asymptotics of M³C²L

In a recent arXival, Blazej Miasojedow, Wojciech Niemiro and Wojciech Rejchel establish the convergence of a maximum likelihood estimator based on an MCMC approximation of the likelihood function. As in intractable normalising constants. The main result in the paper is a Central Limit theorem for the M³C²L estimator that incorporates an additional asymptotic variance term for the Monte Carlo error. Where both the sample size n and the number m of simulations go to infinity. Independently so. However, I do not fully perceive the relevance of using an MCMC chain to target an importance function [which is used in the approximation of the normalising constant or otherwise for the intractable likelihood], relative to picking an importance function h(.) that can be directly simulated.

a paradox about likelihood ratios?

Aware of my fascination for paradoxes (and heterodox publications), Ewan Cameron sent me the link to a recent arXival by Louis Lyons (Oxford) on different asymptotic distributions of the likelihood ratio. Which is full of approximations. The overall point of the note is hard to fathom… Unless it simply plans to illustrate Betteridge’s law of headlines, as suggested by Ewan.

For instance, the limiting distribution of the log-likelihood of an exponential sample at the true value of the parameter τ is not asymptotically Gaussian but almost surely infinite. While the log of the (Wilks) likelihood ratio at the true value of τ is truly (if asymptotically) a Χ² variable with one degree of freedom. That it is not a Gaussian is deemed a “paradox” by the author, explained by a cancellation of first order terms… Same thing again for the common Gaussian mean problem!

talk at Trinity College

Tomorrow noon, I will give a talk at Trinity College Dublin on the asymptotic properties of ABC. (Here are the slides from the talk I gave in Berlin last month.)

SAS on Bayes

Following a question on X Validated, I became aware of the following descriptions of the pros and cons of Bayesian analysis, as perceived by whoever (Tim Arnold?) wrote SAS/STAT(R) 9.2 User’s Guide, Second Edition. I replied more specifically on the point

It [Bayesian inference] provides inferences that are conditional on the data and are exact, without reliance on asymptotic approximation. Small sample inference proceeds in the same manner as if one had a large sample. Bayesian analysis also can estimate any functions of parameters directly, without using the “plug-in” method (a way to estimate functionals by plugging the estimated parameters in the functionals).

which I find utterly confusing and not particularly relevant. The other points in the list are more traditional, except for this one

It provides interpretable answers, such as “the true parameter θ has a probability of 0.95 of falling in a 95% credible interval.”

that I find somewhat unappealing in that the 95% probability has only relevance wrt to the resulting posterior, hence has no absolute (and definitely no frequentist) meaning. The criticisms of the prior selection

It does not tell you how to select a prior. There is no correct way to choose a prior. Bayesian inferences require skills to translate subjective prior beliefs into a mathematically formulated prior. If you do not proceed with caution, you can generate misleading results.

It can produce posterior distributions that are heavily influenced by the priors. From a practical point of view, it might sometimes be difficult to convince subject matter experts who do not agree with the validity of the chosen prior.

are traditional but nonetheless irksome. Once acknowledged there is no correct or true prior, it follows naturally that the resulting inference will depend on the choice of the prior and has to be understood conditional on the prior, which is why the credible interval has for instance an epistemic rather than frequentist interpretation. There is also little reason for trying to convince a fellow Bayesian statistician about one’s prior. Everything is conditional on the chosen prior and I see less and less why this should be an issue.

 

Validity and the foundations of statistical inference

Natesh pointed out to me this recent arXival with a somewhat grandiose abstract:

In this paper, we argue that the primary goal of the foundations of statistics is to provide data analysts with a set of guiding principles that are guaranteed to lead to valid statistical inference. This leads to two new questions: “what is valid statistical inference?” and “do existing methods achieve this?” Towards answering these questions, this paper makes three contributions. First, we express statistical inference as a process of converting observations into degrees of belief, and we give a clear mathematical definition of what it means for statistical inference to be valid. Second, we evaluate existing approaches Bayesian and frequentist approaches relative to this definition and conclude that, in general, these fail to provide valid statistical inference. This motivates a new way of thinking, and our third contribution is a demonstration that the inferential model framework meets the proposed criteria for valid and prior-free statistical inference, thereby solving perhaps the most important unsolved problem in statistics.

Since solving the “most important unsolved problem in statistics” sounds worth pursuing, I went and checked the paper‘s contents.

“To us, the primary goal of the foundations of statistics is to provide a set of guiding principles that, if followed, will guarantee validity of the resulting inference. Our motivation for writing this paper is to be clear about what is meant by valid inference and to provide the necessary principles to help data analysts achieve validity.”

Which can be interpreted in so many ways that it is somewhat meaningless…

“…if real subjective prior information is available, we recommend using it. However, there is an expanding collection of work (e.g., machine learning, etc) that takes the perspective that no real prior information is available. Even a large part of the literature claiming to be Bayesian has abandoned the interpretation of the prior as a serious part of the model, opting for “default” prior that “works.” Our choice to omit a prior from the model is not for the (misleading) purpose of being “objective”—subjectivity is necessary—but, rather, for the purpose of exploring what can be done in cases where a fully satisfactory prior is not available, to see what improvements can be made over the status quo.”

This is a pretty traditional criticism of the Bayesian approach, namely that if a “true” prior is provided (by whom?) then it is optimal to use it. But this amounts to turn the prior into another piece of the sampling distribution and is not in my opinion a Bayesian argument! Most of the criticisms in the paper are directed at objective Bayes approaches, with the surprising conclusion that, because there exist cases where no matching prior is available, “the objective Bayesian approach [cannot] be considered as a general framework for scientific inference.” (p.9)

Another section argues that a Bayesian modelling cannot describe a state of total ignorance. This is formally correct, which is why there is no such thing as a non-informative or the non-informative prior, as often discussed here, but is this truly relevant, in that the inference problem contains one way or another information about the parameter, for instance through a loss function or a pseudo-likelihood.

“This is a desirable property that most existing methods lack.”

The proposal central to the paper thesis is to replace posterior probabilities by belief functions b(.|X), called statistical inference, that are interpreted as measures of evidence about subsets A of the parameter space. If not necessarily as probabilities. This is not very novel, witness the works of Dempster, Shafer and subsequent researchers. And not very much used outside Bayesian and fiducial statistics because of the mostly impossible task of defining a function over all subsets of the parameter space. Because of the subjectivity of such “beliefs”, they will be “valid” only if they are well-calibrated in the sense of b(A|X) being sub-uniform, that is, more concentrated near zero than a uniform variate (i.e., small) under the alternative, i.e. when θ is not in A. At this stage, since this is a mix of a minimax and proper coverage condition, my interest started to quickly wane… Especially because the sub-uniformity condition is highly demanding, if leading to controls over the Type I error and the frequentist coverage. As often, I wonder at the meaning of a calibration property obtained over all realisations of the random variable and all values of the parameter. So for me stability is neither “desirable” nor “essential”. Overall, I have increasing difficulties in perceiving proper coverage as a relevant property. Which has no stronger or weaker meaning that the coverage derived from a Bayesian construction.

“…frequentism does not provide any guidance for selecting a particular rule or procedure.”

I agree with this assessment, which means that there is no such thing as frequentist inference, but rather a philosophy for assessing procedures. That the Gleser-Hwang paradox invalidates this philosophy sounds a bit excessive, however. Especially when the bounded nature of Bayesian credible intervals is also analysed as a failure. A more relevant criticism is the lack of directives for picking procedures.

“…we are the first to recognize that the belief function’s properties are necessary in order for the inferential output to satisfy the required validity property”

The construction of the “inferential model” proposed by the authors offers similarities withn fiducial inference, in that it builds upon the representation of the observable X as X=a(θ,U). With further constraints on the function a() to ensure the validity condition holds… An interesting point is that the functional connection X=a(θ,U) means that the nature of U changes once X is observed, albeit in a delicate manner outside a Bayesian framework. When illustrated on the Gleser-Hwang paradox, the resolution proceeds from an arbitrary choice of a one-dimensional summary, though. (As I am reading the paper, I realise it builds on other and earlier papers by the authors, papers that I cannot read for lack of time. I must have listned to a talk by one of the authors last year at JSM as this rings a bell. Somewhat.) In conclusion of a quick Sunday afternoon read, I am not convinced by the arguments in the paper and even less by the impression of a remaining arbitrariness in setting the resulting procedure.

ABC for big data

“The results in this paper suggest that ABC can scale to large data, at least for models with a xed number of parameters, under the assumption that the summary statistics obey a central limit theorem.”

In a week rich with arXiv submissions about MCMC and “big data”, like the Variational consensus Monte Carlo of Rabinovich et al., or scalable Bayesian inference via particle mirror descent by Dai et al., Wentao Li and Paul Fearnhead contributed an impressive paper entitled Behaviour of ABC for big data. However, a word of warning: the title is somewhat misleading in that the paper does not address the issue of big or tall data per se, e.g., the impossibility to handle the whole data at once and to reproduce it by simulation, but rather the asymptotics of ABC. The setting is not dissimilar to the earlier Fearnhead and Prangle (2012) Read Paper. The central theme of this theoretical paper [with 24 pages of proofs!] is to study the connection between the number N of Monte Carlo simulations and the tolerance value ε when the number of observations n goes to infinity. A main result in the paper is that the ABC posterior mean can have the same asymptotic distribution as the MLE when ε=o(n^-1/4). This is however in opposition with of no direct use in practice as the second main result that the Monte Carlo variance is well-controlled only when ε=O(n^-1/2).

Something I have (slight) trouble with is the construction of an importance sampling function of the f_ABC(s|θ)^α when, obviously, this function cannot be used for simulation purposes. The authors point out this fact, but still build an argument about the optimal choice of α, namely away from 0 and 1, like ½. Actually, any value different from 0,1, is sensible, meaning that the range of acceptable importance functions is wide. Most interestingly (!), the paper constructs an iterative importance sampling ABC in a spirit similar to Beaumont et al. (2009) ABC-PMC. Even more interestingly, the ½ factor amounts to updating the scale of the proposal as twice the scale of the target, just as in PMC.

Another aspect of the analysis I do not catch is the reason for keeping the Monte Carlo sample size to a fixed value N, while setting a sequence of acceptance probabilities (or of tolerances) along iterations. This is a very surprising result in that the Monte Carlo error does remain under control and does not dominate the overall error!

“Whilst our theoretical results suggest that point estimates based on the ABC posterior have good properties, they do not suggest that the ABC posterior is a good approximation to the true posterior, nor that the ABC posterior will accurately quantify the uncertainty in estimates.”

Overall, this is clearly a paper worth reading for understanding the convergence issues related with ABC. With more theoretical support than the earlier Fearnhead and Prangle (2012). However, it does not provide guidance into the construction of a sequence of Monte Carlo samples nor does it discuss the selection of the summary statistic, which has obviously a major impact on the efficiency of the estimation. And to relate to the earlier warning, it does not cope with “big data” in that it reproduces the original simulation of the n sized sample.

Statistics slides (4)

Here is the fourth set of slides for my third year statistics course, trying to build intuition about the likelihood surface and why on Earth would one want to find its maximum?!, through graphs. I am yet uncertain whether or not I will reach the point where I can teach more asymptotics so maybe I will also include asymptotic normality of the MLE under regularity conditions in this chapter…