19 dubious ways to compute the marginal likelihood

Posted in Books, Statistics with tags , , , , , , , , , , on December 11, 2018 by xi'an

A recent arXival on nineteen different [and not necessarily dubious!] ways to approximate the marginal likelihood of a given topology of a philogeny tree that reminded me of our San Antonio survey with Jean-Michel Marin. This includes a version of the Laplace approximation called Laplus (!), accounting for the fact that branch lengths on the tree are positive but may have a MAP at zero. Using a Beta, Gamma, or log-Normal distribution instead of a Normal. For importance sampling, the proposals are derived from either the Laplus (!) approximate distributions or from the variational Bayes solution (based on an Normal product). Harmonic means are still used here despite the obvious danger, along with a defensive version that mixes prior and posterior. Naïve Monte Carlo means simulating from the prior, while bridge sampling seems to use samples from prior and posterior distributions. Path and modified path sampling versions are those proposed in 2008 by Nial Friel and Tony Pettitt (QUT). Stepping stone sampling appears like another version of path sampling, also based on a telescopic product of ratios of normalising constants, the generalised version relying on a normalising reference distribution that need be calibrated. CPO and PPD in the above table are two versions based on posterior predictive density estimates.

When running the comparison between so many contenders, the ground truth is selected as the values returned by MrBayes in a massive MCMC experiment amounting to 7.5 billions generations. For five different datasets. The above picture describes mean square errors for the probabilities of split, over ten replicates [when meaningful], the worst case being naïve Monte Carlo, with nested sampling and harmonic mean solutions close by. Similar assessments proceed from a comparison of Kullback-Leibler divergences. With the (predicatble?) note that “the methods do a better job approximating the marginal likelihood of more probable trees than less probable trees”. And massive variability for the poorest methods:

The comparison above does not account for time and since some methods are deterministic (and fast) there is little to do about this. The stepping steps solutions are very costly, while on the middle range bridge sampling outdoes path sampling. The assessment of nested sampling found in the conclusion is that it “would appear to be an unwise choice for estimating the marginal likelihoods of topologies, as it produces poor approximate posteriors” (p.12). Concluding at the Gamma Laplus approximation being the winner across all categories! (There is no ABC solution studied in this paper as the model likelihood can be computed in this setup, contrary to our own setting.)

computational statistics and molecular simulation [18w5023]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on November 19, 2018 by xi'an

The last day of the X fertilisation workshop at the casa matematicà Oaxaca, there were only three talks and only half of the participants. I lost the subtleties of the first talk by Andrea Agazzi on large deviations for chemical reactions, due to an emergency at work (Warwick). The second talk by Igor Barahona was somewhat disconnected from the rest of the conference, working on document textual analysis by way of algebraic data analysis (analyse des données) methods à la Benzécri. (Who was my office neighbour at Jussieu in the early 1990s.) In the last and final talk, Eric Vanden-Eijden made a link between importance sampling and PDMP, as an integral can be expressed via a trajectory of a path. A generalisation of path sampling, for almost any ODE. But also a competitor to nested sampling, waiting for the path to reach an Hamiltonian level, without some of the difficulties plaguing nested sampling like resampling. And involving continuous time processes. (Is there a continuous time version of ABC as well?!) Returning unbiased estimators of mean (the original integral) and variance. Example of a mixture example in dimension d=10 with k=50 components using only 100 paths.

subset sampling

Posted in Statistics with tags , , , , , , , , on July 13, 2018 by xi'an

A paper by Au and Beck (2001) was mentioned during a talk at MCqMC 2018 in Rennes and I checked Probabilistic Engineering Mechanics for details. There is no clear indication that the subset simulation advocated therein is particularly effective. The core idea is to obtain the probability to belong to a small set A by a cascading formula, namely the product of the probability to belong to A¹, then the conditional probability to belong to A² given A¹, &tc. When the subsets A¹, A², …, A constitute a decreasing embedded sequence. The simulation conditional on being in one of the subsets $A^i$ is operated by a random-walk Metropolis-within-Gibbs scheme, with an additional rejection when the value is not in the said subset. (Surprisingly, the authors re-establish the validity of this scheme.) Hence the proposal faces similar issues as nested sampling, except that the nested subsets here are defined quite differently as they are essentially free, provided they can be easily evaluated. Each of the random walks need be scaled, the harder a task because this depends on the corresponding subset volume. The subsets $A^i$ themselves are rarely defined in a natural manner, except when being tail events. And need to be calibrated so that the conditional probability of falling into each remains large enough, the cost of free choice. The Markov chain on the previous subset $A^i$ can prove useful to build the next subset $A^{i+1}$, but there is no general principle behind this remark. (If any, this is connected with X entropy.) But else, the past chains are very much wasted, compared with, say, an SMC treatment of the problem. The paper also notices that starting a Markov chain in the set $A^{i+1}$ means there is no burnin time and hence that the probability estimators are thus unbiased. (This creates a correlation between successive Markov chains, but I think it could be ignored if the starting point was chosen at random or after a random number of extra steps.) The authors further point out that the chain may fail to be ergodic, if the proposal distribution lacks energy to link connected regions of the current subset $A^i$. They suggest using multiple chains with multiple starting points, which alleviates the issue only to some extent, as it ultimately depends on the spread of the starting points. As acknowledged in the paper.

parallelizable sampling method for parameter inference of large biochemical reaction models

Posted in Books, Statistics with tags , , , , , , , , on June 18, 2018 by xi'an

I came across this older (2016) arXiv paper by Jan Mikelson and Mustafa Khammash [antidated as of April 25, 2018] as another version of nested sampling. The novelty of the approach is in applying nested sampling for approximating the likelihood function in the case of involved hidden Markov models (although the name itself does not appear in the paper). This is an interesting proposal, even though there is a fairly large and very active literature on computational approaches to such objects, from sequential Monte Carlo (SMC) to particle MCMC (pMCMC), to SMC².

“We found a way to efficiently sample parameter vectors (particles) from the super level set of the likelihood (sets of particles with a likelihood equal to or higher than some threshold) corresponding to an increasing sequence of thresholds” (p.2)

The approach here is an aggregate of nested sampling and particle filters (SMC), filters that are paradoxically employed in approximating the likelihood function itself, thus called repeatedly as the value of the parameter θ changes, unless I am confused, when it seems to me that, once started with particle filters, the authors could have used them all the way to the upper level (through, again, SMC²). Instead, and that brings a further degree of (uncorrected) approximation to the procedure, a Dirichlet process prior is used to estimate Gaussian mixture approximations to the true posterior distribution(s) on the (super) level sets. Now, approximating a distribution that is zero outside a compact set [the prior restricted to the likelihood being larger than by a distribution with an infinite support does not a priori sound like a particularly enticing idea. Note also that there is no later correction for using the mixture approximation to the restricted prior. (The method also involves an approximation of the (Lebesgue) volume of the level sets that may be poor in higher dimensions.)

“DP-GMM estimations work very well in high dimensional spaces and since we use rejection sampling to obtain samples from the level set by sampling from the DP-GMM estimation, the estimation error does not get propagated through iterations.” (p.13)

One aspect of the paper that puzzles me is the use of a rejection sampler to produce new parameters simulations from a given (super) level set, as this involves a lower bound M on the Gaussian mixture approximation over this level set. If a Gaussian mixture approximation is available, there is apparently no need for this as it can be sampled directly and values below the threshold can be disposed of. It is also unclear why the error does not propagate from one level to the next, if only because of the connection between the successive particle approximations.

the [not so infamous] arithmetic mean estimator

Posted in Books, Statistics with tags , , , , , , , , , on June 15, 2018 by xi'an

“Unfortunately, no perfect solution exists.” Anna Pajor

Another paper about harmonic and not-so-harmonic mean estimators that I (also) missed came out last year in Bayesian Analysis. The author is Anna Pajor, whose earlier note with Osiewalski I also spotted on the same day. The idea behind the approach [which belongs to the branch of Monte Carlo methods requiring additional simulations after an MCMC run] is to start as the corrected harmonic mean estimator on a restricted set A as to avoid tails of the distributions and the connected infinite variance issues that plague the harmonic mean estimator (an old ‘Og tune!). The marginal density p(y) then satisfies an identity involving the prior expectation of the likelihood function restricted to A divided by the posterior coverage of A. Which makes the resulting estimator unbiased only when this posterior coverage of A is known, which does not seem realist or efficient, except if A is an HPD region, as suggested in our earlier “safe” harmonic mean paper. And efficient only when A is well-chosen in terms of the likelihood function. In practice, the author notes that P(A|y) is to be estimated from the MCMC sequence and that the set A should be chosen to return large values of the likelihood, p(y|θ), through importance sampling, hence missing somehow the double opportunity of using an HPD region. Hence using the same default choice as in Lenk (2009), an HPD region which lower bound is derived as the minimum likelihood in the MCMC sample, “range of the posterior sampler output”. Meaning P(A|y)=1. (As an aside, the paper does not produce optimality properties or even heuristics towards efficiently choosing the various parameters to be calibrated in the algorithm, like the set A itself. As another aside, the paper concludes with a simulation study on an AR(p) model where the marginal may be obtained in closed form if stationarity is not imposed, which I first balked at, before realising that even in this setting both the posterior and the marginal do exist for a finite sample size, and hence the later can be estimated consistently by Monte Carlo methods.) A last remark is that computing costs are not discussed in the comparison of methods.

The final experiment in the paper is aiming at the marginal of a mixture model posterior, operating on the galaxy benchmark used by Roeder (1990) and about every other paper on mixtures since then (incl. ours). The prior is pseudo-conjugate, as in Chib (1995). And label-switching is handled by a random permutation of indices at each iteration. Which may not be enough to fight the attraction of the current mode on a Gibbs sampler and hence does not automatically correct Chib’s solution. As shown in Table 7 by the divergence with Radford Neal’s (1999) computations of the marginals, which happen to be quite close to the approximation proposed by the author. (As an aside, the paper mentions poor performances of Chib’s method when centred at the posterior mean, but this is a setting where the posterior mean is meaningless because of the permutation invariance. As another, I do not understand how the RMSE can be computed in this real data situation.) The comparison is limited to Chib’s method and a few versions of arithmetic and harmonic means. Missing nested sampling (Skilling, 2006; Chopin and X, 2011), and attuned importance sampling as in Berkoff et al. (2003), Marin, Mengersen and X (2005), and the most recent Lee and X (2016) in Bayesian Analysis.

unbiased consistent nested sampling via sequential Monte Carlo [a reply]

Posted in pictures, Statistics, Travel with tags , , , , , , , , on June 13, 2018 by xi'an

Rob Salomone sent me the following reply on my comments of yesterday about their recently arXived paper.

Our main goal in the paper was to show that Nested Sampling (when interpreted a certain way) is really just a member of a larger class of SMC algorithms, and exploring the consequences of that. We should point out that the section regarding calibration applies generally to SMC samplers, and hope that people give those techniques a try regardless of their chosen SMC approach.
Regarding your question about “whether or not it makes more sense to get completely SMC and forego any nested sampling flavour!”, this is an interesting point. After all, if Nested Sampling is just a special form of SMC, why not just use more standard SMC approaches? It seems that the Nested Sampling’s main advantage is its ability to cope with problems that have “phase transition’’ like behaviour, and thus is robust to a wider range of difficult problems than annealing approaches. Nevertheless, we hope this way of looking at NS (and showing that there may be variations of SMC with certain advantages) leads to improved NS and SMC methods down the line.
Regarding your post, I should clarify a point regarding unbiasedness. The largest likelihood bound is actually set to infinity. Thus, for the fixed version of NS—SMC, one has an unbiased estimator of the “final” band. Choosing a final band prematurely will of course result in very high variance. However, the estimator is unbiased. For example, consider NS—SMC with only one strata. Then, the method reduces to simply using the prior as an importance sampling distribution for the posterior (unbiased, but often high variance).
Comments related to two specific parts of your post are below (your comments in italicised bold):
“Which never occurred as the number one difficulty there, as the simplest implementation runs a Markov chain from the last removed entry, independently from the remaining entries. Even stationarity is not an issue since I believe that the first occurrence within the level set is distributed from the constrained prior.”
This is an interesting point that we had not considered! In practice, and in many papers that apply Nested Sampling with MCMC, the common approach is to start the MCMC at one of the randomly selected “live points”, so the discussion related to independence was in regard to these common implementations.
Regarding starting the chain from outside of the level set. This is likely not done in practice as it introduces an additional difficulty of needing to propose a sample inside the required region (Metropolis–Hastings will have non—zero probability of returning a sample that is still outside the constrained region for any fixed number of iterations). Forcing the continuation of MCMC until a valid point is proposed I believe will be a subtle violation of detailed balance. Of course, the bias of such a modification may be small in practice, but it is an additional awkwardness introduced by the requirement of sample independence!
“And then, in a twist that is not clearly explained in the paper, the focus moves to an improved nested sampler that moves one likelihood value at a time, with a particle step replacing a single  particle. (Things get complicated when several particles may take the very same likelihood value, but randomisation helps.) At this stage the algorithm is quite similar to the original nested sampler. Except for the unbiased estimation of the constants, the  final constant, and the replacement of exponential weights exp(-t/N) by powers of (N-1/N)”
Thanks for pointing out that this isn’t clear, we will try to do better in the next revision! The goal of this part of the paper wasn’t necessarily to propose a new version of nested sampling. Our focus here was to demonstrate that NS–SMC is not simply the Nested Sampling idea with an SMC twist, but that the original NS algorithm with MCMC (and restarting the MCMC sampling at one of the “live points’” as people do in practice) actually is a special case of SMC (with the weights replaced with a suboptimal choice).
The most curious thing is that, as you note, the estimates of remaining prior mass in the SMC context come out as powers of (N-1)/N and not exp(-t/N). In the paper by Walter (2017), he shows that the former choice is actually superior in terms of bias and variance. It was a nice touch that the superior choice of weights came out naturally in the SMC interpretation!
That said, as the fixed version of NS-SMC is the one with the unbiasedness and consistency properties, this was the version we used in the main statistical examples.

unbiased consistent nested sampling via sequential Monte Carlo

Posted in pictures, Statistics, Travel with tags , , , , , , , , on June 12, 2018 by xi'an

“Moreover, estimates of the marginal likelihood are unbiased.” (p.2)

Rob Salomone, Leah South, Chris Drovandi and Dirk Kroese (from QUT and UQ, Brisbane) recently arXived a paper that frames the nested sampling in such a way that marginal likelihoods can be unbiasedly (and consistently) estimated.

“Why isn’t nested sampling more popular with statisticians?” (p.7)

A most interesting question, especially given its popularity in cosmology and other branches of physics. A first drawback pointed out in the c is the requirement of independence between the elements of the sample produced at each iteration. Which never occurred as the number one difficulty there, as the simplest implementation runs a Markov chain from the last removed entry, independently from the remaining entries. Even stationarity is not an issue since I believe that the first occurrence within the level set is distributed from the constrained prior.

A second difficulty is the use of quadrature which turns integrand into step functions at random slices. Indeed, mixing Monte Carlo with numerical integration makes life much harder, as shown by the early avatars of nested sampling that only accounted for the numerical errors. (And which caused Nicolas and I to write our critical paper in Biometrika.) There are few studies of that kind in the literature, the only one I can think of being [my former PhD student] Anne Philippe‘s thesis twenty years ago.

The third issue stands with the difficulty in parallelising the method. Except by jumping k points at once, rather than going one level at a time. While I agree this makes life more complicated, I am also unsure about the severity of that issue as k nested sampling algorithms can be run in parallel and aggregated in the end, from simple averaging to something more elaborate.

The final blemish is that the nested sampling estimator has a stopping mechanism that induces a truncation error, again maybe a lesser problem given the overall difficulty in assessing the total error.

The paper takes advantage of the ability of SMC to produce unbiased estimates of a sequence of normalising constants (or of the normalising constants of a sequence of targets). For nested sampling, the sequence is made of the prior distribution restricted to an embedded sequence of level sets. With another sequence restricted to bands (likelihood between two likelihood boundaries). If all restricted posteriors of the second kind and their normalising constant are known, the full posterior is known. Apparently up to the main normalising constant, i.e. the marginal likelihood., , except that it is also the sum of all normalising constants. Handling this sequence by SMC addresses the four concerns of the four authors, apart from the truncation issue, since the largest likelihood bound need be set for running the algorithm.

When the sequence of likelihood bounds is chosen based on the observed likelihoods so far, the method becomes adaptive. Requiring again the choice of a stopping rule that may induce bias if stopping occurs too early. And then, in a twist that is not clearly explained in the paper, the focus moves to an improved nested sampler that moves one likelihood value at a time, with a particle step replacing a single particle. (Things get complicated when several particles may take the very same likelihood value, but randomisation helps.) At this stage the algorithm is quite similar to the original nested sampler. Except for the unbiased estimation of the constants, the final constant, and the replacement of exponential weights exp(-t/N) by powers of (N-1/N).

The remainder of this long paper (61 pages!) is dedicated to practical implementation, calibration and running a series of comparisons. A nice final touch is the thanks to the ‘Og for its series of posts on nested sampling, which “helped influence this work, and played a large part in inspiring it.”

In conclusion, this paper is certainly a worthy exploration of the nested sampler, providing further arguments towards a consistent version, with first and foremost an (almost?) unbiased resolution. The comparison with a wide range of alternatives remains open, in particular time-wise, if evidence is the sole target of the simulation. For instance, the choice of this sequence of targets in an SMC may be improved by another sequence, since changing one particle at a time does not sound efficient. The complexity of the implementation and in particular of the simulation from the prior under more and more stringent constraints need to be addressed.