**I**n 2018, Panayiota Touloupou, research fellow at Warwick, and her co-authors published a paper in Bayesian analysis that somehow escaped my radar, despite standing in my first circle of topics of interest! They construct an importance sampling approach to the approximation of the marginal likelihood, the importance function being approximated from a preliminary MCMC run, and consider the special case when the sampling density (i.e., the likelihood) can be represented as the marginal of a joint density. While this demarginalisation perspective is rather usual, the central point they make is that it is more efficient to estimate the sampling density based on the auxiliary or latent variables than to consider the joint posterior distribution of parameter and latent in the importance sampler. This induces a considerable reduction in dimension and hence explains (in part) why the approach should prove more efficient. Even though the approximation itself is costly, at about 5 seconds per marginal likelihood. But a nice feature of the paper is to include the above graph that includes both computing time and variability for different methods (the blue range corresponding to the marginal importance solution, the red range to RJMCMC and the green range to Chib’s estimate). Note that bridge sampling does not appear on the picture but returns a variability that is similar to the proposed methodology.

## Archive for marginal likelihood

## marginal likelihood with large amounts of missing data

Posted in Books, pictures, Statistics with tags Bayesian Analysis, Chib's approximation, evidence, harmonic mean estimator, importance sampling, marginal likelihood, normalising constant, reversible jump, University of Warwick on October 20, 2020 by xi'an## marginal likelihood as exhaustive X validation

Posted in Statistics with tags Bayesian inference, Biometrika, coherence, Cross Validation, exchangeability, intrinsic Bayes factor, likelihood-free inference, marginal likelihood, pseudo-Bayes factors on October 9, 2020 by xi'an*I*n the June issue of Biometrika (for which I am deputy editor) Edwin Fong and Chris Holmes have a short paper (that I did not process!) on the validation of the marginal likelihood as the unique coherent updating rule. Marginal in the general sense of Bissiri et al. (2016). Coherent in the sense of being invariant to the order of input of exchangeable data, if in a somewhat self-defining version (Definition 1). As a consequence, marginal likelihood arises as the *unique* prequential scoring rule under coherent belief updating in the Bayesian framework. (It is unique given the prior or its generalisation, obviously.)

*“…we see that 10% of terms contributing to the marginal likelihood come from out-of-sample predictions, using on average less than 5% of the available training data.”*

The paper also contains the interesting remark that the log marginal likelihood is the average leave-*p*-out X-validation score, across all values of *p*. Which shows that, provided the marginal can be approximated, the X validation assessment is feasible. Which leads to a highly relevant (imho) spotlight on how this expresses the (deadly) impact of the prior selection on the numerical value of the marginal likelihood. Leaving outsome of the least informative terms in the X-validation leads to exactly the log geometric intrinsic Bayes factor of Berger & Pericchi (1996). Most interesting connection with the Bayes factor community but one that depends on the choice of the dismissed fraction of *p*‘s.

## estimating the marginal likelihood (or an information criterion)

Posted in Books, pictures, Statistics, University life with tags BayesComp 2020, BIC, DIC, estimating constants, marginal likelihood, mixture Bayesian estimation, power likelihood, real log canonical threshold, sBIC, thermodynamic integration, University of Warwick, WBIC on December 28, 2019 by xi'an**T**ory Imai (from Kyoto University) arXived a paper last summer on what first looked like a novel approximation of the marginal likelihood. Based on the variance of thermodynamic integration. The starting argument is that there exists a power 0<t⁰<1 such that the expectation of the logarithm of the product of the prior by the likelihood to the power t⁰ or t⁰-powered likelihood is equal to the standard log-marginal

when the expectation is under the posterior corresponding to the t⁰-powered likelihood (rather than the full likelihood). By an application of the mean value theorem. Watanabe’s (2013) WBIC replaces the optimum t⁰ with 1/log(n), n being the sample size. The issue in terms of computational statistics is of course that the error of WBIC (against the true log m(x)) is only characterised as an order of n.

The second part of the paper is rather obscure to me, as the motivation for the real log canonical threshold is missing, even though the quantity is connected with the power likelihood. And the DIC effective dimension. It then goes on to propose a new approximation of sBIC, where *s* stands for *singular*, of Drton and Plummer (2017) which I had missed (and may ask my colleague Martin later today at Warwick!). Quickly reading through the later however brings explanations about the real log canonical threshold being simply the effective dimension in Schwarwz’s BIC approximation to the log marginal,

(as derived by Watanabe), where m is called the multiplicity of the real log canonical threshold. Both λ and m being unknown, Drton and Plummer (2017) estimate the above approximation in a Bayesian fashion, which leads to a double indexed marginal approximation for a collection of models. Since this thread leads me further and further from a numerical resolution of the marginal estimation, but brings in a different perspective on mixture Bayesian estimation, I will return to this highly in a later post. The paper of Imai discusses a different numerical approximation to sBIC, With a potential improvement in computing sBIC. (The paper was proposed as a poster to BayesComp 2020, so I am looking forward discussing it with the author.)

## an arithmetic mean identity

Posted in Books, pictures, R, Statistics, Travel, University life with tags arithmetic mean, Bayesian Analysis, conjugate normal model, harmonic mean estimator, HPD region, JSM 2009, marginal likelihood, MaxEnt2009, mixture example, Monte Carlo approximations, nested sampling, R, Université Paris Dauphine on December 19, 2019 by xi'an**A **2017 paper by Ana Pajor published in Bayesian Analysis addresses my favourite problem [of computing the marginal likelihood] and which I discussed on the ‘Og, linking with another paper by Lenk published in 2012 in JCGS. That I already discussed here last year. Lenk’s (2009) paper is actually using a technique related to the harmonic mean correction based on HPD regions Darren Wraith and myself proposed at MaxEnt 2009. And which Jean-Michel and I presented at Frontiers of statistical decision making and Bayesian analysis in 2010. As I had only vague memories about the arithmetic mean version, we discussed the paper together with graduate students in Paris Dauphine.

The arithmetic mean solution, representing the marginal likelihood as the *prior* average of the likelihood, is a well-known approach used as well as the basis for nested sampling. With the improvement consisting in restricting the simulation to a set Ð with sufficiently high posterior probability. I am quite uneasy about P(Ð|y) estimated by 1 as the shape of the set containing all posterior simulations is completely arbitrary, parameterisation dependent, and very random since based on the extremes of this posterior sample. Plus, the set Ð converges to the entire parameter space with the number of posterior simulations. An alternative that we advocated in our earlier paper is to take Ð as the HPD region or a variational Bayes version . But the central issue with the HPD regions is how to construct these from an MCMC output and how to compute both P(Ð) and P(Ð|y). It does not seem like a good idea to set P(Ð|x) to the intended α level for the HPD coverage. Using a non-parametric version for estimating Ð could be in the end the only reasonable solution.

As a test, I reran the example of a conjugate normal model used in the paper, based on (exact) simulations from both the prior and the posterior, and obtained approximations that were all close from the true marginal. With Chib’s being exact in that case (of course!), and an arithmetic mean surprisingly close without an importance correction:

> print(c(hame,chme,came,chib)) [1] -107.6821 -106.5968 -115.5950 -115.3610

Both harmonic versions are of the right order but not trustworthy, the truncation to such a set Ð as the one chosen in this paper having little impact.

## back to Ockham’s razor

Posted in Statistics with tags Bayesian model choice, candidate's formula, David MacKay, evidence, flexibility, frequentist inference, marginal likelihood, marginal likelihood identity, Ockham's razor, William of Ockham on July 31, 2019 by xi'an

“All in all, the Bayesian argument for selecting the MAP model as the single ‘best’ model is suggestive but not compelling.”

**L**ast month, Jonty Rougier and Carey Priebe arXived a paper on Ockham’s factor, with a generalisation of a prior distribution acting as a regulariser, R(θ). Calling on the late David MacKay to argue that the evidence involves the correct penalising factor although they acknowledge that his central argument is not absolutely convincing, being based on a first-order Laplace approximation to the posterior distribution and hence “dubious”. The current approach stems from the candidate’s formula that is already at the core of Sid Chib’s method. The log evidence then decomposes as the sum of the maximum log-likelihood minus the log of the posterior-to-prior ratio at the MAP estimator. Called the flexibility.

“Defining model complexity as flexibility unifies the Bayesian and Frequentist justifications for selecting a single model by maximizing the evidence.”

While they bring forward rational arguments to consider this as a measure model complexity, it remains at an informal level in that other functions of this ratio could be used as well. This is especially hard to accept by non-Bayesians in that it (seriously) depends on the choice of the prior distribution, as all transforms of the evidence would. I am thus skeptical about the reception of the argument by frequentists…