Estimating means of bounded random variables by betting

Ian Waudby-Smith and Aaditya Ramdas are presenting next month a Read Paper to the Royal Statistical Society in London on constructing a conservative confidence interval on the mean of a bounded random variable. Here is an extended abstract from within the paper:

For each m ∈ [0, 1], we set up a “fair” multi-round game of statistician
against nature whose payoff rules are such that if the true mean happened
to equal m, then the statistician can neither gain nor lose wealth in
expectation (their wealth in the m-th game is a nonnegative martingale),
but if the mean is not m, then it is possible to bet smartly and make
money. Each round involves the statistician making a bet on the next
observation, nature revealing the observation and giving the appropriate
(positive or negative) payoff to the statistician. The statistician then plays
all these games (one for each m) in parallel, starting each with one unit of
wealth, and possibly using a different, adaptive, betting strategy in each.
The 1 − α confidence set at time t consists of all m 2 [0, 1] such that the
statistician’s money in the corresponding game has not crossed 1/α. The
true mean μ will be in this set with high probability.

I read the paper on the flight back from Venice and was impressed by its universality, especially for a non-asymptotic method, while finding the expository style somewhat unusual for Series B, with notions late into being defined if at all defined. As an aside, I also enjoyed the historical connection to Jean Ville‘s 1939 PhD thesis (examined by Borel, Fréchet—his advisor—and Garnier) on a critical examination of [von Mises’] Kollektive. (The story by Glenn Shafer of Ville’s life till the war is remarkable, with the de Beauvoir-Sartre couple making a surprising and rather unglorious appearance!). Himself inspired by a meeting with Wald while in Berlin. The paper remains quite allusive about Ville‘s contribution, though, while arguing about its advance respective to Ville’s work… The confidence intervals (and sequences) depend on a supermartingale construction of the form

which allows for a universal coverage guarantee of the derived intervals (and can optimised in λ). As I am getting confused by that point about the overall purpose of the analysis, besides providing an efficient confidence construction, and am lacking in background about martingales, betting, and sequential testing, I will not contribute to the discussion. Especially since ChatGPT cannot help me much, with its main “criticisms” (which I managed to receive while in Italy, despite the Italian Government banning the chabot!)

However, there are also some potential limitations and challenges to this approach. One limitation is that the accuracy of the method is dependent on the quality of the prior distribution used to set the odds. If the prior distribution is poorly chosen, the resulting estimates may be inaccurate. Additionally, the method may not work well for more complex or high-dimensional problems, where there may not be a clear and intuitive way to set up the betting framework.

and

Another potential consequence is that the use of a betting framework could raise ethical concerns. For example, if the bets are placed on sensitive or controversial topics, such as medical research or political outcomes, there may be concerns about the potential for manipulation or bias in the betting markets. Additionally, the use of betting as a method for scientific or policy decision-making may raise questions about the appropriate role of gambling in these contexts.

being totally off the radar… (No prior involved, no real-life consequence for betting, no gambling.)

International Women’ Day: EU solidarity [reposted]

EP President Roberta Metsola said: ‘‘I want to start by commending the incredibly brave women of Ukraine who are fighting, forced to shelter their loved ones in bunkers, giving birth in metro stations and leading on the frontline. It is a testament to their courage, strength and resilience even in the worst circumstances.’’

Women’s Rights and Gender Equality Committee Chair Robert Biedroń (S&D, PL), declared: ‘‘A few days ago, in the very heart of Europe, Russia launched a brutal war on Ukraine, entirely in breach of international agreements, human rights and our values. I condemn this aggression that has led to the senseless deaths of many Ukrainian soldiers (20% of whom are women) and civilians. Over a million refugees have crossed the border now, mostly women (some of them pregnant) and children. We need to ensure proper assistance is provided to them.’’

French Minister Delegate for Gender Equality, Diversity and Equal Opportunities Élisabeth Moreno, added: ‘‘I salute the courage and determination of the Ukrainian people, of these women and men who are facing a wave of unanimously condemned violence. Let us remember that women and girls are always the first victims of crises. Despite this, they are excluded from strategic discussions and exchanges on security and peace.’’

The Prime Minister of Iceland Katrín Jakobsdóttir declared: ”We must put pressure on Russia to immediately stop this war. Our thoughts are with the people of Ukraine. Gender equality should always be on our agenda, even on times like these. The world would be more peaceful with more women in the lead, I have no doubt about that.”

Ivanna Klympush-Tsintsadze, Chair of the Verkhovna Rada’s Committee for Ukraine’s integration into the EU, made the following statement: ‘‘I would have loved to join you today to discuss gender equality but I will talk about war. Thousands of women with children are trying to relocate to a safer place. Some have difficulties accessing basic products like baby food, while others are giving birth in shelters. Elderly people are sometimes left alone at home with no access to medicine or food. I urge your governments to declare a no-fly zone over Ukraine, to provide massive military support and to engage right now in providing humanitarian corridors for women, children and elderly people. Please do stand with Ukraine and help us preserve ourselves and the values we are standing for.’’

a journal of the plague year² [600+]

Returned to Warwick for the first time in 600 days!, most pleasantly reconnecting with my colleagues there and realising some were almost as freshly back to the department as myself. But also noticed a strong difference with France in terms of wearing mask and practicing social distanciation among the students (at the University) and the general population (in the local Tesco or the train to the airport). Which may explain for the persistently high number of contaminations, when compared with neighbouring countries. Despite its high vaccination rate.

Read the second volume of Baru Cormorant, after enjoying so much the first instalment (under the Corsican relentless sun). However, it was such a disappointment, as it seemed written by a completely different author, including the style, with the story being more broken, more difficult to follow, and the characters becoming shallow and uninteresting. This is particularly true of Baru, who sent from a sharp focus on her goal in Traitor, to a purposeless floatsam in Monster. With a highly artificial rescue of her ethics in [warning, spoiler ahead!] killing her lover Tain Hu (and hundreds of others) in the first volume. I currently doubt I will buy the third one… A stormy night kept me awake and as a result help me finish the last hundred pages!

Watched Mute, dubbed a “Netflix disaster” in The Guardian! (and following the appalling Warcraft!). Which postulate of a mute character could have been rewarding, had not the characters be of cardboard consistency. And the plot so transparent most of the scenes had to be shot at night. And the cheap plagiarism of Blade Runner is simply ludicrous. I also watched Bong Joon-Ho’s Okja, a satire about the meat industry producing hippo-like pigs that I found very heavy-handed, especially in its characters.Started cooking bread on a regular basis once again, as weather is turning colder and baking also warms up our drafty kitchen! Now turning to a heavier type of loaf, mixed wheat and rye, and reducing the amount of water to make it last longer. And also made a first attempt at Okonomyaki, this tortilla-like dish made of shredded cabbage and flour that I enjoyed very much in Osaka. The result was pleasant enough but very, very far from the original, maybe due to my using (soba) buckwheat rather than plain flour. Or using regular stock rather than daishi stock.

Einstein postdoc fellowship in Bayesian Berlin

Najda Klein , Professor at HU Berlin, is pointing out the availability of the Einstein postdoc fellowships to work with her on Bayesian topics next year, with deadline for the application being 26 March 2021. (Interested PhDs should contact her before this date.)

Berliner Schnauze