Archive for loss function
Bayesian privacies [slides]
Posted in Books, Statistics, Travel, University life with tags adversarial privacy, Bayesian privacy, differential privacy, inference, loss function, multi-agent decision theory, optimal decision, persuasive privacy, seminar, slides, slideshare, summary statistics, Università degli studi di Padova, utility on April 4, 2026 by xi'an
the three i’s of poverty
Posted in Books, pictures, Statistics, Travel, University life with tags Gibbs sampling, loss function, Marseille, mixture of distributions, thesis defence, three i's of poverty on September 15, 2019 by xi'an
Today I made a “quick” (10h door to door!) round trip visit to Marseille (by train) to take part in the PhD thesis defense (committee) of Edwin Fourrier-Nicolaï, which title was Poverty, inequality and redistribution: an econometric approach. While this was mainly a thesis in economics, meaning defending some theory on inequalities based on East German data, there were Bayesian components in the thesis that justified (to some extent!) my presence in the jury. Especially around mixture estimation by Gibbs sampling. (On which I started working almost exactly 30 years ago, when I joined Paris 6 and met Gilles Celeux and Jean Diebolt.) One intriguing [for me] question stemmed from this defense, namely the notion of a Bayesian estimation of a three i’s of poverty (TIP) curve. The three i’s stand for incidence, intensity, and inequality, as, introduced in Jenkins and Lambert (1997), this curve measure the average income loss from the poverty level for the 100p% lower incomes, when p varies between 0 and 1. It thus depends on the distribution F of the incomes and when using a mixture distribution its computation requires a numerical cdf inversion to determine the income p-th quantile. A related question is thus on how to define a Bayesian estimate of the TIP curve. Using an average over the values of an MCMC sample does not sound absolutely satisfactory since the upper bound in the integral varies for each realisation of the parameter. The use of another estimate would however require a specific loss function, an issue not discussed in the thesis.
admissible estimators that are not Bayes
Posted in Statistics with tags admissibility, Bayes estimators, Cornell University, decision theory, exponential families, hypothesis testing, loss function on December 30, 2017 by xi'an
A question that popped up on X validated made me search a little while for point estimators that are both admissible (under a certain loss function) and not generalised Bayes (under the same loss function), before asking Larry Brown, Jim Berger, or Ed George. The answer came through Larry’s book on exponential families, with the two examples attached. (Following our 1989 collaboration with Roger Farrell at Cornell U, I knew about the existence of testing procedures that were both admissible and not Bayes.) The most surprising feature is that the associated loss function is strictly convex as I would have thought that a less convex loss would have helped to find such counter-examples.
MAP as Bayes estimators
Posted in Books, Kids, Statistics with tags Bayesian inference, counterexample, loss function, MAP estimators, posterior distribution on November 30, 2016 by xi'an
Robert Bassett and Julio Deride just arXived a paper discussing the position of MAPs within Bayesian decision theory. A point I have discussed extensively on the ‘Og!
“…we provide a counterexample to the commonly accepted notion of MAP estimators as a limit of Bayes estimators having 0-1 loss.”
The authors mention The Bayesian Choice stating this property without further precautions and I completely agree to being careless in this regard! The difficulty stands with the limit of the maximisers being not necessarily the maximiser of the limit. The paper includes an example to this effect, with a prior as above, associated with a sampling distribution that does not depend on the parameter. The sufficient conditions proposed therein are that the posterior density is almost surely proper or quasiconcave.
This is a neat mathematical characterisation that cleans this “folk theorem” about MAP estimators. And for which the authors are to be congratulated! However, I am not very excited by the limiting property, whether it holds or not, as I have difficulties conceiving the use of a sequence of losses in a mildly realistic case. I rather prefer the alternate characterisation of MAP estimators by Burger and Lucka as proper Bayes estimators under another type of loss function, albeit a rather artificial one.
ISBA 2016 [#3]
Posted in pictures, Running, Statistics, Travel, University life, Wines with tags ABC, approximate likelihood, Calasetta, ISBA 2016, j-ISBA, loss function, restricted inference, San' Antioco, Sardinia, Statistical Science, summary statistics on June 16, 2016 by xi'an
Among the sessions I attended yesterday, I really liked the one on robustness and model mispecification. Especially the talk by Steve McEachern on Bayesian inference based on insufficient statistics, with a striking graph of the degradation of the Bayes factor as the prior variance increases. I sadly had no time to grab a picture of the graph, which compared this poor performance against a stable rendering when using a proper summary statistic. It clearly relates to our work on ABC model choice, as well as to my worries about the Bayes factor, so this explains why I am quite excited about this notion of restricted inference. In this session, Chris Holmes also summarised his two recent papers on loss-based inference, which I discussed here in a few posts, including the Statistical Science discussion Judith and I wrote recently. I also went to the j-ISBA [section] session which was sadly under-attended, maybe due to too many parallel sessions, maybe due to the lack of unifying statistical theme.
