[more than] everything you always wanted to know about marginal likelihood

Earlier this year, F. Llorente, L. Martino, D. Delgado, and J. Lopez-Santiago have arXived an updated version of their massive survey on marginal likelihood computation. Which I can only warmly recommend to anyone interested in the matter! Or looking for a base camp to initiate a graduate project. They break the methods into four families

1. Deterministic approximations (e.g., Laplace approximations)
2. Methods based on density estimation (e.g., Chib’s method, aka the candidate’s formula)
3. Importance sampling, including sequential Monte Carlo, with a subsection connecting with MCMC
4. Vertical representations (mostly, nested sampling)

Besides sheer computation, the survey also broaches upon issues like improper priors and alternatives to Bayes factors. The parts I would have done in more details are reversible jump MCMC and the long-lasting impact of Geyer’s reverse logistic regression (with the noise contrasting extension), even though the link with bridge sampling is briefly mentioned there. There is even a table reporting on the coverage of earlier surveys. Of course, the following postnote of the manuscript

The Christian Robert’s blog deserves a special mention , since Professor C. Robert has devoted several entries of his blog with very interesting comments regarding the marginal likelihood estimation and related topics.

does not in the least make me less objective! Some of the final recommendations

• use of Naive Monte Carlo [simulate from the prior] should be always considered [assuming a proper prior!]
  
• a multiple-try method is a good choice within the MCMC schemes
• optimal umbrella sampling estimator is difficult and costly to implement , so its best performance may not be achieved in practice
• adaptive importance sampling uses the posterior samples to build a suitable normalized proposal, so it benefits from localizing samples in regions of high posterior probability while preserving the properties of standard importance sampling
  
• Chib’s method is a good alternative, that provide very good performances [but is not always available]
• the success [of nested sampling] in the literature is surprising.

SADA’16

mostly nuisance, little interest

Sorry for the misleading if catchy (?) title, I mean mostly nuisance parameters, very few parameters of interest! This morning I attended a talk by Eric Lesage from CREST-ENSAI on non-responses in surveys and their modelling through instrumental variables. The weighting formula used to compensate for the missing values was exactly the one at the core of the Robins-Wasserman paradox, discussed a few weeks ago by Jamie in Varanasi. Namely the one with the estimated probability of response at the denominator: The solution adopted in the talk was obviously different, with linear estimators used at most steps to evaluate the bias of the procedure (since researchers in survey sampling seem particularly obsessed with bias!)

On a somehow related topic, Aris Spanos arXived a short note (that I read yesterday) about the Neyman-Scott paradox. The problem is similar to the Robins-Wasserman paradox in that there is an infinity of nuisance parameters (the means of the successive pairs of observations) and that a convergent estimator of the parameter of interest, namely the variance common to all observations, is available. While there exist Bayesian solutions to this problem (see, e.g., this paper by Brunero Liseo), they require some preliminary steps to bypass the difficulty of this infinite number of parameters and, in this respect, are involving ad-hocquery to some extent, because the prior is then designed purposefully so. In other words, missing the direct solution based on the difference of the pairs is a wee frustrating, even though this statistic is not sufficient! The above paper by Brunero also my favourite example in this area: when considering a normal mean in large dimension, if the parameter of interest is the squared norm of this mean, the MLE ||x||² (and the Bayes estimator associated with Jeffreys’ prior) is (are) very poor: the bias is constant and of the order of the dimension of the mean, p. On the other hand, if one starts from ||x||² as the observation (definitely in-sufficient!), the resulting MLE (and the Bayes estimator associated with Jeffreys’ prior) has (have) much nicer properties. (I mentioned this example in my review of Chang’s book as it is paradoxical, gaining in efficiency by throwing away “information”! Of course, the part we throw away does not contain true information about the norm, but the likelihood does not factorise and hence the Bayesian answers differ…)

I showed the paper to Andrew Gelman and here are his comments:

Spanos writes, “The answer is surprisingly straightforward.” I would change that to, “The answer is unsurprisingly straightforward.” He should’ve just asked me the answer first rather than wasting his time writing a paper!

The way it works is as follows. In Bayesian inference, everything unknown is unknown, they have a joint prior and a joint posterior distribution. In frequentist inference, each unknowns quantity is either a parameter or a predictive quantity. Parameters do not have probability distributions (hence the discomfort that frequentists have with notation such as N(y|m,s); they prefer something like N(y;m,s) or f_N(y;m,s)), while predictions do have probability distributions. In frequentist statistics, you estimate parameters and you predict predictors. In this world, estimation and prediction are different. Estimates are evaluated conditional on the parameter. Predictions are evaluated conditional on model parameters but unconditional on the predictive quantities. Hence, mle can work well in many high-dimensional problems, as long as you consider many of the uncertain quantities as predictive. (But mle is still not perfect because of the problem of boundary estimates, e.g., here..

WSC 2011, Phoenix

WSC stands for Winter Simulation Conference. I was not aware of those conferences that have been running since 1971 but I have been invited to give a tutorial on simulation in statistics at the 2011 edition of WSC in Phoenix, Arizona. Since it means meeting a completely new community of people using simulation, mostly outside statistics, I accepted the invitation and will thus spend a few days in December in Arizona… (If the culture gap proves too wide, I can always bike, run, or even climb in South Mountain Park!) In connection with this tutorial, I was requested to write a short paper that I just posted on arXiv. There is not much originality in the survey as it is mostly inspired from older chapters written for handbooks. I wished I had more space to cover particle MCMC in a few pages but 12 pages was the upper limit.

Creationism in UK schools?!

“It was found that Britons were almost three times more likely than Egyptians to want creationism and intelligent design to be included in the teaching of evolution.” The Guardian, October 25, 2009

I was reading the Guardian on the flight to Madrid and there was this terrible statistics that 54% of the UK public wanted creationism to be “taught” in public schools. This is worse than in the US… Quite an appaling statistic for the Darwin year and Darwin’s country! Of course, this kind of result should be taken with a pinch of salt since the same Guardian reports that four Britons out of five repudiate creationism, along with a paraodical region-by-region map where each region has more than 25% of the people in favour of creationism or “intelligent” design! (The quote above is rather dumb as well! Why should Egyptians be more in favour of creationism?!) Here is a blog reporting more clearly on a similar study, conducted by Science, with France coming almost on top of Darwinian countries! Except that Britain is fairly close to the top as well. So this may end up being a catchy title making too much of a limited or poorly conducted survey. To conclude that “More than half of British adults think that intelligent design and creationism should be taught alongside evolution in schoolscience lessons” based on 973 Britons is extrapolation to the third power…