## taking advantage of the constant

Posted in Books, Kids, pictures, R, Statistics, University life with tags , , , , , , , , on May 19, 2022 by xi'an

A question from X validated had enough appeal for me to procrastinate about it for ½ an hour: what difference does it make [for simulation purposes] that a target density is properly normalised? In the continuous case, I do not see much to exploit about this knowledge, apart from the value potentially leading to a control variate (in a Gelfand and Dey 1996 spirit) and possibly to a stopping rule (by checking that the portion of the space visited so far has mass close to one, but this is more delicate than it sounds).

In a (possibly infinite) countable setting, it seems to me one gain (?) is that approximating expectations by Monte Carlo no longer requires iid simulations in the sense that once visited,  atoms need not be visited again. Self-avoiding random walks and their generalisations thus appear as a natural substitute for MC(MC) methods in this setting, provided finding unexplored atoms proves manageable. For instance, a stopping rule is always available, namely that the cumulated weight of the visited fraction of the space is close enough to one. The above picture shows a toy example on a 500 x 500 grid with 0.1% of the mass remaining at the almost invisible white dots. (In my experiment, neighbours for the random exploration were chosen at random over the grid, as I assumed no global information was available about the repartition over the grid either of mass function or of the function whose expectation was seeked.)

## put the data aside [SCOTUS v. evidence]

Posted in Statistics with tags , , , , , , , , , , , , on May 18, 2022 by xi'an

Reposted from a Nature editorial:

(…) Moving in the opposite direction runs contrary to 50 years of research from around the world showing that abortion access is a crucial component of health care and is important for women’s equal participation in society. After the Supreme Court agreed to hear Mississippi’s case last year, Nature covered some of this evidence, submitted to the court by US scientific societies and more than 800 US researchers in public health, reproductive health, social sciences and economics, to the court in advance of the case’s hearing in December.

Some outcomes of outlawing abortion can be predicted by what’s known. Researchers expect overall infant and maternal health to decline in the United States in the wake of abortion bans, because more unintended pregnancies will be brought to term. Unintended pregnancies are associated with an increased risk of health problems for babies, and often for mothers, for several reasons — including reduced prenatal care.

Maternal health is also expected to decline overall. One straightforward reason is that the risks of dying from pregnancy-related causes are much greater than the risks of dying because of a legal abortion. A predicted rise in maternal mortality among Black women in the United States is particularly distressing, because the rate is already unacceptably high. In one study, sociologist Amanda Stevenson at the University of Colorado Boulder modelled a hypothetical situation in which abortions are banned throughout the United States, and found that the lifetime risk of dying from pregnancy-related causes for non-Hispanic Black women would rise from 1 in 1,300 to 1 in 1,000.

One claim made by abortion opponents in this case is that abortions no longer benefit women and even cause them harm, but dozens of studies contradict this. In just one, health economist Sarah Miller at the University of Michigan in Ann Arbor and her colleagues assessed around 560 women of comparable age and financial standing who sought abortions. They found that, five years after pregnancy, women who were denied the procedure had experienced a substantial increase in debt, bankruptcies, evictions and other dire financial events — whereas the financial standing of women who had received an abortion had remained stable or improved. A primary reason that women give for wanting an abortion is an inability to afford to raise the child, and this study suggests that they understand their own situations.

Abortion bans will extract an unequal toll on society. Some 75% of women who choose to have abortions are in a low income bracket and nearly 60% already have children, according to one court brief submitted ahead of the December hearing and signed by more than 150 economists. Travelling across state lines to receive care will be particularly difficult for people who do not have the funds for flights or the ability to take time off work, or who struggle to find childcare.

Unfortunately, some of the justices seem to be disregarding these data. At the December hearing, Julie Rikelman, a lawyer at the non-profit Center for Reproductive Rights, headquartered in New York City, brought up studies presented in the economists’ brief; Roberts interrupted her and suggested “putting that data aside”. In the leaked draft opinion, Alito also elides a body of research on abortion policy, writing that it’s “hard for anyone — and in particular for a court — to assess” the effect of the right to abortion on women’s lives.

Such an attitude suggests that the justices see research as secondary to the question of whether the US Constitution should protect abortion. But the outcome of this ruling isn’t an academic puzzle. The Supreme Court needs to accept that the consensus of research, knowledge and scholarship — the evidence on which societies must base their laws — shows how real lives hang in the balance. Already, the United States claims the highest rate of maternal and infant mortality among wealthy nations. Should the court overturn Roe v. Wade, these grim statistics will only get worse.

Posted in Books, Statistics, Travel, University life with tags , , , , , on April 4, 2022 by xi'an

The Royal Statistical Society is launching a series of discussions linked with the UK Government handling of the COVID-19 pandemic (and of the related data):

• Communication during the pandemic: Data, statistical analyses and modelling, 5 April
Organising panel of David Spiegelhalter, Tom Chivers and Jen Rogers
Register for the in-person or online event
• Governments’ statistical resources, 3 May
Organising panel of Simon Briscoe and Gavin Freeguard
• Evidence and policy, 21 June
Organising panel of Sylvia Richardson, Dani De Angelis and John Aston
• Evaluation, 12 July
Organising panel of Sheila Bird, Christl Donnelly and Max Parmar.

## robust inference using posterior bootstrap

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , on February 18, 2022 by xi'an

The famous 1994 Read Paper by Michael Newton and Adrian Raftery was entitled Approximate Bayesian inference, where the boostrap aspect is in randomly (exponentially) weighting each observation in the iid sample through a power of the corresponding density, a proposal that happened at about the same time as Tony O’Hagan suggested the related fractional Bayes factor. (The paper may also be equally famous for suggesting the harmonic mean estimator of the evidence!, although it only appeared as an appendix to the paper.) What is unclear to me is the nature of the distribution g(θ) associated with the weighted bootstrap sample, conditional on the original sample, since the outcome is the result of a random Exponential sample and of an optimisation step. With no impact of the prior (which could have been used as a penalisation factor), corrected by Michael and Adrian via an importance step involving the estimation of g(·).

At the Algorithm Seminar today in Warwick, Emilie Pompe presented recent research, including some written jointly with Pierre Jacob, [which I have not yet read] that does exactly that inclusion of the log prior as penalisation factor, along with an extra weight different from one, as motivated by the possibility of a misspecification. Including a new approach to cut models. An alternative mentioned during the talk that reminds me of GANs is to generate a pseudo-sample from the prior predictive and add it to the original sample. (Some attendees commented on the dependence of the later version on the chosen parameterisation, which is an issue that had X’ed my mind as well.)

## distributed evidence

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , , , on December 16, 2021 by xi'an

Alexander Buchholz (who did his PhD at CREST with Nicolas Chopin), Daniel Ahfock, and my friend Sylvia Richardson published a great paper on the distributed computation of Bayesian evidence in Bayesian Analysis. The setting is one of distributed data from several sources with no communication between them, which relates to consensus Monte Carlo even though model choice has not been particularly studied from that perspective. The authors operate under the assumption of conditionally conjugate models, i.e., the existence of a data augmentation scheme into an exponential family so that conjugate priors can be used. For a division of the data into S blocks, the fundamental identity in the paper is

$p(y) = \alpha^S \prod_{s=1}^S \tilde p(y_s) \int \prod_{s=1}^S \tilde p(\theta|y_s)\,\text d\theta$

where α is the normalising constant of the sub-prior exp{log[p(θ)]/S} and the other terms are associated with this prior. Under the conditionally conjugate assumption, the integral can be approximated based on the latent variables. Most interestingly, the associated variance is directly connected with the variance of

$p(z_{1:S}|y)\Big/\prod_{s=1}^S \tilde p(z_s|y_s)$

under the joint:

“The variance of the ratio measures the quality of the product of the conditional sub-posterior as an importance sample proposal distribution.”

Assuming this variance is finite (which is likely). An approximate alternative is proposed, namely to replace the exact sub-posterior with a Normal distribution, as in consensus Monte Carlo, which should obviously require some consideration as to which parameterisation of the model produces the “most normal” (or the least abnormal!) posterior. And ensures a finite variance in the importance sampling approximation (as ensured by the strong bounds in Proposition 5). A problem shared by the bridgesampling package.

“…if the error that comes from MCMC sampling is relatively small and that the shard sizes are large enough so that the quality of the subposterior normal approximation is reasonable, our suggested approach will result in good approximations of the full data set marginal likelihood.”

The resulting approximation can also be handy in conjunction with reversible jump MCMC, in the sense that RJMCMC algorithms can be run in parallel on different chunks or shards of the entire dataset. Although the computing gain may be reduced by the need for separate approximations.