matrices, which is proportional to $\widehat{\beta} \widehat{\beta}'$, and the lower bound $n(X'X)^{-1}$.

\begin{proposition} \label{prop:wmatrix}
For $n > p+p_0$, the solution to the optimization problem
\begin{align*}
&\mathrm{maximize} \, \,  m_W(y) \\  &\mathrm{subject \, to}  \, \,  W \succeq n(X'X)^{-1}
\end{align*}
can be written as
\begin{align*}
	\widehat{W} &= a \,\widehat{\beta} \widehat{\beta}' + n(X'X)^{-1} \\
	a &= \max \{0, (n-p_0-1)/\mathsf{SSE}-(n+1)/\mathsf{SSR}\} \\
	\mathsf{SSR} &= \widehat{\beta}'X'X \widehat{\beta} \,.
\end{align*}
\end{proposition}
In the following subsections, we study the properties of the type II ML prior on $\beta$ that takes $\widehat{W}$ as its design matrix in model selection and uncertainty, estimation, and prediction.

\subsection{Model uncertainty and selection} \label{sec:modunc}
Let $X_i$ be the design matrix that includes a subset of $p_i$ out of the $p$ predictors in $X$, with $i \in \{1, 2, \, ... \, , 2^p\}$ ($p_i$ can be 0, which corresponds to the null model), and let $\mathcal{M}_i$ be the model $Y = X_0 \beta_0 + X_i \beta_i + \epsilon_i$, $\epsilon_i \sim N_n(0,\sigma^2 I_n)$, with $X_0' X_i = 0_{p_0 \times p_i}$. Throughout, we set prior covariance matrices locally -- that is, each $\mathcal{M}_i$ is assigned its own $\widehat{W}_i$. The local approach to empirical Bayes model selection is justified through information-theoretical arguments in \cite{hansen2003minimum} and
makes intuitive sense: each model gets to have its marginal likelihood maximized. We perform model selection using null-based Bayes factors, namely
$$
\mathsf{BF}_{i0} =  \frac{\int N_n(Y \mid X_0 \beta_0 + X_i \beta, \sigma^2 I_n) \pi_{\mathsf{ML}}(\beta_0, \beta_i, \sigma^2) \pi_0(\beta_0, \sigma^2)  \, d(\beta_0, \beta_i, \sigma^2)}{\int N_n(Y \mid X_0 \beta_0, \sigma^2 I_n) \pi_0(\beta_0, \sigma^2) \, d(\beta_0, \sigma^2)} = \frac{m_i(Y)}{m_0(Y)}.
$$

Combining the result in Proposition \ref{prop:wmatrix} with the Sherman-Morrison formula and the matrix determinant lemma, it is straightforward to see that the null-based Bayes factor of $\mathcal{M}_i$ under the type II ML covariance matrix is
\begin{equation} \label{eq:bf}
  \mathsf{BF}_{i0} = \begin{cases}  (n+1)^{\frac{n-p_0-p_i}{2}} \left[ n (1-R^2_i) + 1 \right]^{-(n-p_0)/2} &  \text{ if } R_i^2 \le \frac{n+1}{2n-p_0} \\ \varphi(n)^{-1/2} \left( R_i^2\right)^{-1/2} (1-R^2_i)^{-(n-p_0-1)/2} &  \text{ if } R_i^2 > \frac{n+1}{2n-p_0},\end{cases}
\end{equation}
where $\varphi(n) =  \left( \frac{(n+1)^{p_i-1}(n-p_0)^{n-p_0}}{(n-p_0-1)^{n-p_0-1}}\right)$ and $R_i^2 = 1-\mathsf{SSE}_i/\lVert Y - X_0 \widehat{\beta}_0 \rVert^2$. The first case corresponds to the null-based Bayes factor with the lower bound $\widehat{W}_i = \sigma^2 n(X_i'X_i)^{-1}$.

Before we study the properties of the prior in more detail, we present an example with $p = 2$ predictors to introduce some geometric intuition. In addition, the example  will help us highlight that the lower bound prior $\pi_{\mathsf{LB}}(\beta_i \mid \sigma^2) = N_{p_i}(\beta_i \mid 0_{p_i}, \sigma^2 n(X_i'X_i)^{-1})$ has a particular asymmetry with respect to the sign of the correlation between the predictors. It also serves as motivation to compare $\pi_{\mathsf{LB}}$ and the type II ML prior $\pi_{\mathsf{ML}}$ to the Bayesian Information Criterion (BIC; \cite{schwarz1978estimating}), which is defined as
$$
-2 \log N_n(Y \mid X_0 \widehat{\beta}_0 + X_i \widehat{\beta}_i, \widehat{\sigma}^2 I_n) + p_i \log n,
$$
 where $\widehat{\beta}_0 = (X_0' X_0)^{-1} X_0' Y, \widehat{\beta}_i = (X_i' X_i)^{-1} X_i' Y$, and $\widehat{\sigma}^2 = \lVert Y - X \widehat{\beta} \rVert^2/n$ are the maximum likelihood estimators of $\beta_0, \beta_i$ and $\sigma^2$, respectively. Throughout, we treat $\exp(-\mathsf{BIC}/2)$ as an approximate marginal likelihood with the understanding that the ``BIC'' of the null model is $-2 \log (Y \mid X_0 \widehat{\beta}_0, \widehat{\sigma}^2_0 I_n)$, where $\widehat{\sigma}_0^2 = \lVert Y - X_0 \widehat{\beta}_0 \rVert^2/n$ . These choices lead to the following null-based Bayes factors:
$$
\mathsf{BF}_{i0, \mathsf{BIC}} = n^{-p_i/2} (1-R^2_i)^{-n/2}.
$$
\cite{raftery1995bayesian} observed that $\exp(-\mathsf{BIC}/2)$ is an excellent approximation to the marginal likelihood arising
from $\pi_{\mathsf{BIC}}(\beta_i \mid \sigma^2) = N_{p_i}({\beta}_i \mid \hat{\beta}_i , \sigma^2 n(X_i'X_i)^{-1})$, which is
Zellner's $g$-prior with $g = n$, but centered at $\hat{\beta}_i$ instead of $0_{p_i}$. Indeed, under such type II ML prior, the null-based Bayes factor is
$$
\mathsf{BF}_{i0, \widehat{\beta}} = (n+1)^{-p_i/2} (1-R_i^2)^{-(n-p_0)/2},
$$
which is almost identical to $\mathsf{BF}_{i0, \mathsf{BIC}}$. Another prior we will be considering in numerical comparisons is the Zellner-Siow prior, which is $\mathrm{Cauchy}_{p_i}(0_{p_i}, \sigma^2 n(X_i'X_i)^{-1})$, since this is one of the most commonly recommended model uncertainty priors.

\begin{example} (Correlated predictors) \label{ex:twopreds} Consider a model with 2 standardized (centered and scaled) predictors and an intercept, $Y = 1_n \alpha + X \beta + \epsilon$ where $\beta = (\beta_1, \, \beta_2)'$ and $\epsilon \sim N_n(0_n, I_n)$. Since the predictors are standardized, their (uncorrected) sample correlation is the off-diagonal entry of $(X'X)/n$, which we denote $r$. The prior covariance between $X_1$ and $X_2$ implied by the prior $\beta \mid \sigma^2 = 1 \sim N_2(0_2, n(X'X)^{-1})$ is $-r/(1-r^2)$ \citep{ghosh2015}. Therefore, if $X_1$ and $X_2$ are positively correlated, the prior covariance between $\beta_1$ and $\beta_2$ induced by the prior is negative (and conversely for negative correlations).

We set $n = 10$, $\beta = (5,5)'$ and consider two cases: $r = 0.9$ and $r = -0.9$. In order to isolate the effect of changing the sign of $r$ as much as possible, we use the same random $\epsilon$ in both cases and the same $N_1(0,1)$ random numbers for generating the design matrices before transforming them (deterministically, via principal component scores times the Choleski matrix square-root of the target sample covariance) to correlated predictors with the desired $r$.

Figure~\ref{fig:contours} shows contours of $N_2(0_2, n(X'X)^{-1})$ (solid blue) and $N_2(0, \widehat{W})$ (solid green; setting $\sigma^2 = 1$), the type II ML prior. It also shows the contours of $N_2(\widehat{\beta}, n(X'X)^{-1})$ (dashed red), the ``BIC prior''; note that the likelihood function (a function of $\beta$) is proportional to $N_p(\widehat{\beta}, (X'X)^{-1})$, so it has the same shape. When $r = -0.9$, the marginal likelihood of the true model is high with all the priors. If $r = 0.9$, the highest density regions of the likelihood of the true model are assigned relatively low probability density under $N_2(0_2, n(X'X)^{-1})$.

Table~\ref{tab:twopredictors} confirms this geometric intuition -- for sample sizes ranging from 5 to 15 and after 1000 simulations, the average posterior probability that the lower bound LB ($g$-prior with $g=n$) assigns to the true model is lower than with BIC  or the type II ML prior (ML). The Zellner-Siow (ZS) prior is less sensitive to the sign of $r$ than the lower bound, despite the fact that they are both centered at $0_p$ and have the same prior scale.

This intuition can be supported mathematically. If $\sigma^2$ is known,
\begin{align} \label{eq:knowns2}
\log \left( \frac{\mathsf{BF}_{i0,\mathsf{BIC}}}{\mathsf{BF}_{i0,\mathsf{LB}}} \right) = \frac{p_i}{2} \log \left(\frac{n+1}{n} \right) + \frac{1}{2 \sigma^2} \left(1 - \frac{n}{n+1} \right) \mathsf{SSR}_{i} \ge 0,
\end{align}
which depends on the data only through $\mathsf{SSR}_i$. If the full model is true, $
\mathbb{E}[\mathsf{SSR}_i] = 2 \sigma^2  + n[\beta_1^2 + 2 \beta_1 \beta_2 r + \beta_2^2]$. Since in our example $\beta_1$ and $\beta_2$ are positive, $\mathbb{E}[\log (\mathsf{BF}_{i0,\mathsf{BIC}}/\mathsf{BF}_{i0,\mathsf{LB}})]$ increases as $r$ increases.
\end{example}

\begin{figure}[h!]
  \centering
    \includegraphics[width=\textwidth]{contours.pdf}
    \caption{Highest probability density regions (20\%, 50\%, 95\%) of the lower bound (g-prior) $N_p(0_p, n(X'X)^{-1})$ (solid blue), ``BIC prior'' $N_p(\widehat{\beta}, n(X'X)^{-1})$ (dashed red), and the type II ML prior $N_p(0_p, \widehat{W})$ (solid green). The MLE is indicated with a $\widehat{\beta}$ symbol. }
    \label{fig:contours}
\end{figure}

\begin{table}[h!]
\centering
\caption{Average posterior probability assigned to the true model (full model), $B = 1000$ simulations.}
\begin{tabular}{rrrrr|rrrr}
 &   \multicolumn{4}{c}{$r= -0.9$} & \multicolumn{4}{c}{$r= 0.9$} \\
  \hline
$n$ & BIC & ML & LB  & ZS & BIC & ML & LB  & ZS \\
  \hline
    5 & 0.954 & 0.797 & 0.665 & 0.503 & 0.978 & 0.911 & 0.361 & 0.310  \\
  10 &  0.997 & 0.983 & 0.976 & 0.952 & 0.999 & 0.995 & 0.605 & 0.964  \\
  15 &  1.000 & 0.999 & 0.999 & 0.997 & 1.000 & 1.000 & 0.874 & 0.998 \\
  20 & 1.000 & 1.000 & 1.000 & 1.000 &  1.000 & 1.000 & 0.979 & 1.000 \\
   \hline
   \hline
\end{tabular}
\label{tab:twopredictors}
\end{table}

The intuition, from the figure and table, that ML is between BIC and the lower bound (LB; i.e. a $g$-prior with $g=n$), is shown formally below.
\begin{proposition} \label{prop:compromise}
If $\mathcal{M}_j \supset \mathcal{M}_i$ (that is, if $\mathcal{M}_j$ contains all the predictors in $\mathcal{M}_i$), then
$$
\mathsf{BF}_{ji, \mathsf{BIC}} \ge \mathsf{BF}_{ji, \mathsf{ML}}\ge \mathsf{BF}_{ji, \mathsf{LB}},
$$
where  $\mathsf{BF}_{ji}= \mathsf{BF}_{j0}/\mathsf{BF}_{i0}$.  Let $\mathcal{M}_f$ be the full model (which includes all $p$ predictors) and $\mathcal{M}_0$ be the null model. If the prior on the model space is the same in all cases, the inequality above implies
\begin{align*}
\mathbb{P}_{\mathsf{BIC}}( \mathcal{M}_f \mid Y) \ge \mathbb{P}_{\mathsf{ML}}( \mathcal{M}_f \mid Y) \ge \mathbb{P}_{\mathsf{LB}}( \mathcal{M}_f \mid Y) \\
\mathbb{P}_{\mathsf{BIC}}( \mathcal{M}_0 \mid Y) \le \mathbb{P}_{\mathsf{ML}}( \mathcal{M}_0 \mid Y) \le \mathbb{P}_{\mathsf{LB}}( \mathcal{M}_0 \mid Y).
\end{align*}
\end{proposition}

If the true model is the full model, BIC assigns more probability to the truth than the type II ML prior and the lower bound; on the other hand, if the true model is the null model, the lower bound ($g$-prior) assigns more probability to the truth than the type II ML prior and BIC. However, there is yet another interesting asymmetry. When the true model is the null model, the differences between the lower bound and BIC tend to be small, whereas if the true model is the full model the differences can be rather large. We can provide some mathematical support to this claim. First, assume that $\sigma^2$ is known, so that the expression for $\log (\mathsf{BF}_{i0,\mathsf{BIC}}/\mathsf{BF}_{i0,\mathsf{LB}})$ is given in Equation~\ref{eq:knowns2}. If $\mu_\ast = X_0 \beta_{0\ast} + X_\ast \beta_\ast$ is the true value of the linear predictor, we can write $\mathbb{E}[\mathsf{SSR}_i] = p_i \sigma^2 + \lVert \mathsf{P}_{{X}_i} \mu^\ast \rVert^2$. For fixed $X_i$, $\mathbb{E}[\log (\mathsf{BF}_{i0,\mathsf{BIC}}/\mathsf{BF}_{i0,\mathsf{LB}})]$ is minimized when $\beta_{\ast} = 0_{p_\ast}$ (i.e. when the null model is true). Also note that $\mathbb{E}[\mathsf{SSR}_i]$ is increasing in $p_i$, which implies that the expected (log) differences between the lower bound and BIC grow as the number of predictors grows. For unknown $\sigma^2$, $\log (\mathsf{BF}_{i0,\mathsf{BIC}}/\mathsf{BF}_{i0,\mathsf{LB}})$
is increasing in $R^2_i$, which is also consistent with our intuition.

At the beginning of this section, we mentioned that a type II ML prior that has been previously studied is the $g$-prior $N_p(\beta \mid 0_p, \widehat{g} \,  \sigma^2 (X'X)^{-1})$, where $\widehat{g}$ is set locally by maximizing the marginal likelihood subject to $g \ge 0$ \citep{george2000calibration, hansen2003minimum, liang2008mixtures}. This prior has undesirable features that are a byproduct of not maximizing the marginal likelihood subject to a lower bound on $g$ that is bounded away from 0. One of them is that the resulting null-based Bayes factors are always greater or equal to 1 (which leads to inconsistency if the null model is true), and another one is that the Bayes factor between any two models can be equal to 1 with positive probability in cases where $n > p + p_0$ (especially when $n \approx p + p_0$), which cannot occur (with positive probability) with proper priors or our restricted type II prior.

We close this subsection by studying whether the type II ML prior satisfies the desiderata in \cite{bayarri2012criteria} for objective priors in model selection.
\begin{enumerate}
\item \textbf{Basic criterion:} The basic criterion is satisfied if the prior is proper, which the type II ML prior satisfies directly because of the restriction.
\item \textbf{Model selection consistency:} Let the true model be $\mathcal{M}_\ast: N_n( Y \mid X_0 \beta_0 + X_\ast \beta_\ast, \sigma^2 I_n)$. Then, model selection consistency is satisfied if $\mathrm{plim}_{n \rightarrow \infty} \mathbb{P}(\mathcal{M}_\ast \mid Y) = 1$. The type II ML prior is model-selection consistent under the following regularity condition, which is commonly made in the literature \citep{fernandez2001benchmark,liang2008mixtures,guo2009bayes,maruyama2011fully,bayarri2012criteria,som2014block}. For any model $\mathcal{M}_j$ that doesn't nest the true model, assume that
$$
\lim_{n \rightarrow \infty} \frac{\beta_\ast' X_\ast' (I_n-\mathsf{P}_{X_j})X_\ast \beta_\ast}{n} = b_j \in (0, \infty).
$$
The assumption can be interpreted as that the models have design matrices that can be differentiated in the limit \citep{bayarri2012criteria}.
\item \textbf{Information consistency:} Suppose that, for a fixed $n$, $\lVert \widehat{\beta}_i \rVert \rightarrow \infty$, which implies $R^2_i \rightarrow 1$. This is a situation where there is overwhelming evidence in favor of $\mathcal{M}_i$ \citep{liang2008mixtures}. Information consistency holds if $\mathsf{BF}_{i0} \rightarrow \infty$, which is satisfied by the type II ML prior.
\item \textbf{Intrinsic consistency:} A prior satisfies intrinsic consistency if, as $n$ grows, it converges to a proper prior which does not depend on model-specific parameters or $n$. In general, this criterion isn't satisfied for the type II ML prior. To see this, assume that $(X_i'X_i)/n \rightarrow \Xi_i$ for a positive definite matrix $\Xi_i$, which holds if there is a fixed design or the covariates are drawn independently from a distribution with finite second moments \citep{bayarri2012criteria}. Then, the prior covariance $\widehat{W}_\ast$ for the true model has the limiting behavior
$$
\mathrm{plim}_{n \rightarrow \infty} \, \widehat{W}_\ast = \begin{cases}  \Xi_\ast^{-1} & \text{if } \, \beta_\ast'\Xi_\ast \beta_\ast \le \sigma^2_\ast \\ \left( \frac{1}{\sigma^2_\ast} - \frac{1}{\beta_\ast' \Xi_\ast \beta_\ast} \right) \beta_\ast \beta_\ast' + \Xi_\ast^{-1} & \text{if } \, \beta_\ast'\Xi_\ast \beta_\ast > \sigma^2_\ast \end{cases},
$$
which depends on $\beta_\ast$ and $\sigma^2_\ast$.
\item \textbf{Null and dimensional predictive matching:} In both cases, the notion of minimal training sample size is central to the definition. For any model $\mathcal{M}_i$, the minimal training sample size is the smallest sample size $n^\ast_i$ such that the marginal likelihood of the model is finite. Null predictive matching is achieved if, for any model $\mathcal{M}_i$, we have $\mathsf{BF}_{i0} = 1$ when the sample size is equal to the minimal training sample size $n^\ast_i$. Dimensional predictive matching is achieved if, for any pair models of the same dimension $\mathcal{M}_i$ and $\mathcal{M}_j$, we have $\mathsf{BF}_{ij} = 1$ whenever $n^\ast_i = n^\ast_j$. The type II ML prior isn't null or dimensional predictive matching. For $p > 1$, the minimal training sample size for the type II ML prior is $n = p +p_0+1$. [If $p=1$, the marginal likelihood doesn't depend on the choice of $W$.] When $n = p+p_0$, the marginal likelihood is finite for any given $W$, but one can choose $W \succeq n (X'X)^{-1}$ so that the marginal goes to $\infty$ (this is shown in the supplementary material). Null predictive matching isn't satisfied: in fact, $\mathsf{BF}_{i0}$ goes to $\infty$ as $R^2_i \rightarrow 1$ when $n = p + p_0 +1$. Similarly, it is easy to see that dimensional predictive matching isn't satisfied, either; different models will have different $R^2_i$, yielding Bayes factors that are different than 1.
\item \textbf{Invariance:} The type II ML prior is invariant with respect to linear transformations of the design matrix (in particular, this is true for changes of measurement units). More explicitly, let $A$ be an invertible $p \times p$ matrix and $\tilde{X} = X A$. Let $\beta$ and $\tilde{\beta}$ be the regression coefficients of the linear model if the design matrices are $X$ and $\tilde{X}$, respectively. If the type II ML prior is put on $\beta$ and $\tilde{\beta}$, then $\beta$ and $A \tilde{\beta}$ are equal in distribution.
\end{enumerate}

 Table~\ref{tab:properties} compares the properties of the type II ML prior with those of BIC, the lower bound LB ($g$-prior with $g=n$), the Zellner-Siow prior (ZS), and the type II ML $g$-prior where $g$ is set locally by maximizing the marginal likelihood subject to $g \ge 0$, which we denote $\widehat{g}$. Our type II ML prior is model-selection consistent, whereas the $\widehat{g}$-prior isn't under the null model; however, the $\widehat{g}$-prior is predictive matching, while our type II ML prior isn't. According to the definition above, it doesn't make sense to assert that BIC is invariant to linear transformation (since it isn't a prior), but it depends on the data only through $R^2$, which is invariant to invertible linear transformations.

 \begin{table}
\centering
\caption{Comparison of model selection desiderata for different approaches.}
    \begin{tabular}{l|lllll}
    ~                       & ML & BIC            &  LB & ZS  &  $\widehat{g}$                 \\
\hline
    Proper                  & yes                         & -            & yes   & yes & yes                  \\
    Model selection consistency   & yes                         & yes            & yes   & yes & no                   \\
    Information consistency & yes                         & yes            & no    & yes & yes                  \\
    Intrinsic consistency   & no         & -             & yes   & yes & no  \\
    Predictive matching     & no                          & no             & yes   & yes & yes                   \\
   	Invariance  			& yes                         & -            & yes   & yes & yes                  \\
    Closed form Bayes factors  & yes    & yes               &yes & no  &yes \\
    \hline
    \end{tabular}
    \label{tab:properties}
\end{table}

It is not a surprise that data-dependent priors lack some of the desirable properties of real priors. One sacrifices some Bayesian
 features when leaving the pure Bayesian domain.

\subsection{Estimation and prediction} \label{sec:estimation}

\subsubsection{The type II ML posterior mean}

For simplicity, we omit model subscripts and assume that the model is $Y \sim N_n(X_0 \beta_0 + X \beta, \sigma^2 I_n)$, $X_0'X = 0_{p_0 \times p}$. If we put the right-Haar prior $\pi(\beta_0, \sigma^2) \propto 1/\sigma^2$ on the common parameters and the type II ML prior on $\beta \mid \sigma^2$, the posterior mean of $\beta$ is
$$
\tilde{\beta} = \mathbb{E}(\beta \mid Y) = \begin{cases}    \frac{n}{n+1} \widehat{\beta} & \text{if } R^2 \le \frac{n+1}{2n-p_0} \\ \left(1-\frac{1-R^2}{(n-p_0-1)R^2}\right) \widehat{\beta} & \text{if } R^2 > \frac{n+1}{2n-p_0}. \end{cases}
$$
The expression can be derived by applying the Sherman-Morrison formula twice to $\mathbb{E}(\beta \mid Y) = [\widehat{W}^{-1}+X'X]^{-1} X' Y$. The properties of an analogous estimator in the normal means problem (for known $\sigma^2$) are studied in \cite{dasgupta1988frequentist}, where it is shown that it is minimax with respect to squared error loss. Proposition~\ref{prop:minimax} shows that $\mathbb{E}(\beta \mid Y)$ is also minimax with respect to a (scaled) predictive loss because it belongs to the class of minimax estimators characterized in \cite{strawderman1973proper} (a proof can be found in the supplementary material).

\begin{proposition} \label{prop:minimax}
Let $p \ge 3$ and $n > p + p_0$. The estimator $\tilde{\beta} = \mathbb{E}(\beta \mid Y)$ is minimax with respect to the (scaled) squared predictive loss
$$
L(\beta, \delta) = (\beta-\delta)' (X'X) (\beta - \delta)/\sigma^2 \,.
$$
\end{proposition}

 The mean squared error of the posterior mean of the lower bound prior (Zellner's $g$-prior, where $g = n$) is increasing in $\lVert \beta \rVert$. On the other hand, the mean squared error of $\hat{\beta}$ is constant in $\lVert \beta \rVert$. The estimator $\tilde{\beta}$ is equal to the posterior mean of the lower bound when $R^2$ is small, and close to $\widehat{\beta}$ when $R^2$ is large. Therefore, $\tilde{\beta}$ avoids ``selecting'' the lower bound in cases where it has high mean squared error (that is, whenever $\lVert \beta \rVert$ and $R^2$ are large).

 \subsubsection{A simulation with correlated predictors} \label{sec:simstudies}

To gain further insight into the differences between the type II ML prior, the lower bound (LB) prior ($g$-prior with $g = n$), the Zellner Siow (ZS) prior and BIC,
we simulate data from $Y = 1_n \, \alpha + X \beta + \epsilon$, $\epsilon \sim N_n(0_n, \sigma^2 I_n)$, where $n = 50$, $\alpha = 2$, $\sigma^2 =1$, and $\beta$ is $8$-dimensional with $k$ nonzero elements, for $k \in \{0,1,2, \, ... \, , 8\}$. We consider 2 different types of correlation between the predictors: the orthogonal case $X'X = I_p$ and an AR(1) structure
$$
\frac{1}{n-1}(X'X) = \left( \begin{array}{ccccc}
1 & \rho & \rho^2 & ... & \rho^p \\
\rho & 1 & \rho & ... & \rho^{p-1} \\
\rho^2 & \rho & 1 &  ... & \rho^{p-2} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\rho^p & \rho^{p-1} & \rho^{p-2} & ... & 1 \end{array} \right)
$$
for $\rho = 0.9$. For all $k$, we generate $\beta_k \sim N_k(0_k, g I_k)$. The location of the $k$ zeros in the $\beta$ vector is drawn at random (according to the uniform distribution). We use $g \in \{5,25\}$ as in \cite{cui2008empirical} and \cite{liang2008mixtures}, representing weak and strong signal-to-noise ratios, and evaluate performance with respect to the predictive squared loss function
$
L(\beta, \delta) = \lVert X \beta - X \delta \rVert^2,
$
where $\delta$ is an estimator of $\beta$. [This is the loss function that was used in the simulation studies in \cite{cui2008empirical} and \cite{liang2008mixtures}]. The estimators that are considered for the various priors are the posterior means (and $\widehat{\beta}$ in the case of BIC) of the highest probability model (HPM) and the median probability model (MPM), and the estimate arising from Bayesian model averaging (BMA).
We do $1000$ simulations for all scenarios, and the results are reported in Figures~\ref{fig:ortho} and~\ref{fig:ar1}.

In the orthogonal case, BIC, the type II ML prior, LB ($g$-prior with $g=n$) and ZS behave similarly when $g = 5$. When $g = 25$, we can observe more differences: in particular, LB is progressively worse than the rest as the number of true predictors increases, ZS is slightly better than BIC and the type II ML prior when not all predictors are active, and the difference between ZS and BIC and the type II ML prior narrows as the number of true predictors increases.

 The results with the AR(1) correlation structure show bigger discrepancies. As the number of true predictors increases, the loss of the LB is substantially higher than the loss with any other prior, especially when $g=25$. When $g = 5$, both LB and ZS are outperformed by BIC and the type II ML prior. When $g = 25$, ZS has similar losses as BIC and the type II ML prior when the number of true predictors is between 0 and 6, but is outperformed when the true number of predictors is 7 or 8 (in which case, the true model is the full model).

 In the cases where the LB is clearly outperformed, its posterior distribution over the model space is closer to the uniform distribution than the other posteriors, as evidenced in the first panel in Figure~\ref{fig:simstplots}, which shows the average entropy of the posterior distributions over the model space. Additionally, ZS induces a noticeabily less entropic (more concentrated) posterior distribution over the model space, especially when few predictors are active.  ZS and the LB select HPMs and MPMs with fewer predictors than BIC and the type II ML prior (see second and third panel in Figure~\ref{fig:simstplots}, which show the percentage of times the MPM equals the true model and the average size of the MPM, respectively). When the true model is the full model, an interesting phenomenon occurs: ZS is the prior where the MPM is equal to the true model less often, but the average predictive loss of the prior stays competitive with BIC and ML. Upon further inspection in our simulations, this is due to the fact that when some of the true coefficients are non-zero but rather small, ZS does not include their predictors in its MPM, but that does not worsen the predictive loss by much. The HPM and MPM with BIC and the type II ML prior tend to be the same model, and they coincide with the models selected with the LB in the cases where the signal is low, as expected. On the other hand, when the signal is high, the LB assigns more probability to wrong models than the other approaches, and sometimes the HPM and MPM end up being an egregiously bad model, resulting in a substantially higher average loss. Note that ZS, which also has $n(X'X)^{-1}$ as its prior scale but has thicker tails, does not seem to be nearly as affected by this issue as the LB, especially when the signal is high enough (i.e. $g=25$).

\section{High-dimensional ANOVA} \label{sec:ANOVA}

In this section, we revisit the one-way ANOVA problem that was studied in \cite{berger2003approximations}. In this example, the number of predictors $p$ grows to infinity.

Suppose we have observations
$$
y_{ij} = \mu_i + \epsilon_{ij}, \, \epsilon_{ij} \stackrel{\mathrm{iid}}{\sim} N_1(0,1)
$$
where $i \in \{1,2, \, ... \, , p\}$ (groups) and $j \in \{1,2, \, ... \, , r\}$ (replicates). We assume that $r$ is fixed and $p$ grows to infinity.

We only consider the null model and the full model
\begin{align*}
\mathcal{M}_1 : \mu = 0_p, \, \, \mathcal{M}_2 : \mu \neq 0_p.
\end{align*}
If the true model is $\mathcal{M}_2$, we assume  that $\mathrm{lim}_{p \rightarrow \infty} \lVert{\mu}\rVert^2/p = \tau^2 > 0.$

Let $\ell$ be the log-likelihood function of the full model and $\widehat{\mu}$ the maximum likelihood estimate of $\mu$. If BIC is defined as $-2 \ell(\widehat{\mu}) +p \log n$,
it is inconsistent under $\mathcal{M}_2$ \citep{stone1979comments}. \cite{berger2003approximations} show that, if the prior on $\mu$ under $\mathcal{M}_2$ is $\mu \mid g \sim N_p(0, g I_p)$ with a mixing density over $g$ (which doesn't depend on $n$) with support $(0, +\infty)$, consistency holds. Alternatively, if $g$ has restricted support $(0, T)$ for $T < \infty$, there is a region of inconsistency under $\mathcal{M}_2$.

In this problem, the prior scale has to be chosen carefully. A naive parallel of the type II ML prior in Section~\ref{sec:linmod} would have as lower bound for the prior covariance $n (X'X)^{-1} = \frac{n}{r} I_p = p I_p.$
However, it is straightforward to show that any normal prior whose scale goes to infinity is inconsistent under $\mathcal{M}_2$. Since the effective sample size of $\mu$ in this problem is $r$ instead of $n$ \citep{berger2014effective}, we study the properties of a prior whose covariance is $r (X'X)^{-1} = I_p.$

In the same vein, BIC can be defined appropriately by taking $\log r$ as the penalty instead of $\log n$. The asymptotic behavior of both approaches can be summarized as follows:
\begin{itemize}
	\item Normal prior with $I_p$ as prior covariance: Under $\mathcal{M}_1$, consistency for all $r$ and $\tau^2$. Under $\mathcal{M}_2$, inconsistency if $\tau^2 \le (1+r) \log (1+r)/r^2 -1/r$ and consistency otherwise. For example, if $\tau^2 = 0.25$, consistency holds under $\mathcal{M}_2$ for $r \ge 5$, and consistency holds for all $r$ if $\tau^2 > 2 \log 2 -1$.
	\item BIC with $\log r$ as penalty: Under $\mathcal{M}_1$, inconsistency if $r \in \{1,2\}$ and consistency otherwise. Under $\mathcal{M}_2$, inconsistency if $\tau^2 \le (\log r -1)/r$ and consistency otherwise. The condition is most stringent at $r = e^2$, so consistency holds for all $r$ if $\tau^2 > 1/e^2$.
\end{itemize}
Under $\mathcal{M}_2$, the region of inconsistency of BIC is contained in the region of inconsistency of the normal prior; however, BIC can be inconsistent under $\mathcal{M}_1$.

The type II ML prior
\begin{align*}
\mu &\sim N_p( 0_p, \widehat{W}) \\
\widehat{W} &= \mathrm{arg \, max}_{W \succeq I_p} m(Y) = I_p + \max\{ 0, 1-(r+1)/(r\lVert \widehat{\mu} \rVert^2) \} \widehat{\mu} \widehat{\mu}'
\end{align*}
yields the Bayes factor
$$
\mathsf{BF}_{10} = \begin{cases}
(r+1)^{-p/2} \exp \left\{ \frac{r^2 \lVert \widehat{\mu} \rVert^2 }{2(r+1)} \right\}  & \text{if } \lVert \widehat{\mu} \rVert^2 \le  1 + 1/r  \\
\left( \frac{r \lVert \widehat{\mu} \rVert^2}{r+1} \right)^{-1/2} (r+1)^{-p/2} \exp \left\{ \frac{(r \lVert \widehat{\mu} \rVert^2 - 1)}{2} \right\}  & \text{if } \lVert \widehat{\mu} \rVert^2 > 1 + 1/r
\end{cases} \,.
$$
The type II ML Bayes factor is consistent under $\mathcal{M}_1$ and is consistent for all $r \ge 1$ and consistent under $\mathcal{M}_2$ for $\tau^2 > [\log(r+1)-1]/r$. Again, the type II ML prior acts as a compromise between the normal prior and BIC but, unfortunately, it still has a region of inconsistency which the mixtures of normal priors avoid. The ML Bayes factor is available in closed form, however, whereas the mixture of
normals Bayes factors generally are not.

\section{Incorporating prior information} \label{sec:shibata}

The constraints we have placed on the type II ML prior have been basic constraints, preventing the prior from becoming too concentrated.
It is also possible to use constraints that incorporate available prior information, which can lead to improved inferences. We illustrate this possibility by revisiting
the example in \cite{shibata1983asymptotic}, which was also studied in \cite{barbieri2004optimal}.

The goal in the Shibata example is to estimate the function $f(x) = - \log(1-x)$, $-1 \le x \le 1$ from independent observations $y_i = f(x_i) + \varepsilon_i$, where the $\varepsilon_i$ are independent  $\varepsilon_i \sim N_1(0, \sigma^2 I_n)$ and $\sigma^2$ is known. The function $f$ can be expressed in an orthogonal series expansion as $
f(x) = \sum_{i = 1}^\infty \beta_i \phi_i(x),
$
where $\phi_i(x)$ are the Chebyshev polynomials of the first kind. We approximate $f$ with a finite series expansion, modeling $y_i = \sum_{i = 1}^j \beta_i \phi_i(x) + \varepsilon_i$. We consider different truncation points $j$, ranging from $1$ to $k$, so our model space consists of a sequence of nested models
$$
\mathcal{M}_j : Y \mid \alpha, \beta_j, \sigma^2 \sim N_n(1_n \alpha + X_j \beta_j, \sigma^2 I_n)
$$
for $j \in \{1,2,3, \, ... \,  , k\}$, where the design matrices $X_j$ have dimension $n \times j$ and the columns are given by the Chebyshev polynomials of the first kind evaluated at the knots $x_i = \cos ( \pi (n-i+1/2)/n)$, for $i \in \{1,2, \, ... , n\}$. The true coefficients in an infinite orthogonal expansion are $\alpha = \log 2$ and $\beta_j = 2/j$. The design matrices are orthogonal with $X_j ' X_j = (n/2) I_{j}$ and $1_n' X_j = 0_j'$. [See \cite{barbieri2004optimal} for a more detailed explanation.]

We consider $n = 30, k=29, \sigma^2 = 1$, $n = 100, k=79, \sigma^2 = 1$, and $n = 2000, k = 79, \sigma^2 = 3$ and put a uniform prior (i.e., 1/29 or 1/79) on the size of the nested models. We utilize two local type II ML priors based on $\beta \sim N(0, \sigma^2 A)$:
\begin{itemize}
\item The unit information constraint $A \succeq  n (X'X)^{-1}$.
\item In polynomial regression, the true coefficients often decrease at polynomial rate. With that in mind, we define a type II ML prior whose covariance matrix is diagonal, with diagonal elements decreasing according to some power law. That is, $A = diag(d_1, d_2, \, ... \, , d_k)$ with $d_i = c i^{-a}$ for $i \in \{1, 2, \, ... \, , k\}$. The parameters $c, a \ge 0$ are found by maximizing the marginal likelihood.
\end{itemize}
We will compare these three methods on Shibata's example, utilizing squared predictive loss $L(f, \widehat{f}) = \int_{-1}^{1} (f(x) - \widehat{f}(x))^2 \, \mathrm{d} x$, as in \cite{barbieri2004optimal}.
We also consider AIC ($-2\ell(\widehat{\beta}_j) + j$) and BIC ($-2\ell(\widehat{\beta}_j) +j \log n$), treating $\exp(-\mathrm{AIC}/2)$ and $\exp(-\mathrm{BIC}/2)$ as approximate marginal likelihoods. We compare the predictive loss of Bayesian model averaging (BMA), the median probability model (MPM; \cite{barbieri2004optimal}), and the highest probability model (HPM). Note that the AIC and BIC columns for the HPM
correspond to use of the actual AIC and BIC criteria, since maximizing the posterior probability is
equivalent to minimizing the criterion. The MPM and BMA columns utilize AIC and BIC by converting them to approximate
marginal likelihoods and utilizing the relevant Bayesian theory.

The results are summarized in Table~\ref{tab:shibata}. BIC and $\widehat{W}$ behave similarly in all cases, as we have seen in previous sections. The informative type II ML priors outperform the others. AIC is somewhat better than BIC, and their Bayesian implementations (MPM and BMA)
outperform use of the raw criteria (HPM).

 All across the board, BMA outperforms the rest (as expected), followed by the MPM and the HPM; the MPM
 is the best single predictive model in nested model scenarios, as shown in \cite{barbieri2004optimal}.

\begin{table}[h!]
\centering
\caption{Predictive loss, based on $N = 1000$ simulations. Average model sizes in square brackets.}
\label{tab:shibata}
\begin{tabular}{l|lllll}
    \textbf{HPM}  & $c i^{-a}$ & $\widehat{W}$   & AIC  & BIC  \\
\hline
$n = 30, k = 29, \sigma^2 = 1$  & \textbf{0.904} [10]  & 1.141 [4] & 1.076 [7] & 1.131 [4] \\
$n = 100, k = 79, \sigma^2 = 1$  & \textbf{0.471} [23] & 0.693 [7]  & 0.582 [13] & 0.692 [7]  \\
$n = 2000, k =79, \sigma^2 = 3$ & \textbf{0.136} [57] & 0.295 [13]  & 0.188 [36] & 0.295 [13]  \\
\textbf{MPM}    &   &   &     &   &   \\
\hline
$n = 30, k = 29, \sigma^2 = 1$  &  \textbf{0.839} [16] & 1.093 [4] & 1.027 [7] & 1.089 [4] \\
$n = 100, k = 79, \sigma^2 = 1$   & \textbf{0.441} [44] & 0.680 [7]  & 0.566 [13] & 0.679 [7]  \\
$n = 2000, k =79, \sigma^2 = 3$   & \textbf{0.134} [59] & 0.289 [13]  & 0.185 [37] & 0.289 [13]  \\
\textbf{BMA}   &  &  &    &  & \\
\hline
$n = 30, k = 29, \sigma^2 = 1$   & \textbf{0.837} & 0.990  & 0.921 & 0.983  \\
$n = 100, k = 79, \sigma^2 = 1$  & \textbf{0.437} & 0.623  & 0.521 & 0.621 \\
$n = 2000, k = 79, \sigma^2 = 3$   & \textbf{0.133}  & 0.275   & 0.170 & 0.275   \\
\end{tabular}
\end{table}

\section{Conclusions} \label{sec:conclusions}

Conceptually, the type II ML priors that were studied offer an attractive compromise between conventional priors that might seem overly concentrated at the null model and priors
that are centered at the model likelihood function (such as the prior leading to BIC). The importance of constraining
the prior so that it does not overly concentrate was highlighted, and the need to carefully choose the constraint
in high-dimensional situations was discussed.

The surprise of the analysis was that the recommended type II ML prior gives remarkably similar answers to BIC. Indeed,
the paper could be viewed as primarily providing a new justification of BIC in normal linear
models, suggesting that BIC need not just be viewed as an approximation but as something that corresponds quite
closely to Bayesian analysis with sensible priors (and not just with priors that sit on top of the model likelihoods).

Finally, we revisited the nonparametric regression example in \cite{shibata1983asymptotic}, showing how
prior information could be incorporated into the constraints defining type II ML priors, leading
to considerably improved performance (when the prior information is correct). This is perhaps the most
promising practical venue for type II ML priors: embed available structural information about the prior
into the class of priors, and then use type II ML.

\begin{figure}[h]
    \centering
    \includegraphics[width=\textwidth]{lossortho.pdf}
    \caption{Simulation study, under an orthogonal design, comparing the performance in estimation of the lower bound prior LB ($g$-prior with $g=n$), the type II ML prior (ML), the Zellner-Siow prior (ZS) and BIC. The considered estimates are the posterior means ($\widehat{\beta}$ for BIC)  from the highest probability model (HPM) and the median probability model (MPM), and the Bayesian model averaging (BMA) estimate. }
    \label{fig:ortho}
\end{figure}

\begin{figure}[h]
    \centering
    \includegraphics[width=\textwidth]{lossAR1.pdf}
    \caption{Simulation study, under an AR(1) correlation structure design, comparing the performance in estimation of the lower bound prior LB  ($g$-prior with $g=n$), the type II ML prior (ML), the Zellner-Siow prior (ZS) and BIC. The considered estimates are the posterior means ($\widehat{\beta}$ for BIC)  from the highest probability model (HPM) and the median probability model (MPM), and the Bayesian model averaging (BMA) estimate.
     }
    \label{fig:ar1}
\end{figure}

\begin{figure}[h]
    \centering
    \includegraphics[width=0.7\textwidth]{extraAR1.pdf}
    \caption{First panel: Average entropy of the posterior distribution over the model space; second panel: \% of times the median probability model equals the true model; third panel: average size of median probability model. AR(1) correlation structure in design matrix. }
    \label{fig:simstplots}
\end{figure}

\bibliographystyle{chicago}

redpajama_set_name: RedPajamaArXiv


text:
Home Model Trains	train model trees | Model Train Express
train model trees | Model Train Express
Some had wheels that turned, but these needed to be pushed or pulled. A few of the early 19th-century push toy trains were made of tinplate, such as the big, durable, stylized locomotive toys in the U.S., which were painted red and gold and decorated with hearts and flowers. train model trees
Around 1875, technological advancements in materials and manufacturing allowed tin to be scraped, cut, rolled, and lithographed faster than previously.
Toy trains were altered when Märklina German firm which specialized in doll house accessories, sought to make an equivalent toy for boys in which a constant revenue stream could be ensured by selling add-on accessories for decades after the initial purchase. In addition to boxed sets containing a train and track, Märklin offered additional track, rolling stock, and buildings sold separately, creating the predecessor to the modern model train layout featuring scenery and buildings along with a working train.
Electric trains adopted, with the first appearing in 1897, produced by the U.S. company Carlisle & Finch. As home use of electricity became more prevalent in the early 20th century, electrical trains gained popularity and as time went on, these electric trains grew in sophistication, gaining light, the ability to change direction, to emit a whistling noise, to smoke, to remotely couple and uncouple cars and even load and unload cargo. Toy trains by the first half of the 20th century were frequently made of lithographed tin; later trains were often made mostly of plastic.
Pull toys and wind-up trains were marketed towards children, while electric trains were marketed towards teens, especially teenaged boys.
Now, S gauge and O gauge railroads continue to be considered toy trains even by their adherents and are often accessorized with semi-scale model buildings by Plasticville or even K-Line (who owns the rights to the Plasticville-like buildings created by Marx in the 1950s to the 1970s). However, due to their high cost, one is more inclined to locate an HO scale or N scale train set in a toy store than a O scale collection.
Many modern electric toy trains comprise sophisticated electronic equipment that emit digitized sound effects and allow the operator to safely and easily run multiple remote control trains on a single loop of course. In the last few decades, many toy train operators may operate a train with a TV camera in the front part of the motor and hooked up to a screen, such as computer monitor. This will show a picture, like that of a real (smaller size) railroad)
Thanks for your interest in train model trees