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About Us > Staff
Steve Schneider
Senior Program Director, Science, Technology, Engineering, & Mathematics
650.381.6410650.381.6401	Contact by email	-	400 Seaport Court, Suite 222,	Redwood City, CA	94063-2767
Steve Schneider is the Senior Program Director of the Science, Technology, Engineering, & Mathematics program at WestEd.
He also serves as the Principal Investigator (PI) of the National Science Foundation's (NSF's) Center for Assessment and Evaluation of Student Learning (CAESL). As a result of WestEd's work, vast numbers of teachers and graduate students have received training and/or advanced degrees in the use of formative classroom assessment and measurement.
In addition, Schneider directs the National Center on Cognition and Mathematics Instruction, and serves as the Principal Investigator and Content Expert for the Science Review Team for the U.S. Department of Education's What Works Clearinghouse.
He previously served as the evaluation chairperson for the National Network of Eisenhower Mathematics and Science Consortia and Clearinghouse, and represented the Network on the Department of Education's Mathematics and Science Expert Panels.
Schneider has published numerous articles on science, mathematics, and technology education, professional development, and teacher preparation. He was invited to testify before Congress on the role of the federal government in STEM education and serves as Content Expert for the Science Review Team for the U.S. Department of Education's What Works Clearinghouse.
In 2006, Schneider and his fellow NAEP Framework Team members received WestEd's Paul D. Hood Award for Distinguished Contribution to the Profession. In 2004, Schneider received this award for his creation of collaborative partnerships that have advanced math and science education in the nation including curricula, professional development, assessment, and evaluation.
Schneider received a BA in biology from the University of California, Berkeley, and a PhD in the design and evaluation of educational programs with an emphasis in science, mathematics, and technology education from Stanford University. He has a State of California Life Teaching Credential from California State University, San Jose.
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The single Green Fan 24V ducted kit system is designed to extract & reduce musty smells, to prevent growth of fungi, mould, timber decay and accumulation of dampness. The air is extracted from the subfloor area and ducted to the grill in the outside wall. Green fans run off an adjustable transformer depending on speed and noise level preferences.
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Would you recommend It's the Bit to a friend?
Dan Dascalescu has never used this product.
Hey guys. We are sharing It's the Bit with the great Product Hunt community today. It's the Bit is a new service that we've tested with a small group of beta testers. It allows anybody to request any service (that is not illegal or a super-small project), anywhere in the U.S., anytime. Depending on what is requested, we'll either handle the service in-house or use vetted professionals to handle the service. We are confident that in most cases we can deliver (and, in fact, over-deliver). The website is simple and there isn't much to it other than a form because we want to keep your request and information private (and potentially anonymous if that is what you want). The magic happens behind the scenes. We will likely add some more to the website over time, but right now we are focused on the underlying services. I stand behind it personally, and it is our goal to always far outperform your expectations. After all, we won't last long if we don't prove our value on every gig. I'm around all day to answer any questions! Update 10 days after being featured: Business is great at It's the Bit, thanks in no small part to the exposure we received here. Thanks again to this great community!
Humbled to be in the top spot so far today. But know this doesn't mean much if we don't execute. That's why we kept the website so simple to begin with (in many ways, taking a page out of the Product Hunt playbook). I was reluctant to even post it on PH today initially, because the website itself is not much at all at this point, but we promise to deliver. Thanks guys!
Mateo D.@mateolovesearth · Failing entrepreneur, but I keep trying.
Cool! Checking this out. Will report back!
Looks solid! Looking forward to trying it out (my wife can tell you, my Honey-Do list is plenty long... Forgive me if I'm being dense, but I'm curious, is BIT's primary difference from Taskrabbit that you run on Bitcoin, or am I missing another differentiator?
@jeff_osborn there are a few differentiators, but we value service above all else and use professionals that are incentivized to perform their best. It's also possible to do entirely anonymous projects through our service - for example, something you want done for someone else. It's the Bit is basically our vetted layering on top of Hire.Bid, our marketplace that allows anybody to bid on anybody else's available time. And we use a variety of methods to find the best professional for your gig through It's the Bit, even if the professional is not on Hire.Bid. It's the Bit is also for any task you can dream up (rather than limited to home services, etc.). We are 100% focused on client satisfaction, rather than commoditized work.


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First day of Humanitarian Summit ends with 'grand bargain'
World Humanitarian Summit
Grand Bargain
governemts
humanitarian agencies
G7 leaders slammed for not attending Humanitarian Summit
First World Humanitarian Summit to be held in Istanbul
TRT World Forum to begin with focus on global peace and security
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The first ever World Humanitarian Summit opened on Monday in Istanbul, Turkey with UN member states and humanitarian agencies in attendance.
World Leaders and representatives of the participant countries pose for a family photo during the World Humanitarian Summit in Istanbul. (TRT World and Agencies)
The opening speech of the summit was given by United Nations Secretary General Ban Ki-moon.
In his speech, Ban urged world leaders to share the responsibility of helping people and close the gap in humanitarian funding.
"We need to provide more direct funding to local people and communities and fix the persistent humanitarian funding gap and investing in building stable and inclusive societies.
"We declare we are one humanity with shared responsibilities. Let us resolve, ourselves, here and now, not only to keep people alive but to [give] people a chance at life in dignity."
He also called on the international community to help reduce the number of internally displaced people by 2030 and drew attention to the necessity of increasing humanitarian spending in order to reduce the risk of potential humanitarian crises.
"I urge you to commit to cutting in half the number of internally displaced people by 2030 and to find better long term solutions for refugees and displaced people."
United Nations Secretary General Ban Ki-moon, delivers a speech at the opening ceremony of the World Humanitarian Summit, in Istanbul, Monday, May 23, 2016. (AP)
Governments and humanitarian agencies agreed on a grand bargain to cut the cost of administering aid by nearly $1 billion a year, and make sure that more of that funding goes to help people in crisis.
At the summit, 21 donor countries and 16 aid organisations agreed to decrease overheads and help fill an estimated $15 billion annual funding gap in order to provide it to more than 125 million people who are in urgent need.
.@KGeorgievaEU "Can I say something to all those who worked on the Grand Bargain? We did it! Thanks to you!" pic.twitter.com/xL57KdqzRH
— Agenda4Humanity (@agenda4humanity) May 23, 2016
It's expected that the new approach will create "a virtuous circle, drawing in more resources" from a wider range of donors.
According to the UN Secretary General, the deal would enable improved humanitarian response to conflicts and natural disasters.
"It should lead to faster action, better engagement with people affected by crises, more funding for national organisations, greater efficiency and more accountability," he said at the launch event in Istanbul.
"These are all fundamental to improving support for the people we serve."
The US, Britain, Germany, France and Japan, who supported the bargain, will follow its implementation.
Governments and humanitarian agencies represented at the summit agreed that they would assess required needs in the case of a disaster and provide supplies accordingly without relying on an outside analysis.
Backdropped by a projected picture of refugees, actors perform during the opening ceremony of the World Humanitarian Summit, in Istanbul, Monday, May 23, 2016. (AP)
Elhadj As Sy, Secretary General of the International Federation of Red Cross and Red Crescent Societies, supported the bargain because it would provide 25 percent of humanitarian funding directly to local and national agencies by 2020.
"There is a recognition of those who are on the frontline. They're in places where there is no doctor, no school, where women have to walk the last mile to the water point, often paved with danger."
According to World Vision International head, Kevin Smith, the bargain will not solve all the problems related to the aid system, but it is "a serious and realistic way forward."
"Making change will cost, but staying the same will cost more," he added.
Source: TRTWorld and agencies
Turkiye media mogul buys UK football club in 'dream come true' deal
Turkiye to run 'charity train' to Afghanistan as hunger worsens
Turkiye restarts crude flows for key Iraq pipeline

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: after pressing home keyboard is not hiding I use following code to show keyboard
InputMethodManager imm = (InputMethodManager)getSystemService(Context.INPUT_METHOD_SERVICE);
        imm.toggleSoftInput(InputMethodManager.SHOW_FORCED,
        InputMethodManager.HIDE_IMPLICIT_ONLY);

I use following code to hide keyboard getWindow().setSoftInputMode(WindowManager.LayoutParams.SOFT_INPUT_STATE_ALWAYS_HIDDEN);
to hide keyboard, and
also tried for this
InputMethodManager imm =(InputMethodManager)getSystemService(Context.INPUT_METHOD_SERVICE);
imm.hideSoftInputFromWindow(myEditText.getWindowToken(), 0);

but after pressing the home key. apps closes but keyboard remain same on the screen.
please guide me. what am I doing wrong?
I have put my hiding code.
I put hide in onDestroy(), onBackPressed() and also onOptionsItemSelected(MenuItem item)  Sorry For my code formatting.
protected void onCreate(Bundle savedInstanceState) {
    // TODO Auto-generated method stub
    super.onCreate(savedInstanceState);
    setContentView(R.layout.mylayout);

    textTv=(EditText)findViewById(R.id.textview1);
    InputMethodManager imm = (InputMethodManager)getSystemService(Context.INPUT_METHOD_SERVICE);
    imm.toggleSoftInput(InputMethodManager.SHOW_FORCED,
    InputMethodManager.HIDE_IMPLICIT_ONLY);

    if(getIntent().getExtras()!=null)
    {
        Bundle extra=getIntent().getExtras();
        if(extra!=null)
        {
            // code
        }
    }
}

public void onBackPressed() {
    // TODO Auto-generated method stub
    InputMethodManager imm = (InputMethodManager)getSystemService(Context.INPUT_METHOD_SERVICE);
    imm.hideSoftInputFromWindow(noteTv.getWindowToken(), 0);
    }

public boolean onOptionsItemSelected(MenuItem item) {
    // TODO Auto-generated method stub

    getWindow().setSoftInputMode(WindowManager.LayoutParams.SOFT_INPUT_STATE_ALWAYS_HIDDEN);

    switch (item.getItemId()) {
    case R.id.menu1:
    //code
    break;
    case R.id.menu2:
    //code
    break;

}

A: Try
@Override
protected void onPause() {
    final InputMethodManager imm = (InputMethodManager) getSystemService(Context.INPUT_METHOD_SERVICE);
    imm.hideSoftInputFromWindow(noteTv.getWindowToken(), 0);
    super.onPause();
}

redpajama_set_name: RedPajamaStackExchange


text:
Colorado > Pueblo > Schools > School Profile
1327 East 9th Street, Pueblo, CO 81001
Pueblo City 60 School District
I removed my child from this school after a bulling incident when the principal told me "My God... more
This school is rated below average in school quality compared to other schools in Colorado. Students here are making about average year-over-year academic improvement, ... More this school has below average results in how well it's serving disadvantaged students, and students perform below average on state tests.
Students at this school are making average academic progress given where they were last year, compared to similar students in the state.
Average progress with low test scores means... that students are starting at a low point but keeping up with their peers.
Test scores at this school fall far below the state average. This suggests that students at this school are likely not performing at grade level.
Test scores are important to look at because they indicate where students are performing compared to grade-level standards. Ask the school what... More it's doing to help students who are behind. Understand what on-track learning looks like, and explore higher-performing schools in your area to find the best opportunities for your child to succeed.
MR FLOYD GALLEGOS - Principal


text:
Ball State football: Cardinals blown out at Ohio as Riley Neal leaves with injury
The second straight ugly performance from Ball State.
Ball State football: Cardinals blown out at Ohio as Riley Neal leaves with injury The second straight ugly performance from Ball State. Check out this story on thestarpress.com: https://www.thestarpress.com/story/sports/college/ball-state/2018/10/25/ball-state-football-ohio-cardinals-bobcats-score-riley-neal-james-gilbert-mike-neu-nathan-rourke/1767522002/
Ryan O'Gara, Muncie Star Press Published 6:39 p.m. ET Oct. 25, 2018 | Updated 9:41 a.m. ET Oct. 26, 2018
Ball State's James Gilbert, shown here against Eastern Michigan, ran for 52 yards on nine carries Thursday at Ohio.(Photo: Jordan Kartholl / Star Press)Buy Photo
Ball State falls to 3-6 overall and 2-3 in MAC)play.
MUNCIE, Ind. — The Ball State football team hit a new low point of this 2018 season. One week after a poor showing on Homecoming, the Cardinals were even worse Thursday night at Ohio.
Ball State was blown out by the Bobcats 52-14, allowing scores on eight straight possessions at one point. After taking a 7-0 lead, the wheels fell off.
This season, which started so promising, is slipping away and appears to be long gone as Ball State (3-6, 2-3 MAC) has been blown out twice in six days. For a second straight week, Ball State was short on execution and effort. And that's surprising after the Cardinals grinded out a road win at Central Michigan.
"I want guys who are interested in sticking together and want to fight for a solution," Ball State head coach Mike Neu told the Star Press said after the game. "There are no shortcuts. I only know one way, and that's to not to make excuses. I'm interested in guys who are getting on the bus and handling things the right way.
"I'm looking for guys who embrace that. I didn't see any negative vibes or energy on the sideline, that's just what I'm emphasizing. There is going to be noise, there is going to be those who have an opinion or a solution, but we inside the building and inside the program have to stick together. Don't let anyone tear us apart."
►RELATED: Ball State athletic director discusses direction of football program
Riley Neal left the game in the second quarter with a left knee injury, not that it was any excuse as it was already 24-7 at that point. Neal was 7 of 16 for 76 yards and had three rushes for 20 yards. He appeared to get hurt on a hit as he was running out of bounds. Drew Plitt came in and was 9 of 17 for 86 yards and an interception. Plitt added a rushing touchdown late to stop a run of 52 unanswered points.
So where does Ball State go from here? The team seems to be regressing after opening the season with a competitive game at Notre Dame. Ball State would have to win its final three games to be bowl eligible, not that a bowl game seems possible after two lopsided losses in a row and games against Toledo, Western Michigan and Miami still to come.
When asked if Ball State has regressed, Neu didn't think so.
"I understand that. Obviously, I'm disappointed, but I won't say that (we have regressed)," Neu said. "The last two weeks have been challenging, and it's not what anyone want to be part of. We've had some challenges, and we had a setback tonight."
Ohio ran it over, and over, and over again, piling up 411 rushing yards and 554 total yards. The Bobcats played three quarterbacks and moved the ball at will. The last time Ohio reached 400 rushing yards in a game was in 2005 against… Ball State.
This one started well for Ball State, if you can believe it. Christian Albright's interception led to Riley Miller's fourth touchdown of the season and second spectacular grab in as many weeks. But just about nothing went right from there.
Jacob White left in the first half, as well, which certainly didn't help Ball State's cause against preseason MAC favorite. Neu singled out Brandon Martin for playing hard the entire game, and Martin would likely see an increase in playing time without White.
With another short week coming up, Ball State may play without Neal and White.
"It's not what you want," Neu said, "but you have to make sure your guys understand that when number is called, you have to be ready."
Ball State's Riley Miller, shown here against Eastern Michigan, had an 18-yard touchdown catch against Ohio on Thursday. (Photo: Jordan Kartholl / Star Press)
Ohio 52, Ball State 7 (Q3, 4:47)
That's 52 points and counting.
That's 45 points and counting. And Riley Neal has not come back into the game.
That's 38 points and counting. Almost unfathomable with how well Ball State played in the first quarter. But here we are.
Ohio is running it down Ball State's throat, and it doesn't appear to be slowing down soon. The Bobcats are up to 259 rushing yards.
Halftime: Ohio 31, Ball State 7
Just a disastrous second quarter for Ball State, which allows 31 straight points.
For one, Riley Neal was not in on the last drive as he was on the sideline with his helmet off. He appeared to take a hit on a run out of bounds. That's the risk Mike Neu and Ball State takes by running him so frequently. Drew Plitt came on with 1:23 left and was unable to get a first down or run any time off the clock.
Ohio somehow scored twice in the final 1:23 of the half. The last drive was particularly demoralizing. Ohio went 60 yards in 31 seconds, capped by a wide-open 21-yard touchdown pass. And when I say wide open, I mean no one was within 15 yards of Meyer in the center of the end zone.
A couple big mistakes by Bal State, and Ohio takes full advantage. First, a a rare fumble from Riley Miller. Then Myles Hannah led with the crown of the helmet on Nathan Rourke's run on third-and-8 and was called for targeting. Rourke would've been short of the first-down marker, too, but AJ Oulette ran it in from nine yards on the next play.
Ohio has only attempted 10 field goals all season after that 23-yarder. It was the first the Bobcats have attempted in MAC play, too. The Cardinals have said from the beginning, they want to be a bend-but-don't-break type of defense. That definitely fits the mold.
Ball State 7, Ohio 7 (Q2, 14:54)
Ball State caught a huge break when Ohio had a false-start penalty on third-and-goal from the 1, but Ohio ran it in anyways from the 6.
Third time's the charm, as Nathan Rourke once again options right to Maleek Irons, this time for a touchdown!
7-7 in the opening seconds of the 2Q. #BALLvsOHIOpic.twitter.com/fk42JL29Oz
— Hustle Belt (@HustleBelt) October 25, 2018
Ball State 7, Ohio 0 (Q1, 4:06)
Riley Neal to Riley Miller. Have we heard that one before? The Yorktown High School teammates connected for another big play, an 18-yard touchdown on third-and-goal, their second straight game with a really impressive throw/catch. The 32-yard pass from Neal to Justin Hall moved the Cardinals down the field.
As you can see below, some magnificent footwork from Miller. And a high-level throw from Neal.
Talk about a toe tap 👀@BallStateFB gets on the board first here in Athens! #MACtionpic.twitter.com/zk25K5taCh
— #MACtion (@MACSports) October 25, 2018
On the defensive end, Christian Albright is a turnover machine. He is usually forcing fumbles, but on this night, his interception leads to the touchdown.
🤫 @c_albright51 with the interception at midfield late in the 1st quarter.#ItsTime#WeFly#BallState#MACtionpic.twitter.com/NvNzi1FmSL
— Ball State Sports Link (@bsusportslink) October 25, 2018
Some quick injury updates: Starting left guard Zac Ricketts is out with a concussion, and Grant Williamson will start in his place. Running back Caleb Huntley will also miss a third straight game with an Achilles injury, while starting left tackle Kaleb Slaven (foot) and starting safety Lamar Anderson (shoulder) are also out again.
Who's ready for some #MACtion?
Here are the highest graded players in the Ohio - Ball State game tonight. pic.twitter.com/U2Rc6N6qG8
— PFF College (@PFF_College) October 25, 2018
Here are three things to watch:
•Ball State is coming off a subpar performance, a 42-20 loss to Eastern Michigan that was arguably its worst game of the season. Head coach Mike Neu said the Cardinals were in a "vulnerable position" afterward. Jacob White, one of the team's leaders, agreed. How will Ball State respond?
•Ohio quarterback Nathan Rourke is the top passer in the MAC in terms of efficiency and is also the fourth-leading rusher in the conference. Ball State's defense has a tremendous challenge ahead of itself.
•Will Ball State be able to generate offense through the running game outside of Riley Neal? The Cardinals have been especially reliant on the quarterback the last two games as he has attempted 37 rushes after having 46 in the first six games of the season.
Ryan O'Gara covers Ball State and East Central Indiana high schools at the Star Press. Contact him at (765) 213-5829, rogara@muncie.gannett.com or @RyanOGara.
5Qs: Jerry Peirson talks ECI ties, more
Ball State learns Diamond Head Classic fate
Local notes: Basketball, baseball and more


text:
"We recognise the critical importance of advanced manufacturing for our ongoing competitiveness and for the development of our future products. Our collaborative work with the ARTC is therefore completely aligned with our advanced manufacturing strategy. Core elements of this strategy relating to the development and deployment of cyber-physical systems and the broader digital manufacturing technologies are already a significant area of focus in the work we are doing at the centre.
My team and I consider ARTC as a crucial strategic partner. Its growing critical mass of subject matter experts and rapidly improving research capability when combined with the long term strategic commitment of its sponsors and stakeholders in A*STAR, EDB and partner universities make it a very important destination for the development of our future of manufacturing capability and workforce training in the fast-evolving digital environment."
- Dr Hamid Mughal, Director of Global Manufacturing, Rolls-Royce
"We have been in partnership with ARTC since 2017, when we joined the consortium as an Anchor Member. The setup of a fast moving consumer goods (FMCG) consortium provides a platform for collaboration with industry peers, which allows us to address similar problem statements in a pre-competitive environment. In addition, access to the ARTC Model Factory enables us to rapidly testbed new technologies and solutions.
With the consortium and its ecosystem, we are able to quickly identify and work with the right technology vendors without having to screen and negotiate agreements with individual companies. This setup allows us to have flexibility in working with a range of providers, bringing in expertise as needed. We are able to pilot projects such as data analytics for process optimisation and validation of cobots in the workshop floors.
The core projects and member-specific projects are aligned with Nestlé's objectives and strategy for digital transformation. Working with ARTC allows Nestlé to tap on Singapore's progressive R&D ecosystem on a neutral platform. We can carry out small-scale manufacturing experiments and test new concepts very quickly. Through this partnership, we hope to pilot digital technologies in areas such as smart connected devices, cobots and advanced analytics."
- Dr Sze Tan, Managing Director, Nestlé R&D Center Singapore
"voestalpine High Performance Metals has been a member of ARTC for more than five years now and we have benefited a lot from the knowledge and input of the great team of experts at ARTC. The services ARTC can offer through their ecosystem and state-of-the-art equipment broaden our possibilities to explore new technologies and innovative solutions. For voestalpine High Performance Metals, the collaboration with ARTC is the consequent next step of pushing our efforts in additive manufacturing and digitalisation to be one step ahead of our competition."
- Mr Georg Reithofer, Head of Divisional Coordination, Research and Development, voestalpine High Performance Metals
"DMG MORI started its membership with ARTC since 2013. We will continue to work with ARTC and we can gather information on additive manufacturing (AM) and its trends through projects. We are not only machine tool manufacturers, but also providers of the solutions on manufacturing. ARTC also shows the cutting edge technology and solutions on additive manufacturing. We believe that our partnership with ARTC will help us achieve our processes.
ARTC has developed AM process technologies on both selective laser melting (SLM) and directed energy deposition (DED) with members so far. We can look into these process technologies by projects, understanding customers' problems and concerns, and provide AM solutions to customers from the equipment and process point of view."
- Dr Masahiko Mori, President, DMG MORI
"P&G has been a Tier 1 member of ARTC since January 2018, and an Anchor member as of June 2019. During our membership here, we have made tremendous progress on both core and dedicated projects. We have successfully leveraged the Singapore ecosystem, enabling faster and more collaborative innovation. The Industry 4.0 Workshop across Asia was well-received and opened more opportunities to solve business problems, particularly in advanced robotics and smart manufacturing.
The partnership accelerates open innovation for P&G, enabling us to apply advanced manufacturing's best practices across industries into fast moving consumer goods (FMCG). This is more efficient and cost-effective than doing in-house innovation. With the quickened pace of innovation, growing demands of consumers, and the competitive landscape in Asia, the innovation ecosystem and the public-private consortium are critical in accelerating the development and adoption of disruptive technologies in manufacturing.
The partnership with ARTC allows us to serve our customers and consumers faster and more efficiently. It concretes our vision to become a "Manufacturer of the Future", strengthening our foothold as a leader in our industry and accelerating the Industry 4.0 transformation of the FMCG sector as part of Singapore's Smart Nation blueprint."
- Mr Long Quan, Associate Director, Product Supply Engineering, P&G Singapore
Arcstone
"Arcstone joined ARTC in November 2016 as a Tier 2 member, and we upgraded to Tier 1 in December 2019. The value and benefits we get from our investment in time and resources into the ARTC are always returned in excess through all our years. From the research that we do together, to the connections and partnerships that were made and the joint solutions integrated with other members, the ARTC has given Arcstone a significant boost in our growth, scale and reach.
The technical capabilities of the ARTC allowed us to innovate and move faster than industry peers and helps us to deliver more value to our customers with our solutions. In the years to come we will continue to funnel more of our research and integration work with ARTC in member-specific projects that will allow co-development of Industry 4.0 solutions that can integrate and achieve Arcstone's goal of complete manufacturing transparency across the supply chain to realize a true global real-time data driven manufacturing ecosystem"
- Mr Willson Deng, CEO, Arcstone Pte Ltd
Last updated on 15 Sep 2020


text:
USC's Chimeze Metu learns to vary shots
Washington's Carlos Johnson, right, attempts a dunk over Southern California's Chimezie Metu during the first half of an NCAA college basketball game in the first round of the Pac-12 men's tournament Wednesday, March 8, 2017, in Las Vegas. (AP Photo/John Locher)
Washington's David Crisp shoots against Southern California's Chimezie Metu during the first half of an NCAA college basketball game in the first round of the Pac-12 men's tournament Wednesday, March 8, 2017, in Las Vegas. (AP Photo/John Locher)
By Joey Kaufman | jkaufman@scng.com | Orange County Register
PUBLISHED: March 9, 2017 at 1:37 a.m. | UPDATED: September 6, 2017 at 4:42 a.m.
LAS VEGAS >> Chimezie Metu followed a seemingly standard routine last offseason.
Metu attended summer classes and worked out with USC's coaches for about two hours, the allotted practice time, at the Galen Center.
Shooting was a priority for the 6-foot-11 forward.
As a freshman, the bulk of Metu's points came through thrilling slam dunks. He knew he needed to extend his range as he entered his second season with the Trojans, poised for a larger role.
Metu, a sophomore, made a significant leap, his improved scoring ability a primarily reason why he was named the Pac-12's most improved player earlier this week.
Starting all 31 regular season games for the Trojans, Metu averaged 14.4 points, 7.6 rebounds and 1.6 blocks per game prior to the team's Pac-12 Tournament game on Wednesday night against Washington.
"He's becoming an all-around player," USC coach Andy Enfield said, "and we're excited for him."
Last season, Metu averaged 6.4 points and 3.6 rebounds per game off the bench.
The scoring increase hinged on seeing more minutes, but Metu also has relied on a newfound mid-range jump shot and a budding low-post game.
Metu has made 44 percent of two-point jump shots, according to Hoop-Math.com, compared to a 36 percent mark a year ago.
Before to his freshman season, Metu re-tooled his shooting form, bringing the release point forward, but the extra time over the recent offseasons allowed him to become more comfortable.
"When the season came around, it was just kind of second nature," Metu said. "If I was open, I knew I was going to knock it down."
He also became more adept scoring out of the low post, where he said he added a jump hook, a move he worked on with assistant Jason Hart.
It has given USC a reliable post presence.
"It's really important, because we have a lot of guys who space the floor, a lot of shooters," Metu said. "We need somebody down there to throw the ball to, to get a basket, to get fouled."
Enfield expected a jump. For skilled college big men, he contended, their biggest leaps tend to occur between their freshman and sophomore seasons, pointing to Washington State forward Josh Hawkinson.
As a freshman in 2013-14, Hawkinson averaged 1.2 points and 1.6 rebounds per game.
"Since then he's been a double-double machine," Enfield said.
As a sophomore a year, Hawkinson averaged 14.7 points and 10.8 rebounds per game and was named the conference's most improved player as well. He is now a senior for the Cougars.
USC has its own example, too.
Former forward Nikola Vucevic rose his scoring average to 10.7 points and 9.4 rebounds per game (from 2.6 points and 2.7 rebounds per game) as a sophomore in the 2009-10 seasons.
USC guard Jordan Mathews, who missed the team's regular-season finale against Washington while in the concussion protocol, was cleared by team doctors and played in the first-round Pac-12 Tournament game against Washington. … Senior forward Charles Buggs was sidelined against the Huskies on Wednesday night because of a viral illness, a school spokesman said.
Andy Enfield
Joey Kaufman | Reporter
Joey Kaufman is the USC beat writer for the Southern California News Group. Since joining the Orange County Register in 2015, he has also covered Major League Baseball and UCLA athletics. His work has been recognized by the Associated Press Sports Editors and Football Writers Association of America. Kaufman grew up in beautiful downtown Burbank.
jkaufman@scng.com
Follow Joey Kaufman	@joeyrkaufman


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Coming back from a nice, spring run, I was nearing my house when something caught my eye. From a distance this looked like there was a canvas bag hanging in the tree. I am always on the lookout for something unique so I kept my eye on the prize. As I neared the "bag" it seemed to be moving and so I crept up into the neighbor's front yard to get a better look.
WOW! There must be a thousand bees there. They were diving in and buzzing out and swarming. It was cool because of the overnight rain so they were huddled in together to keep warm and didn't seem to mind me standing there at all. I wo nder what the homeowner thought if they saw me through the window snapping pics of her front yard?


text:
European Association for Theoretical Computer Science
European Association for Theoretical Computer Science explained
The European Association for Theoretical Computer Science (EATCS[1]) is an international organization with a European focus, founded in 1972. Its aim is to facilitate the exchange of ideas and results among theoretical computer scientists as well as to stimulate cooperation between the theoretical and the practical community in computer science.
The major activities of the EATCS are:
Organization of ICALP, the International Colloquium on Automata, Languages and Programming;[2]
Publication of the Bulletin of the EATCS;
Publication of a series of monographs[3] [4] and texts[5] on theoretical computer science;
Publication of the journal Theoretical Computer Science;
Publication of the journal Fundamenta Informaticae.
EATCS Award
Each year, the EATCS Award[6] is awarded in recognition of a distinguished career in theoretical computer science. The first award was assigned to Richard Karp in 2000; the complete list of the winners is given below:
2021 Toniann Pitassi ICALP (Glasgow)
2020 Mihalis Yannakakis ICALP (Saarbrücken)
2019 Thomas Henzinger ICALP (Patras)
2018 Noam Nisan ICALP (Prague)
2017 Éva Tardos ICALP (Warsaw)
2016 Dexter Kozen ICALP (Rome)
2015 Christos Papadimitriou ICALP (Kyoto)
2014 Gordon Plotkin ICALP (Copenhagen)
2013 Martin Dyer ICALP (Riga)
2012 Moshe Vardi ICALP (Warwick)
2011 Boris Trakhtenbrot ICALP (Zurich)
2010 Kurt Mehlhorn ICALP (Bordeaux)
2009 Gérard Huet ICALP (Rhodes)
2008 Leslie G. Valiant ICALP (Reykjavík)
2007 Dana S. Scott ICALP (Wroclaw)
2006 Mike Paterson ICALP (Venice)
2005 Robin Milner ICALP (Lisboa)
2004 Arto Salomaa ICALP (Turku)
2003 Grzegorz Rozenberg ICALP (Eindhoven)
2002 Maurice Nivat ICALP (Málaga)
2001 Corrado Böhm ICALP (Creta)
2000 Richard Karp ICALP (Geneva)
Presburger Award
See main article: Presburger Award.
Starting in 2010, the European Association of Theoretical Computer Science (EATCS) confers each year at the conference ICALP the Presburger Award to a young scientist (in exceptional cases to several young scientists) for outstanding contributions in theoretical computer science, documented by a published paper or a series of published papers. The award is named after Mojzesz Presburger who accomplished his path-breaking work on decidability of the theory of addition (which today is called Presburger arithmetic) as a student in 1929. The complete list of the winners[7] is given below:
2021 Shayan Oveis Gharan ICALP (Glasgow)
2020 Dmitriy Zhuk ICALP (Saarbrücken/online)
2019 Karl Bringmann, Kasper Green Larsen ICALP (Patras)
2018 Aleksander Mądry ICALP (Prague)
2017 Alexandra Silva ICALP (Warsaw)
2016 Mark Braverman ICALP (Rome)
2015 Xi Chen ICALP (Kyoto)
2014 David Woodruff ICALP (Copenhagen)
2013 Erik Demaine ICALP (Riga)
2012 Venkatesan Guruswami, Mihai Patrascu ICALP (Warwick)
2011 Patricia Bouyer-Decitre ICALP (Zurich)
2010 Mikołaj Bojańczyk ICALP (Bordeaux)
EATCS Fellows
The EATCS Fellows Program[8] has been established by the Association to recognize outstanding EATCS Members for their scientific achievements in the field of Theoretical Computer Science. The Fellow status is conferred by the EATCS Fellows-Selection Committee upon a person having a track record of intellectual and organizational leadership within the EATCS community. Fellows are expected to be "model citizens" of the TCS community, helping to develop the standing of TCS beyond the frontiers of the community.
Recognized Year
Luca Aceto 2021
Jiri Adamek 2018
Susanne Albers 2014
Rajeev Alur 2021
Giorgio Ausiello 2014
Wilfried Brauer 2014
Artur Czumaj 2015
Pierpaolo Degano 2020
Mariangiola Dezani-Ciancaglini 2015
Josep Diaz 2017
Herbert Edelsbrunner 2014
Zoltán Ésik 2016
Mike Fellows 2014
Fedor Fomin 2019
Yuri Gurevich 2014
Mohammad Hajiaghayi 2020
Magnús M. Halldórsson 2020
David Harel 2016
Monika Henzinger 2014
Thomas A. Henzinger 2015
Giuseppe F. Italiano 2016
Samir Khuller 2021
Dexter Kozen 2015
Marta Kwiatkowska 2017
Stefano Leonardi 2018
Kurt Mehlhorn 2016
Rocco de Nicola 2019
David Peleg 2021
Jean-Éric Pin 2014
Dana Ron 2019
Davide Sangiorgi 2021
Saket Saurabh 2021
Scott A. Smolka 2016
Paul Spirakis 2014
Aravind Srinivasan 2017
Wolfgang Thomas 2014
Moshe Y. Vardi 2015
Moti Yung 2017
EATCS Bulletin
The EATCS Bulletin is a newsletter of the EATCS, published online three times annually in February, June, and October respectively. The Bulletin is a medium for rapid publication and wide distribution of material such as:
EATCS matters;
information about the current ICALP;
technical contributions;
columns;
surveys and tutorials;
reports on conferences;
calendar of events;
reports on computer science departments and institutes;
listings of technical reports and publications;
book reviews;
open problems and solutions;
abstracts of PhD Theses;
information on visitors at various institutions; and
entertaining contributions and pictures related to computer science.
Since 2021 its editor-in-chief has been Stefan Schmid (TU Berlin).
EATCS Young Researchers Schools
Beginning in 2014, the European Association for Theoretical Computer Science (EATCS) established a series of Young Researcher Schools on TCS topics. A brief history of the schools follows below.
2017 ProbProgSchool 2017 – 1st School on Foundations of Programming and Software systems. Probabilistic programming Braga, Portugal
2015 2nd EATCS Young Researchers School – Understanding COMPLEXITY and CONCURRENCY through TOPOLOGY of DATA Camerino, Italy
2014 1st EATCS Young Researchers School – Automata, Logic and Games Telč, Czech Republic
European Association for Theoretical Computer Science website
http://www.acronymfinder.com/European-Association-of-Theoretical-Computer-Science-(EATCS).html What does EATCS stand for? European Association of Theoretical Computer Science
Brauer, Ute; Brauer, Wilfried: European Association for Theoretical Computer Science / About the Association / Silver Jubilee of EATCS
https://www.springer.com/series/776 Monographs in Theoretical Computer Science. An EATCS Series
http://www.informatik.uni-trier.de/~ley/db/series/eatcs/ Monographs in Theoretical Computer Science. An EATCS Series
https://www.springer.com/series/3214 Texts in Theoretical Computer Science. An EATCS Series
http://www.eatcs.org/index.php/eatcs-award EATCS Award
Web site: Presburger Award. European Association for Theoretical Computer Science. 2020-07-23.
https://www.eatcs.org/index.php/eatcs-fellows EATCS Fellows
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "European Association for Theoretical Computer Science".


The software giant is restructuring once again. Its past shakeups have generally paid off.
This column has been updated from June 30 with additional information.
Microsoft Corp. (MSFT - Get Report) is planning a cloud-focused restructuring that will reportedly involve laying off a large number of workers worldwide. That's not too shocking given its past restructurings, and the speed at which cloud apps and service have come to account for a giant portion of the software giant's sales.
Since Steve Ballmer announced a big restructuring in July 2013 -- not long before he announced he was stepping down as CEO -- mid-year reorgs have become a way of life for Microsoft. Much like Alphabet Inc. (GOOGL - Get Report) , Microsoft is willing to frequently overhaul its business units, as well as the people leading them, in order to both pursue strategic goals and inject fresh blood into key divisions.
This approach doesn't work for every major tech company. For example, Apple Inc. (AAPL - Get Report) , though sometimes willing to shake things up, is comfortable keeping its org chart largely unchanged for long periods of time. Facebook Inc. (FB - Get Report) has also been pretty restrained in its restructuring efforts. But for companies like Microsoft and Alphabet, where different product teams can easily feel as if they're working for different companies, there's arguably more incentive to shake things up from time to time -- particularly if it helps different teams play nice with each other.
Jim Cramer and the AAP team hold positions in Apple, Facebook and Alphabet for their Action Alerts PLUS Charitable Trust Portfolio. Want to be alerted before Cramer buys or sells AAPL, FB or GOOGL? Learn more now.
On Friday, June 30th, the Puget Sound Business Journal reported that Microsoft plans to announce a new re-org on July 5th that "will better align the company under its cloud-first mantra." Later in the day, Bloomberg reported the reorg will cover Microsoft's sales teams, and aims to improve their ability to sell cloud software. The shakeup was also said to feature job cuts, as well as impact Microsoft's local marketing efforts in certain countries.
On Monday, July 3rd, Microsoft confirmed the restructuring through an internal e-mail sent by EVP Judson Althoff. The company's business sales teams will be split into an Enterprise group and a Small, Medium and Corporate (SMC) group. The Enterprise group, in turn, will be split into teams for specific accounts, teams focused on adding new business and "customer success units" focused on driving additional consumption (presumably cloud-related) of Microsoft products and services.
Microsoft is also overhauling its attempts to sell to consumers. The company plans to better integrate its consumer and hardware sales efforts with its commercial software and services efforts, and pitch consumers on using the same Microsoft "creativity tools" they use in the office at home.
Althoff's memo didn't mention the layoffs expected to accompany the restructuring. The timing of the announcement is close to that for previous reorgs announced in 2013, 2015 and 2016.
The 2013 reorg, perhaps the biggest of the bunch, put an end to Microsoft divisions created around product groups such as Windows, Office and server software, and replaced them with "functional" divisions covering fields such as engineering (four different engineering groups were set up), finance and HR. Among other things, the overhaul sought to end the turf wars the product divisions were infamous for.
The 2015 reorg, launched by current CEO Satya Nadella a year after he carried out major layoffs, produced the departure of hardware chief and ex-Nokia CEO Stephen Elop, as well as a few other senior execs. It also yielded the merging of Microsoft's OS and hardware engineering teams, and added the company's Dynamics business app developer teams to its Cloud and Enterprise engineering unit.
The 2016 reorg featured the departure of long-time COO/sales chief Kevin Turner, and handed off his responsibilities to five execs who had already been overseeing functions such as corporate strategy, finance and marketing. Nadella's stated goal with this move was to "more deeply integrate" Microsoft's Sales, Marketing and Service Group (SMSG) with the rest of the company.
Ahead of its latest restructuring, Microsoft had already carried out a pair of shakeups this year. Earlier in June, the company created three new software engineering groups: A Cloud AI Platform team responsible for things such as the Azure Machine Learning service and Microsoft's Bot Framework, an Azure + Data Platform Group covering Azure cloud service engineering and several database and analytics products, and a Business Applications Group that covers the Dynamics line and a slew of other business apps. And In January, the company consolidated teams responsible for its relationships with channel partners and third-party software vendors, and also created a Microsoft Digital unit meant to help companies adopt cloud services.
There's clearly a trend towards better integrating engineering teams working on different products that complement each other, and towards having non-engineering teams work with the company as a whole. And as Microsoft's sales mix has dramatically shifted towards cloud apps and services -- the company's Commercial Cloud revenue was on a $15.2 billion annual revenue run rate as of June -- there's also generally a trend towards giving more attention to fast-growing cloud and analytics offerings.
Google can certainly relate to Microsoft's non-stop reorg activity. In addition to its big 2015 restructuring, which created the Alphabet parent company and placed various "moonshot" projects into an "Other Bets" segment, Google has seen leadership changes for products such as Search, YouTube and Android over the last several years. It has also done things such as put most of the company's hardware work under the oversight of former Motorola Mobility hardware chief Rick Osterloh, and both the Google Cloud Platform (GCP) and the G Suite productivity apps (formerly Google Apps) under the oversight of VMware co-founder Diane Greene.
A company like Apple, whose entire business philosophy has long revolved around tightly integrating hardware, software and services, understandably feels less need to be so aggressive with its executive and organizational shakeups. But Microsoft and Google are very different kinds of companies. And given how well each one has performed over the last few years, it's hard to complain about what either is doing.


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text:
\section{Powered Dirichlet-Multinomial distribution}
As a side result, we derive a closed-form expression for the Powered Dirichlet-Multinomial distribution based on our definition of the powered conditional Dirichlet prior:
\begin{equation}
    \begin{split}
        \label{eq-DirMultr}
        DirMult_r(\vec{N} \vert \vec{\alpha}) &= \int_{\vec{p}} Mult(\vec{N}^r \vert \vec{p}) Dir(\vec{p} \vert \vec{\alpha}) d\vec{p}\\
        &= \int_{\vec{p}} \frac{ (\sum_k N_k^r)!}{\prod_k N_k^r! B(\vec{\alpha})} \prod_k p_k^{\alpha_k + N_k^r - 1} d\vec{p}\\
        &=  \frac{ (\sum_k N_k^r)!}{\prod_k N_k^r! B(\vec{\alpha})} B(\vec{\alpha} + \vec{N}^r) \underbrace{\int_{\vec{p}}  Dir(\vec{p} \vert \alpha_k + N_k^r) d\vec{p}}_{=1} \\
        &= \frac{B(\vec{\alpha} + \vec{N}^r) (\sum_k N_k^r)!}{B(\vec{\alpha}) \prod_k N_k^r! }
    \end{split}
\end{equation}

\bibliographystyle{splncs04}

\section{Introduction}
The notion of clustering has been initially introduced by anthropologists Driver and Kroeber in 1932 \cite{Driver1932} in the classification of human psychological traits. It has been later used successfully in a broad range of applications, ranging from scientific research to data compression, marketing, and medicine. Over the past decades, it also became a central problem in machine learning\footnote{As an illustration, scraping Google Scholar shows that the yearly number of publications containing the keyword ``Clustering'' averages to 250.000.}.

The Bayesian clustering approach received broad attention in the last years. A non-exhaustive list of application includes medicine, \cite{SEESdrugdrugInter}, natural language processing \cite{LDA2003,Yin2014}, genetics \cite{mcdowell2018,Jensen2008,Qin2003}, recommender systems \cite{AiroldiMMSBM,AntoniaScalableRS,IMMSBM}, sociology \cite{HumanPrefSBM,SergioSocialDilemma}, etc. The key idea is to simulate a corpus of independent observations by drawing them from a set of latent variables (clusters). Those clusters are each associated with a probability distribution on the observations, whose parameters are drawn from a prior distribution, as we will formulate mathematically later. Now, an often desirable property of Bayesian models is to make them nonparametric. In our case, it means that both the number of clusters and their associated distributions are inferred. A very popular prior that allows this is the Dirichlet process. It incorporates a chance for a new cluster to be created in the prior probability of a distribution  (often when an observation is not likely to be explained by existing clusters). Otherwise, the observation is associated with an existing cluster with a probability proportional to that cluster's population.

However, the Dirichlet process (and the related Pitman-Yor process) comes with a hypothesis on the way observations are allocated to various clusters: the \textit{rich-get-richer} property \cite{Ferguson1973}. As stated before, a new observation belongs to a cluster with a probability proportional to the number of observations already present in the cluster; large clusters have a greater chance to get associated with new observations. While this can be a relevant property in some cases, it implies a strong assumption on the way data is generated. It has already been pointed out \cite{welling2006} that there is a need for more flexible priors.
Imagine sampling topics over time on Twitter: there is no specific reason for a new word to belong to a topic with a probability depending linearly on the topic size as in the regular Dirichlet and Pitman-Yor processes. For instance, due to temporal variations of topic usage: a small news outbreak might go unnoticed using a DP. Since the emerging cluster has a low population, the ``rich-get-richer'' assumption could lead to ignore the outbreak if it is not different enough from another existing topic. Following the same idea in spatial clustering, tiny clusters (at the level of cities, for instance) might go unnoticed at larger scales (world map, for instance) if the ``rich-get-richer'' assumption has too much influence (see Fig.\ref{fig-Antoni}). In these cases, one needs to weaken the ``rich-get-richer'' assumption to control clustering's level of detail --- to make it flatter.

Little effort has been put into exploring alternative forms of priors for nonparameteric Bayesian modeling. In the present work, we offer to address this problem by deriving a more general form of the Dirichlet process that explicitly controls the importance of the ``rich-get-richer'' assumption. Explicitly, we derive the Powered Chinese Restaurant Process (PCRP) that allows control of the ``rich-get-richer'' property while generalizing state-of-the-art works. We show that controlling the ``rich-get-richer'' prior of simple models yields better results on synthetic and real-world datasets.

\section{Background}
\label{SotA}
\subsection{Motivation}
\label{motivation}
This work is motivated by the need to control the ``rich-get-richer'' assumption's importance in Dirichlet process (DP) priors.
The ``rich-get-richer'' property of the DP may not always be the most suitable prior for modeling a given dataset. The usual motivation for using a DP prior is that a new observation has a probability of being assigned to any cluster proportional to its population in the absence of external information (such as inter-points distance in case of spatial clustering, for instance). However, this assumption might be wrong (see Introduction). External information is often more relevant to the clustering than the clusters' population; we then want the DP prior to be flatter so that the model relies less on it.

Most state-of-the-art works rely on tuning a parameter $\alpha$ (see Eq.\ref{eq-CRP}) to get the ``right'' number of clusters (this parameter shifts the distribution of the number of clusters as $\mathbb{E}(K \vert N) \propto \alpha \log N$ with $K$ the number of clusters and $N$ the number of observations). However, we argue this is a bad practice in some cases. Imagine sampling topics over time on Twitter: there is no specific reason for topics to appear at a rate $\alpha \log N$ as in the regular DP. If we later consider more observations on the same dataset, the parameter $\alpha$ would need to be tuned again to correct the likely wrong evolution of $K$ as $\log N$. It makes little sense since the dataset is still about the same data type.
Moreover, there is no specific reason for a new word to belong to a topic with a probability depending linearly on the topic size as in the regular Dirichlet and Pitman-Yor processes. A prior that is too peaky on crowded clusters might then lead to irrelevant results (due to temporal variations of topics, for instance, see Introduction and Fig.\ref{fig-Antoni}). To alleviate those assumptions, we develop a more general form of the DP process allowing a natural control of the ``rich-get-richer'' property.

\subsection{Previous works}
\subsubsection{Dirichlet process}
A well-known metaphor for the Dirichlet process is referred to as ``Chinese restaurant''. The corresponding process is named ``Chinese Restaurant Process'' (CRP). It can be illustrated as follows: if a $n^{th}$ client arrives in a Chinese restaurant, she will sit at one of the $K$ already occupied table with a probability proportional to the number of persons already sat at this table. She can also go to a new table in the restaurant and be the first client to sit there with a probability inversely proportional to the total number of clients already sat at other tables.
It can be written formally as:
\begin{equation}
\label{eq-CRP}
    CRP (C_i = c \vert C_1, C_2, ..., C_{i-1}) =
    \begin{cases}
    \frac{N_c}{\alpha + N} \text{ if c = 1, 2, ..., K}\\
    \frac{\alpha}{\alpha + N} \text{ if c = K+1}
    \end{cases}
\end{equation}
Where $c$ is the cluster chosen by the $i^{th}$ customer, $N_k$ is the population of cluster $k$, $K$ is the number of already occupied tables and $\alpha$ the concentration parameter.
When the number of clients goes to infinity, this process is equivalent to a draw from a Dirichlet distribution over an infinite number of clusters with an identical initial probability to get chosen proportional to $\alpha$. The form of Eq.\ref{eq-CRP} is helpful to understand the underlying dynamics of the process and the contribution of seminal works we will detail now. It can be shown that the expected number of clusters after $N$ observations evolves as $\log N$ \cite{Arratia1992}.

The two best-known variations of the regular Dirichlet process that address the ``rich-get-richer'' property control are the seminal Pitman-Yor process and the Uniform process. Each of them can be expressed in a similar form as Eq.\ref{eq-CRP}.

\subsubsection{Pitman-Yor process}
Following the Chinese Restaurant process metaphor, the Pitman-Yor process \cite{PitmanYor1997,Ishwaran2003} proposed to incorporate a \textit{discount} when a client opens a new table.
Mathematically, the process can be formulated as:
\begin{equation}
\label{eq-PY}
    CRP (C_i = c \vert C_1, C_2, ..., C_{i-1}) =
    \begin{cases}
    \frac{N_c - \beta}{\alpha + N} \text{ if c = 1, 2, ..., K}\\
    \frac{\alpha + \beta K}{\alpha + N} \text{ if c = K+1}
    \end{cases}
\end{equation}

The introduction of the parameter $\beta>0$ increases the probability of creating new clusters. A table with a low number of customers has significantly less chances to gain new ones, while the probability of opening a new table increases significantly. It can be shown that the number of tables evolves with the number of clients $N$ as $N^{\beta}$ \cite{Sudderth2009,Goldwater2011}. However, this process does not control the arguable ``rich-get-richer'' hypothesis \cite{welling2006}, since the relation to the population of a table remains linear; it only shifts this dependence of a value $\beta$. It makes so by creating clusters based on the number of existing clusters and the total number of observations, but not according to the population of already existing clusters. Those play the same role in the Pitman-Yor process as in the DP. The Pitman-Yor process thus comes with two limitations. First, since $\beta>0$, it cannot modify the process to generate fewer clusters. Second, the discount parameter does not modify the linear dependence on previous observations for cluster allocations --- rich still get richer; the prior is as peaky on large clusters as before. The present work offers to address those two limitations.

\subsubsection{Uniform process}
Another process that aims at breaking the ``rich-get-richer'' property is the Uniform process. It has been used in some occasions \cite{Jensen2008,Qin2003} without proper definition. More recently, it has been formalized and studied in comparison with the regular Dirichlet and Pitman-Yor processes \cite{wallach2010}. It can be written as follows:
\begin{equation}
\label{eq-UP}
    CRP (C_i = c \vert C_1, C_2, ..., C_{i-1}) =
    \begin{cases}
    \frac{1}{\alpha + K} \text{ if c = 1, 2, ..., K}\\
    \frac{\alpha}{\alpha + K} \text{ if c = K+1}
    \end{cases}
\end{equation}
This formulation completely gets rid of the ``rich-get-richer'' property. The probability of a new client joining an occupied table is a uniform distribution over the number of occupied tables; it does not depend on the tables' population. In \cite{wallach2010}, it has been shown that the expected number of tables evolves with $N$ as $\sqrt{N}$. Removing the ``rich-get-richer'' property leads to a flat prior. As we show later, our formulation allows to retrieve such flat priors and thus generalizes the Uniform Process.

\subsection{Contributions}
In the present work, we derive the Powered Chinese Restaurant Process (PCRP) that allows controlling the ``rich-get-richer'' property while generalizing state-of-the-art works --- rich-get-no-richer (Uniform process), rich-get-less-richer, ``rich-get-richer'' (DP), and rich-get-more-richer. Doing so, we define the Powered Dirichlet-Multinomial distribution. We detail some key-properties of the Powered Dirichlet Process (convergence, expected number of clusters). Finally, we show that controlling the ``rich-get-richer'' prior of simple models yields better results on synthetic and real-world datasets.

\section{The model}
\subsection{The Dirichlet-Multinomial distribution}
We recall:
\begin{equation}
\label{eq-Mult}
    Dir(\vec{p} \vert \vec{\alpha}) = \frac{\prod_k p_k^{\alpha_k - 1}}{B(\vec{\alpha})} \ \ \ \, \ \ \ \ Mult(\vec{N} \vert N, \vec{p}) = \frac{\Gamma(\sum_k N_k + 1)}{\prod_k \Gamma(N_k + 1)} \prod_k p_k^{N_k}
\end{equation}
With $\vec{N} = (N_1, N_2, ..., N_K)$ where $N_k$ is the integer number of draws assigned to cluster $k$, $N = \sum_k N_k$ the total number of draws, $\Gamma(x)=(x-1)!$ and $B(\vec{x}) = \prod_k \Gamma(x_k)/\Gamma(\sum_k x_k)$.

The regular Dirichlet process can be derived from the Dirichlet-Multinomial distribution. The Dirichlet-Multinomial distribution is defined as follows:
\begin{equation}
\label{eq-DirMult}
\begin{split}
    p(\vec{N} \vert \vec{\alpha}, n) &= \int_{\vec{p}} p(\vec{N}\vert \vec{p}, n)p(\vec{p} \vert \vec{\alpha}) d\vec{p}\\
    &= \frac{(n!)\Gamma(\sum_k \alpha_k)}{\Gamma(n+\sum_k \alpha_k)} \prod_{k=1}^{K}\frac{\Gamma(N_k + \alpha_k)}{(N_k!)\Gamma(\alpha_k)}\\
    \text{where }&\Vec{p} \sim Dir(\vec{p} \vert \vec{\alpha}) \, ; \, \Vec{N} \sim Mult(\vec{N} \vert n, \Vec{p})
\end{split}
\end{equation}
In Eq.\ref{eq-DirMult}, we sample $n$ values over a space of $K$ distinct clusters each with probability $\vec{p}=(p_1, p_2, ..., p_K)$, using a Dirichlet prior with parameter $\vec{\alpha} = (\alpha_1, \alpha_2, ..., \alpha_K)$. As we will show in the next section, we can derive the Dirichlet process equation by iterating the Dirichlet-Multinomial distribution. More precisely, one has to compute a new observation's conditional distribution to belong to any cluster given the allocation of all the previous random variables when $K \rightarrow \infty$.

\subsection{Powered conditional Dirichlet prior}
In the derivation of the standard Dirichlet-Multinomial posterior predictive, one considers a categorical distribution coupled with a Dirichlet prior on its parameter $\vec{p}$. Usually, this prior is linearly dependent on previous draws from the distribution. We propose to modify this assumption by using a Dirichlet prior that depends on the history of draws as:
\begin{equation}
\label{eq-DirMultPrior}
    Dir_r(\vec{p} \vert \vec{\alpha}, \vec{N}) = \frac{1}{B(\vec{\alpha} + \vec{N}^r)} \prod_k p_k^{\alpha_k + N_k^r - 1}
\end{equation}
In Eq.\ref{eq-DirMultPrior}, the vector $\vec{N}^r$ shifts the parameter $\vec{\alpha}$ according to the count of draws allocated to each cluster $k$ up to the n$^{th}$ draw. The parameter $r \in \mathbb{R}^+$ controls the intensity of this shift for each entry of $\vec{X}$.

We demonstrate that the Powered Dirichlet distribution is a conjugate prior of the Multinomial distribution, by writing Eq.\ref{eq-DirMultPrior} as:
\begin{equation}
\begin{split}
\label{eq-DirMultPrior2}
    Dir_r(\vec{p} \vert \vec{\alpha}, \vec{N}) &= \frac{1}{B(\vec{\alpha} + \vec{N}^r)} \prod_k p_k^{\alpha_k - 1} \prod_k p_k^{N_k^r}\\
    &\stackrel{\text{Eqs.\ref{eq-Mult}}}{=} \frac{B(\vec{\alpha}) \prod_k N_k^r!}{B(\vec{\alpha} + \vec{N}^r) (\sum_k N_k^r)!} Dir(\vec{p} \vert \vec{\alpha}) Mult(\vec{N}^r \vert \sum_k N_k^r, \vec{p})\\

\end{split}
\end{equation}
where the prior on vector $\vec{N}$ is a regular Multinomial distribution of parameter $N=\sum_k N_k^r$. Note that for certain values of $r$, the vector $\vec{N}^r$ might not be made of integer values; the resulting Multinomial prior on $\vec{N}^r$ must then be expressed in terms of $\Gamma$ functions (see Eq.\ref{eq-Mult}) to be valid for $\vec{N}^r \in \mathbb{R}^{\vert \vec{N} \vert}$.
Distributions of non-integer counts are not new in the literature \cite{Ghitza2013,McCarthy2012DifferentialEA,Khurshid2005ConfidenceIntervNegBinDist} and are essentially allowed by the generalized definition of the factorial function in terms of the gamma function.
When $r=1$, we recover the standard Dirichlet-Multinomial prior on $\vec{p}$ for the $n^{th}$ draw; the history of draws $\vec{N}$ can be expressed as the result of $N$ independent draws of equal probability $\vec{p}$. When $r \neq 1$, the prior on $\vec{N}$ is sampled from a Multinomial distribution in which the number of samples drawn depends on $r$ as $\sum_k N_k^r$. For instance, let $\vec{N} = (1, 2)$ and $r=2$: the resulting powered conditional Dirichlet prior would then be sampled from a Multinomial distribution $Mult(\vec{N}=(1,4) \vert N=5, \vec{p}=(p, 1-p))$.

\subsection{Posterior predictive}
We now derive the posterior distribution for the $n^{th}$ draw to belong to a cluster $c$ given all previous draws. We assume that $\vec{C_-}$ represents all previous realizations up to $n-1$, that is, the cluster to which each previous draw has been associated. For simplicity of notation, we define the population of a cluster $k$ at time $n-1$ as $N_k = \vert \{C_i \vert i=k\}_{i=1, 2, ..., n-1} \vert$. We are now looking at the probability distribution of its $n^{th}$ draw to belong to $c$. It is expressed as the probability of a draw from the categorical distribution given all previous observations (because there is only one new draw, it is the same as a Multinomial distribution with parameter $N=1$) combined with the powered Dirichlet prior defined Eq.\ref{eq-DirMultPrior}. Then:
\begin{equation}
    \label{eq-DirCatr}
    \begin{split}
        DirCat_r(C_n = c \vert \vec{\alpha}, \vec{C_-}) &= \int_{\vec{p}} Cat(C_n=c \vert \vec{p}) \underbrace{Dir_r(\vec{p} \vert \vec{\alpha}, \vec{N})}_{\textbf{Eq.\ref{eq-DirMultPrior}}} \\
        =& \int_{\vec{p}} \frac{1}{B(\vec{\alpha} + \vec{N}^r)} \prod_k p_k^{c_k + \alpha_k + N_k^r - 1}\\
        &= \frac{B(\vec{c} + \vec{\alpha} + \vec{N}^r)}{B(\vec{\alpha} + \vec{N}^r)}
    \end{split}
\end{equation}
where $\vec{c}$ is a vector of the same length as $\vec{\alpha}$ and $\vec{C}$ whose c$^{th}$ entry equals 1, and 0 anywhere else. Alternative demonstrations of this result are possible \cite{Sethuraman1994,Wilks1992}.

\subsection{Powered Chinese Restaurant process}
We finally derive an expression for the Powered Chinese Restaurant process from Eq.\ref{eq-DirCatr}. We recall that $N_k = \vert \{C_{-i} \vert i=k\}_{i=1, 2, ..., n-1} \vert$. Taking back the conditional probability for the $n^{th}$ observation to belong to cluster $c$ (Eq.\ref{eq-DirCatr}), we have:
\begin{equation}
\label{eq-derivPCRP}
\begin{split}
    p(C_n = c \vert \vec{C_-}, \vec{\alpha}) =& DirCat_r(C_n=c \vert \vec{C_-}, \vec{\alpha})\\
    =& B(\vec{c}+\vec{N^r}+\vec{\alpha}) / B(\vec{N^r}+\vec{\alpha})\\
    =& \Gamma(N_c^r + \alpha_c + 1) \frac{\prod_{k \neq c} \Gamma(N_k^r + \alpha_k)}{\Gamma(1 + \sum_k N_k^r + \alpha_k)} \frac{\Gamma(\sum_k N_k^r + \alpha_k)}{\prod_{k \neq c} \Gamma(N_k^r + \alpha_k)}\\
    =& \frac{(N_c^r + \alpha_c)}{\sum_k N_k^r + \alpha_k} \frac{\prod_{k} \Gamma(N_k^r + \alpha_k)}{\Gamma(\sum_k N_k^r + \alpha_k)} \frac{\Gamma(\sum_k N_k^r + \alpha_k)}{\prod_{k} \Gamma(N_k^r + \alpha_k)}\\
    =& \frac{N_c^r + \alpha_c}{\sum_k N_k^r + \alpha_k}
\end{split}
\end{equation}

Finally, taking the limit $K \rightarrow \infty$ and defining $\sum_k \alpha_k \stackrel{\rightarrow}{K \rightarrow \infty} \alpha$, we find the Powered Chinese Restaurant Process:
\begin{equation}
\label{eq-PowCRP}
    P-CRP (C_i = c \vert C_1, C_2, ..., C_{i-1}) =
    \begin{cases}
    \frac{N_c^r}{\alpha + \sum_k^K N_k^r} \text{ if c = 1, 2, ..., K}\\
    \frac{\alpha}{\alpha + \sum_k^K N_k^r} \text{ if c = K+1}
    \end{cases}
\end{equation}

The formal derivation of the Powered Chinese Restaurant process in Eq.\ref{eq-PowCRP} and the demonstration of its link to the conditional Dirichlet prior on $\vec{p}$ are the first main contribution of this work. Besides, this demonstration uncovers the link between the prior in Eq.\ref{eq-DirMultPrior} and an exotic formulation of the Multinomial distribution, which has never been considered before.
As stated in the introduction, special cases of the process have already been used in some occasions \cite{wallach2010,Qin2003,Jensen2008} but never demonstrated. Furthermore, this formulation generalizes the Uniform process when $r\rightarrow 0$ \cite{wallach2010}, the Dirichlet process when $r \rightarrow 1$ and the Pitman-Yor process when $r_k(N_k) = (log(1-\beta/N_k) + log(N_k))/log(N_k)$ and $\alpha(K) = \alpha + \beta K$ (see Eq.\ref{eq-PY}, we recall that $e^{\log x} = x$). The present expression explicitly allows for controlling the importance of the ``rich-get-richer'' property as well as recovering state-of-the-art processes.

\begin{figure}
    \centering
    \includegraphics[width = 0.8\columnwidth]{Prior.pdf}
    \caption{Effect of $r$ on the Powered Chinese Restaurant process prior probability.}
    \label{fig-prior}
\end{figure}

We illustrate the change on prior probability for an existing cluster to get chosen induced by the Powered Chinese Restaurant process in Fig.\ref{fig-prior} -- we do not plot the prior probability for a new cluster to be created. This figure plots the population of clusters (grey bars) and their associated prior probability of getting chosen. When $r>1$, the most populated clusters are associated with a more significant prior probability than in the standard CRP, whereas the less populated ones have even less chances to get chosen; rich-get-more-richer, the prior on population is more peaky on large clusters. On the other hand, when $r<1$, most populated clusters have less chances to get chosen than in CRP, whereas less populated ones have an increased chance of getting chosen; rich-get-less-richer, the prior on population is flatter across clusters of different sizes. In the limit case $r=0$, the clusters' population does not play any role anymore; rich get-no-richer, the prior is flat over all clusters. Note that if we wanted to represent the Pitman-Yor process prior in this figure, it would correspond to the plot for $r=1$ vertically shifted of $-\beta$ (such as defined Eq.\ref{eq-PY}) leading to an increased probability of creating a new cluster of $\beta K$ (not represented in the plot) \cite{PitmanYor1997}. Varying the parameter $\alpha$ of $\Delta \alpha$ plays a similar role as $\beta$ in this situation. It would uniformly shift the prior probability for each existing cluster to get chosen by $\frac{\Delta \alpha}{K}$ and increase the probability of creating a new one by $\Delta \alpha$. For Both Pitman-Yor and Dirichlet processes, the linear dependence of each cluster's population does not change.

In Fig\ref{fig-prior}, we understand that the Powered Chinese Restaurant process allows for defining priors from clusters population that are not possible when tuning the Chinese Restaurant or Pitman-Yor processes. Introducing non-linearity in the dependence on previous observations allows giving any importance to the ``rich-get-richer'' property.

\section{Properties of the Powered Chinese Restaurant process}
We will now investigate some key properties of the Powered Chinese Restaurant process. We recall that $N_k$ is the population of the cluster $k$, and $N = \sum_k N_k$.
\subsection{Convergence}
\begin{prop}
\label{th-statio}
For $N \rightarrow \infty$, the Powered Chinese Restaurant process converges towards a stationary distribution. When $r<1$, it converges towards a uniform distribution over all the possible clusters, and when $r>1$, it converges towards a Dirac distribution on a single cluster.
\end{prop}
\begin{proof}
We consider a simple situation where only 2 clusters are involved. The generalization to the case where $K$ clusters are involved is straightforward. When clusters' population is large enough, we make the following Taylor approximation:
\begin{equation}
\label{eq-taylorapprox}
    \begin{split}
        (N_i+1)^r &= N_i^r(1 + \frac{1}{N_i^r}) = N_i^r + r N_i^{r-1} + \mathcal{O}(N^{r-2})
    \end{split}
\end{equation}

Since the population of a cluster $N_i$ is a non-decreasing function of $N$, we assume that first order Taylor approximation holds when $N \rightarrow \infty$. Given clusters population at the $N^{th}$ observation, we perform a stability analysis of the gap between probabilities $\Delta p(N) = p_1(N)-p_2(N)$. We recall that the probability for cluster $i$ to get chosen is $p_i(N) = N_i^r / (\sum_k N_k^r)$ and that either of the clusters is chosen with this probability at the next step (at step $N+1$, $\Delta p(N+1) = p_1(N+1)-p_2(N)$ with probability $p_1(N)$ and $\Delta p(N+1) = p_1(N)-p_2(N+1)$ with probability $p_2(N)$). Explicitly the variation of the gap between probabilities when $N$ grows is written as:
\begin{equation}
\label{eq-varGap}
    \begin{split}
        &\frac{p_1(N) (p_1(N+1) - p_2(N)) +  p_2(N) (p_1(N) - p_2(N+1)) - \Delta p(N)}{\Delta p(N)}\\
        \stackrel{\text{Eq.\ref{eq-taylorapprox}}}{\approx} &\frac{1}{p_1(N) - p_2(N)} \times \left( p_1(N) \frac{N_1^r - N_2^r + r N_1^{r-1}}{N_1^r + N_2^r + r N_1^{r-1}}  + p_2(N) \frac{N_1^r - N_2^r - r N_2^{r-1}}{N_1^r + N_2^r + r N_2^{r-1}} \right)\\
        = &\frac{2 r N_1^r N_2^r}{(N_1^r+N_2^r+rN_1^{r-1})(N_1^r+N_2^r+rN_2^{r-1})}\left( \frac{N_1^{r-1} - N_2^{r-1}}{N_1^r-N_2^r} \right)
    \end{split}
\end{equation}

We see in Eq.\ref{eq-varGap} that the sign of the variation of the gap between probabilities depend only on the term $\frac{N_1^{r-1} - N_2^{r-1}}{N_1^r-N_2^r}$. We can therefore perform a stability analysis of the Powered Chinese Restaurant process using only this expression.

When $0<r<1$, the following relation holds: $N_1^{r-1} - N_2^{r-1} < 0 \Leftrightarrow N_1^{r} - N_2^{r} > 0 \ \forall N_1, N_2$; that makes right hand side of Eq.\ref{eq-varGap} negative. Therefore adding a new observation statistically reduces the gap between the probabilities of the two clusters. We could forecast this prediction from Eq.\ref{eq-taylorapprox} by seeing that adding a new observation to a large cluster increases its probability to get chosen lesser than for a small cluster -- rich-get-less-richer. Moreover, we see from Eq.\ref{eq-taylorapprox} that a crowded cluster (such as $N_1^r \gg N_2^r$) see its probability evolve as $N^{r-1}$. Asymptotically, the only fixed point of Eq.\ref{eq-varGap} when $N \rightarrow \infty$ is $N_1 \rightarrow N_2$, which implies a uniform distribution.

On the contrary, when $r>1$ we have the following relation: $N_1^{r-1} - N_2^{r-1} > 0 \Leftrightarrow N_1^{r} - N_2^{r} > 0 \ \forall N_1, N_2$; ; that makes right hand side of Eq.\ref{eq-varGap} positive. Adding a new observation statistically increases the gap between probabilities. From Eq.\ref{eq-taylorapprox}, we see that adding an observation to a large cluster increases its probability with its population -- rich-get-more-richer. In this case, Eq.\ref{eq-varGap} has $K+1$ fixed points, with $K$ the number of clusters. The uniform distribution is an unstable fixed point, while $K$ Dirac distributions (each on one cluster) are stable fixed points of the system. It means the gap converges to $1$, that is a probability of 1 for one cluster and a probability of 0 for the others.

When $r=1$, the right hand side of Eq.\ref{eq-varGap} is null. It means the gap remains statistically constant $\forall N_i$, which is a classical result for the regular Dirichlet process. This convergence has already been studied on many occasions \cite{Ferguson1973,Arratia1992}.

We note that as $r \rightarrow 0$, Eq.\ref{eq-varGap} is not defined anymore. That is because the probability for a cluster to be chosen does not depend on its population anymore. In this case, $p_1(N) - p_2(N) \propto N_1^0 - N_2^0 = 0$: the probability for any cluster to be chosen is equal, hence the Uniform process -- ``rich-get-no-richer''.

\qed
\end{proof}

\subsection{Expected number of tables}

\begin{prop}
\label{th-slowVarp}
When $N$ is large, $\sum_k N_k^r$ varies with $N$ as $N^{\frac{r^2+1}{2}}$ when $r<1$, and with $N^{r}$ when $r\geq 1$.
\end{prop}
\begin{proof}
Taking back Eq.\ref{eq-PowCRP}, we are interested in the variation of $p_i = \frac{N_i^r}{\sum_k N_k^r}$ according to $N$ when $N_i^r$ is large:
\begin{equation}
\label{eq-slowVarp}
\begin{split}
p_i(N+1)-p_i(N) \approx \begin{cases}
\frac{rN_i^{r-1} + \mathcal{O}(N^{r-2})}{\sum_k N_k^r} &\text{ if $N_i$ grows}\\
0 &\text{ else}
\end{cases}
\end{split}
\end{equation}

We see in Eq.\ref{eq-slowVarp} that for $r<1$, the larger $N_i$ the slower the variation of $p_i$. It means that for large $N_i^r$, we can write $N_i \propto N p_i$, with $p_i$ a constant of $N$. Since $N_i$ is either way a non-decreasing function of $N$, we reformulate the constraint $N_i^r$ large in $N^r$ large.

For $r>1$, the probability $p_i$ varies greatly with $N$ and quickly converges to 1 for large $N$ (see Proposition \ref{th-statio}), and so $N_i \approx N$ for cluster $i$ and $N_{j \neq i} \ll N_i \ \forall j$.

Since the sum $\sum_k N_k^r$ essentially varies according to large $N_k$, we can approximate $\sum_k N_k^r \approx N^r \sum_k p_k^r$ for large $N^r$.

Besides, we showed in Proposition \ref{th-statio} that for large $N$ the process converges towards a uniform distribution for $r<1$ and towards a Dirac distribution when $r>1$. Therefore, we can express $\sum_k^K p_k^r$ as:
\begin{equation}
    \sum_k^K p_k^r \stackrel{N \gg 1}{\approx}
    \begin{cases}
    K^{1-r} &\text{for $r<1$}\\
    1 &\text{for $r \geq 1$}
    \end{cases}
\end{equation}

Based on the demonstration of Eq.4 in \cite{wallach2010}, we suppose that $K$ evolves with $N$ as $N^{\frac{1-r}{2}}$ when $r<1$. We verify that this assumption holds in the Experiment section.

Therefore, we can write:
\begin{equation}
    \sum_k N_k^r \approx N^r \sum_k^K p_k^r \approx \begin{cases}
    N^{r} \left( N^{\frac{1-r}{2}} \right)^{1-r} = N^{\frac{1+r^2}{2}} &\text{for $r<1$}\\
    N^{r} &\text{for $r \geq 1$}\\
    \end{cases}
\end{equation}

\qed
\end{proof}

\begin{prop}
\label{th-expK}
The expected number of tables of the Powered Chinese Restaurant process evolves with $N \gg 1$ as $H_{\frac{r^2+1}{2}}(N)$ for $r<1$ and as $H_{r}(N)$ when $r \geq 1$, where $H_m(n)$ is the generalized harmonic number.
\end{prop}
\begin{proof}
In general, the expected number of clusters at the $N^{th}$ step can be written as:
\begin{equation}
\label{eq-expK}
    \mathbb{E}(K \vert N, r) = \sum_1^N \frac{\alpha}{\sum_k N_k^r + \alpha} \stackrel{N^r \gg 1}{\propto} \sum_1^N \frac{1}{\sum_k N_k^r}
\end{equation}

We showed in Proposition \ref{th-slowVarp} that we can rewrite $\sum_k N_k^r \propto N^{\frac{r^2 + 1}{2}}$ when $r<1$ and $\sum_k N_k^r \propto N^r$ when $r\geq 1$.
Injecting this result in Eq.\ref{eq-expK} for $r$, we get:
\begin{equation}
    \label{eq-expK-fin}
    \mathbb{E}(K \vert N, r) \stackrel{N^r \gg 1}{\propto}
    \begin{cases} \sum_1^N\frac{1}{N^{\frac{r^2 + 1}{2}}} = H_{\frac{r^2+1}{2}}(N)\\
    \sum_1^N\frac{1}{N^r} = H_{r}(N)
    \end{cases}
\end{equation}

\qed
\end{proof}

For $r=1$, $\mathbb{E}(K \vert N, r=1) \propto H_1(N) \approx \gamma + \log(N)$ where $\gamma$ is the Euler–Mascheroni constant, which is a classical result for the regular Dirichlet process.

When $r>1$ and $N \rightarrow \infty$, the term $H_{\frac{r^2+1}{2}}(N)$ converges towards a finite value and the sum $\sum_k p_k^r$ goes to 1 (see Proposition \ref{th-statio}). By definition $\mathbb{E}(K \vert N, r>1) \stackrel{N \rightarrow \infty}{\propto} \zeta(\frac{r^2+1}{2})$, where $\zeta$ is the Riemann Zeta function.

When $r < 1$, we can approximate the harmonic number in a continuous setting. We rewrite Eq.\ref{eq-expK-fin} as:
\begin{equation}
    \begin{split}
    \mathbb{E}(K \vert N, r) &\stackrel{N^r \gg 1}{\propto} \sum_{n=1}^N\frac{1}{n^{\frac{r^2 + 1}{2}}}
    \stackrel{N^r \gg 1}{\approx} \int_1^N n^{-\frac{r^2 + 1}{2}} dn
    = \frac{2}{1 - r^2}(N^{\frac{1-r^2}{2}}-1)
    \end{split}
\end{equation}
One can show that $\frac{N^{1-x}-1}{1-x} = H_x(N) + \mathcal{O}(\frac{1}{N^x})$. Therefore, the Powered Chinese Restaurant process exhibits a power-law behaviour similar to the Pitman-Yor process Eq.\ref{eq-PY} for $r = \sqrt{1 - 2\beta}$ for $0<r<1$. For values of $r>1 \Leftrightarrow \beta<0$, the equivalent Pitman-Yor process is not defined unlike the Powered Chinese Restaurant process. Note that there is \textit{a priori} no reason for $r$ to be constrained in the domain of real number. Complex analysis of the process might be an interesting lead for future works.

\section{Experiments}
\label{Experiments}
\subsection{Numerical validation of propositions}
First of all, we present numerical confirmations of propositions stated above (Propositions \ref{th-statio}, \ref{th-slowVarp}, \ref{th-expK}) by simulating 100 independent Powered Chinese Restaurant processes with parameter $\alpha=1$ for various values of $r$. We present the results of numerical simulations in Fig.\ref{fig-validTh}.

\begin{figure}
    \centering
    \includegraphics[width = \textwidth]{Ths_light.png}
    \caption{Numerical validation of Propositions \ref{th-statio} (left), \ref{th-slowVarp} (middle), \ref{th-expK} (right). In the third and fourth plots, the theoretical results are the solid lines and the associated numerical results are the transparent lines of same color. Except for small $N$, the difference between theory and experiments is almost indistinguishable.}
    \label{fig-validTh}
\end{figure}

On the \textbf{left} part, we plot the evolution of the probability for each cluster to be chosen as $N$ grows for $r=0.2$ for one run. We see that the probabilities do not remain constant but instead diminish as the number of clusters grows. The figure suggests they all converge to a common value (a uniform probability) as shown in Proposition \ref{th-statio}. The black line shows the probability of a uniform distribution. We chose not to show the results for $r>1$; in this case, one probability goes to 1 as the other fades to 0 as $N$ grows, as expected.

In the \textbf{middle} part of the figure, we plot the expression for $\sum_k N_k^r$ derived in Proposition \ref{th-slowVarp} (solid lines) versus the value of the sum from experimental results (transparent lines), averaged over 100 runs. Note that plots are in a log-log scale and that curves have been shifted vertically for visualization purposes. As assumed in Proposition \ref{th-slowVarp}, the approximation holds for all values of $r$.

Finally in the \textbf{right} picture, we plot the evolution of the number of clusters $K$ versus $N$ according to Proposition \ref{th-expK} (solid lines) and experiments (transparent lines). The error bars correspond to the standard deviation over the 100 runs. We see that the expression derived in Proposition \ref{th-expK} accounts well for the evolution of the number of clusters. Note that plots are in a log-log scale and that curves have been shifted vertically for visualization purposes. We must point out that there is a constant shift from experiments to the theory that does not appear on the plot (because of the rescaling). This shift comes from the approximation of large $N^r$ which is not valid at the beginning of the process. However, it does not play any role in the evolution of $K$ as $N$ grows large enough.

\subsection{Use case: infinite Gaussian mixture model}
We now illustrate the usefulness of a prior that alleviates the ``rich-get-richer'' property with specific synthetic datasets and with a real-world application. We choose to consider as an illustration its use as a prior in the infinite Gaussian mixture model\footnote{All codes and datasets can be found at https://anonymous.4open.science/r/91ea587e-fba6-4ba0-887e-79d87abf0b31/}. We choose this application to ease visual understanding of the implications of the P-CRP, but the argument holds for other models using DP priors as well (text modeling, gene expression clustering, etc.).

We consider a classical infinite Gaussian mixture model coupled with a Powered Dirichlet process prior. We fit the data using a standard collapsed Gibbs sampling algorithm for IGMM \cite{Rasmussen1999,wallach2010,Yin2014}, with a Normal Inverse Wishart prior on the Gaussians' parameters. The input data is shuffled at each iteration to reduce the ordering bias from the dataset. Note that we cannot completely get rid of the bias because the Powered Dirichlet Process is not exchangeable for all $r$. The problem has been addressed on numerous occasions (Uniform process \cite{wallach2010}, distance-dependent CRP \cite{Blei2011,Ghosh2014}, spectral CRP \cite{Socher2011}) and shown to induce negligible variations of results in the case of Gibbs sampling. We stop the sampler once the likelihood of the model reaches stability
; we repeat this procedure 100 times for each value of $r$.
Finally, the parameter $\alpha$ is set to 1 in all experiments (see Section \ref{motivation}).

\subsubsection{Synthetic data}
\begin{figure}
    \centering
    \includegraphics[width = \columnwidth]{FigFin.png}
    \caption{Application for synthetic data. (Top) Original datasets used for the experiments. (Bottom) Results for various values of $r$; the x and y axes all the same. The dashed line indicates the regular DP prior as $r=1$. The error correspond to the standard error of the mean over all runs.}
    \label{fig-Results}
\end{figure}
We present the results on synthetic data in Fig.\ref{fig-Results}. We consider standard metrics in clustering evaluation with a non-fixed number of clusters: mutual information score and rand index both adjusted for chance, normalized variation of information, Fowlkes-Mallow score, marginal likelihood (normalized for visualization) and absolute relative variation of the inferred number of clusters according to the number used in the generation process. Note that we purposely chose stereotypical cases to illustrate the argument better. The dataset on the \textbf{left} of Fig.\ref{fig-Results} is informative about the change induced by $r$. Here, clusters are distributed at various scales in the dataset; we see that the lower the value of $r$, the better the results. Indeed, when $r$ is small, the model can distinguish clusters in the dense area better, whereas when $r$ is closer to 1, the clusters in the dense area are put together in a larger cluster. The same happens with the dataset in the \textbf{middle} of Fig.\ref{fig-Results}, where clusters are distributed according to two different scales. Finally, on the \textbf{right} part of Fig.\ref{fig-Results}, we see an optimum $r$ exists to distinguish the clusters distributed on a grid; it makes sense since only one scale in clusters distribution is involved in this dataset.

\subsubsection{Real data}
\begin{figure}
    \centering
    \includegraphics[width = \columnwidth]{Antoni-150-200.pdf}
    \caption{Application to spatial clustering on geolocated data for $r=0.8$ (left), $r=1$ (right) and $r=0.9$ (middle). We see that the Powered Chinese Restaurant process for $r=0.9$ and $r=0.8$ describes the data better than the classical one for $r=1$.}
    \label{fig-Antoni}
\end{figure}
We now illustrate the interest of using an alternate form of prior for the Infinite Gaussian Mixture model on real-world data. We consider a dataset of 4.300 roman sepulchral inscriptions comprising the substring ``Antoni'' that have been dated between 150AC and 200AC and assigned with map coordinates. The dates correspond to the reign of Antoninus Pius over the Roman empire. The dataset is available on Clauss-Slaby repository\footnote{http://www.manfredclauss.de/fr/index.html}. It was common to give children or slaves the name of the emperor; the dataset gives a global idea of the main areas of the roman empire at that time \cite{Hanson2017}. The task here is to discover spatial clusters of individuals named after the emperor. We expect to find geographical clusters around: Italy, Egypt, Gauls, Judea, and all along the \textit{limes} (borders of the roman empire, which concentrate lots of sepulchral inscriptions for war-related reasons) \cite{Hanson2016}. We present the results for various values of $r$ in Fig.\ref{fig-Antoni}.

We see that when $r=1$, the classical CRP prior is not fit for describing this dataset, as it misses most of the clusters. On the other hand, when $r=0.9$, the infinite Gaussian mixture model retrieves the expected clusters. It also makes some clusters that were not expected, such as the north Italian cluster or the long cluster going through Spain and France that corresponds to roman roads layout (via Augusta and via Agrippa; it was common to bury the dead on roads edges). Finally, when $r=0.8$, we get even more detail: some of the main clusters are broken into smaller ones (Italy breaks into Rome, North Italy, and South Italy; Britain becomes an independent cluster, etc.). In this case, changing $r$ controls the level of details of the clustering. We see how different results can be according to the extent the model relies on the ``rich-get-richer'' prior and how it is needed to control it to make modeling relevant to every situation.

\section{Conclusion}
In this article, we discuss the necessity of controlling the ``rich-get-richer'' property that arises from the common Chinese Restaurant Process usual formulation. We discuss cases where this modeling hypothesis must be alleviated or strengthened to describe data more accurately. To this end, we derive the Powered Chinese Restaurant Process from a powered version of the Dirichlet-Multinomial distribution. This formulation allows reducing the expected number of clusters, which is not possible in the standard Pitman-Yor processes, while generalizing the standard Dirichlet process and the Uniform process. The principal feature of this formulation is that it allows for direct control of the ``rich-get-richer'' priors' importance. We derive elementary results on convergence and the expected number of clusters of the new process. Finally, we show that it yields better results on synthetic data and illustrates a possible use case with real-world data. For future works, it might be interesting to investigate cases where $r$ takes non-positive values (which might lead to a ``poor-get-richer'' kind of process) or complex values (for $r=a+ib$ the prior probability $N^r$ would have an amplitude $N^a$ and a phase $b \log N$).

The regular Chinese Restaurant process has been used for decades as a powerful prior in many real-world applications. However, alternate forms for this prior have been little explored. It would be interesting to consider the changes brought to state-of-the-art models by varying the importance of the ``rich-get-richer'' prior as proposed in this paper.

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