ABC for copula estimation

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , on March 23, 2015 by xi'an

Clara Grazian and Brunero Liseo (di Roma) have just arXived a note on a method merging copulas, ABC, and empirical likelihood. The approach is rather hybrid and thus not completely Bayesian, but this must be seen as a consequence of an ill-posed problem. Indeed, as in many econometric models, the model there is not fully defined: the marginals of iid observations are represented as being from well-known parametric families (and are thus well-estimated by Bayesian tools), while the joint distribution remains uncertain and hence so does the associated copula. The approach in the paper is to proceed stepwise, i.e., to estimate correctly each marginal, well correctly enough to transform the data by an estimated cdf, and then only to estimate the copula or some aspect of it based on this transformed data. Like Spearman’s ρ. For which an empirical likelihood is computed and aggregated to a prior to make a BCel weight. (If this sounds unclear, each BEel evaluation is based on a random draw from the posterior samples, which transfers some uncertainty in the parameter evaluation into the copula domain. Thanks to Brunero and Clara for clarifying this point for me!)

At this stage of the note, there are two illustrations revolving around Spearman’s ρ. One on simulated data, with better performances than a nonparametric frequentist solution. And another one on a Garch (1,1) model for two financial time-series.

I am quite glad to see an application of our BCel approach in another domain although I feel a tiny bit uncertain about the degree of arbitrariness in the approach, from the estimated cdf transforms of the marginals to the choice of the moment equations identifying the parameter of interest like Spearman’s ρ. Especially if one uses a parametric copula which moments are equally well-known. While I see the practical gain in analysing each component separately, the object created by the estimated cdf transforms may have a very different correlation structure from the true cdf transforms. Maybe there exist consistency conditions on the estimated cdfs… Maybe other notions of orthogonality or independence could be brought into the picture to validate further the two-step solution…

Posted in pictures, Statistics, University life with tags , , , , , , on February 9, 2015 by xi'an

I just arXived my comments about A. Ronald Gallant’s “Reflections on the Probability Space Induced by Moment Conditions with Implications for Bayesian Inference”, capitalising on the three posts I wrote around the discussion talk I gave at the 6th French Econometrics conference last year. Nothing new there, except that I may get a response from Ron Gallant as this is submitted as a discussion of his related paper in Journal of Financial Econometrics. While my conclusion is rather negative, I find the issue of setting prior and model based on a limited amount of information of much interest, with obvious links with ABC, empirical likelihood and other approximation methods.

reflections on the probability space induced by moment conditions with implications for Bayesian Inference [slides]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on December 4, 2014 by xi'an

Here are the slides of my incoming discussion of Ron Gallant’s paper, tomorrow.

reflections on the probability space induced by moment conditions with implications for Bayesian Inference [discussion]

Posted in Books, Statistics, University life with tags , , , , , , on December 1, 2014 by xi'an

[Following my earlier reflections on Ron Gallant’s paper, here is a more condensed set of questions towards my discussion of next Friday.]

“If one specifies a set of moment functions collected together into a vector m(x,θ) of dimension M, regards θ as random and asserts that some transformation Z(x,θ) has distribution ψ then what is required to use this information and then possibly a prior to make valid inference?” (p.4)

The central question in the paper is whether or not given a set of moment equations

$\mathbb{E}[m(X_1,\ldots,X_n,\theta)]=0$

(where both the Xi‘s and θ are random), one can derive a likelihood function and a prior distribution compatible with those. It sounds to me like a highly complex question since it implies the integral equation

$\int_{\Theta\times\mathcal{X}^n} m(x_1,\ldots,x_n,\theta)\,\pi(\theta)f(x_1|\theta)\cdots f(x_n|\theta) \text{d}\theta\text{d}x_1\cdots\text{d}x_n=0$

must have a solution for all n’s. A related question that was also remanent with fiducial distributions is how on Earth (or Middle Earth) the concept of a random theta could arise outside Bayesian analysis. And another one is how could the equations make sense outside the existence of the pair (prior,likelihood). A question that may exhibit my ignorance of structural models. But which may also relate to the inconsistency of Zellner’s (1996) Bayesian method of moments as exposed by Geisser and Seidenfeld (1999).

For instance, the paper starts (why?) with the Fisherian example of the t distribution of

$Z(x,\theta) = \frac{\bar{x}_n-\theta}{s/\sqrt{n}}$

which is truly is a t variable when θ is fixed at the true mean value. Now, if we assume that the joint distribution of the Xi‘s and θ is such that this projection is a t variable, is there any other case than the Dirac mass on θ? For all (large enough) sample sizes n? I cannot tell and the paper does not bring [me] an answer either.

When I look at the analysis made in the abstraction part of the paper, I am puzzled by the starting point (17), where

$p(x|\theta) = \psi(Z(x,\theta))$

since the lhs and rhs operate on different spaces. In Fisher’s example, x is an n-dimensional vector, while Z is unidimensional. If I apply blindly the formula on this example, the t density does not integrate against the Lebesgue measure in the n-dimension Euclidean space… If a change of measure allows for this representation, I do not see so much appeal in using this new measure and anyway wonder in which sense this defines a likelihood function, i.e. the product of n densities of the Xi‘s conditional on θ. To me this is the central issue, which remains unsolved by the paper.

reflections on the probability space induced by moment conditions with implications for Bayesian Inference [refleXions]

Posted in Statistics, University life with tags , , , , , , , , , , on November 26, 2014 by xi'an

“The main finding is that if the moment functions have one of the properties of a pivotal, then the assertion of a distribution on moment functions coupled with a proper prior does permit Bayesian inference. Without the semi-pivotal condition, the assertion of a distribution for moment functions either partially or completely specifies the prior.” (p.1)

Ron Gallant will present this paper at the Conference in honour of Christian Gouréroux held next week at Dauphine and I have been asked to discuss it. What follows is a collection of notes I made while reading the paper , rather than a coherent discussion, to come later. Hopefully prior to the conference.

The difficulty I have with the approach presented therein stands as much with the presentation as with the contents. I find it difficult to grasp the assumptions behind the model(s) and the motivations for only considering a moment and its distribution. Does it all come down to linking fiducial distributions with Bayesian approaches? In which case I am as usual sceptical about the ability to impose an arbitrary distribution on an arbitrary transform of the pair (x,θ), where x denotes the data. Rather than a genuine prior x likelihood construct. But I bet this is mostly linked with my lack of understanding of the notion of structural models.

“We are concerned with situations where the structural model does not imply exogeneity of θ, or one prefers not to rely on an assumption of exogeneity, or one cannot construct a likelihood at all due to the complexity of the model, or one does not trust the numerical approximations needed to construct a likelihood.” (p.4)

As often with econometrics papers, this notion of structural model sets me astray: does this mean any latent variable model or an incompletely defined model, and if so why is it incompletely defined? From a frequentist perspective anything random is not a parameter. The term exogeneity also hints at this notion of the parameter being not truly a parameter, but including latent variables and maybe random effects. Reading further (p.7) drives me to understand the structural model as defined by a moment condition, in the sense that

$\mathbb{E}[m(\mathbf{x},\theta)]=0$

has a unique solution in θ under the true model. However the focus then seems to make a major switch as Gallant considers the distribution of a pivotal quantity like

$Z=\sqrt{n} W(\mathbf{x},\theta)^{-\frac{1}{2}} m(\mathbf{x},\theta)$

as induced by the joint distribution on (x,θ), hence conversely inducing constraints on this joint, as well as an associated conditional. Which is something I have trouble understanding, First, where does this assumed distribution on Z stem from? And, second, exchanging randomness of terms in a random variable as if it was a linear equation is a pretty sure way to produce paradoxes and measure theoretic difficulties.

The purely mathematical problem itself is puzzling: if one knows the distribution of the transform Z=Z(X,Λ), what does that imply on the joint distribution of (X,Λ)? It seems unlikely this will induce a single prior and/or a single likelihood… It is actually more probable that the distribution one arbitrarily selects on m(x,θ) is incompatible with a joint on (x,θ), isn’t it?

“The usual computational method is MCMC (Markov chain Monte Carlo) for which the best known reference in econometrics is Chernozhukov and Hong (2003).” (p.6)

While I never heard of this reference before, it looks like a 50 page survey and may be sufficient for an introduction to MCMC methods for econometricians. What I do not get though is the connection between this reference to MCMC and the overall discussion of constructing priors (or not) out of fiducial distributions. The author also suggests using MCMC to produce the MAP estimate but this always stroke me as inefficient (unless one uses our SAME algorithm of course).

“One can also compute the marginal likelihood from the chain (Newton and Raftery (1994)), which is used for Bayesian model comparison.” (p.22)

Not the best solution to rely on harmonic means for marginal likelihoods…. Definitely not. While the author actually uses the stabilised version (15) of Newton and Raftery (1994) estimator, which in retrospect looks much like a bridge sampling estimator of sorts, it remains dangerously close to the original [harmonic mean solution] especially for a vague prior. And it only works when the likelihood is available in closed form.

“The MCMC chains were comprised of 100,000 draws well past the point where transients died off.” (p.22)

I wonder if the second statement (with a very nice image of those dying transients!) is intended as a consequence of the first one or independently.

“A common situation that requires consideration of the notions that follow is that deriving the likelihood from a structural model is analytically intractable and one cannot verify that the numerical approximations one would have to make to circumvent the intractability are sufficiently accurate.” (p.7)

This then is a completely different business, namely that defining a joint distribution by mean of moment equations prevents regular Bayesian inference because the likelihood is not available. This is more exciting because (i) there are alternative available! From ABC to INLA (maybe) to EP to variational Bayes (maybe). And beyond. In particular, the moment equations are strongly and even insistently suggesting that empirical likelihood techniques could be well-suited to this setting. And (ii) it is no longer a mathematical worry: there exist a joint distribution on m(x,θ), induced by a (or many) joint distribution on (x,θ). So the question of finding whether or not it induces a single proper prior on θ becomes relevant. But, if I want to use ABC, being given the distribution of m(x,θ) seems to mean I can only generate new values of this transform while missing a natural distance between observations and pseudo-observations. Still, I entertain lingering doubts that this is the meaning of the study. Where does the joint distribution come from..?!

“Typically C is coarse in the sense that it does not contain all the Borel sets (…)  The probability space cannot be used for Bayesian inference”

My understanding of that part is that defining a joint on m(x,θ) is not always enough to deduce a (unique) posterior on θ, which is fine and correct, but rather anticlimactic. This sounds to be what Gallant calls a “partial specification of the prior” (p.9).

Overall, after this linear read, I remain very much puzzled by the statistical (or Bayesian) implications of the paper . The fact that the moment conditions are central to the approach would once again induce me to check the properties of an alternative approach like empirical likelihood.

a bootstrap likelihood approach to Bayesian computation

Posted in Books, R, Statistics, University life with tags , , , , , , , , on October 16, 2014 by xi'an

This paper by Weixuan Zhu, Juan Miguel Marín [from Carlos III in Madrid, not to be confused with Jean-Michel Marin, from Montpellier!], and Fabrizio Leisen proposes an alternative to our 2013 PNAS paper with Kerrie Mengersen and Pierre Pudlo on empirical likelihood ABC, or BCel. The alternative is based on Davison, Hinkley and Worton’s (1992) bootstrap likelihood, which relies on a double-bootstrap to produce a non-parametric estimate of the distribution of a given estimator of the parameter θ. Including a smooth curve-fitting algorithm step, for which not much description is available from the paper.

“…in contrast with the empirical likelihood method, the bootstrap likelihood doesn’t require any set of subjective constrains taking advantage from the bootstrap methodology. This makes the algorithm an automatic and reliable procedure where only a few parameters need to be specified.”

The spirit is indeed quite similar to ours in that a non-parametric substitute plays the role of the actual likelihood, with no correction for the substitution. Both approaches are convergent, with similar or identical convergence speeds. While the empirical likelihood relies on a choice of parameter identifying constraints, the bootstrap version starts directly from the [subjectively] chosen estimator of θ. For it indeed needs to be chosen. And computed.

“Another benefit of using the bootstrap likelihood (…) is that the construction of bootstrap likelihood could be done once and not at every iteration as the empirical likelihood. This leads to significant improvement in the computing time when different priors are compared.”

This is an improvement that could apply to the empirical likelihood approach, as well, once a large enough collection of likelihood values has been gathered. But only in small enough dimensions where smooth curve-fitting algorithms can operate. The same criticism applying to the derivation of a non-parametric density estimate for the distribution of the estimator of θ. Critically, the paper only processes examples with a few parameters.

In the comparisons between BCel and BCbl that are produced in the paper, the gain is indeed towards BCbl. Since this paper is mostly based on examples and illustrations, not unlike ours, I would like to see more details on the calibration of the non-parametric methods and of regular ABC, as well as on the computing time. And the variability of both methods on more than a single Monte Carlo experiment.

I am however uncertain as to how the authors process the population genetic example. They refer to the composite likelihood used in our paper to set the moment equations. Since this is not the true likelihood, how do the authors select their parameter estimates in the double-bootstrap experiment? The inclusion of Crakel’s and Flegal’s (2013) bivariate Beta, is somewhat superfluous as this example sounds to me like an artificial setting.

In the case of the Ising model, maybe the pre-processing step in our paper with Matt Moores could be compared with the other algorithms. In terms of BCbl, how does the bootstrap operate on an Ising model, i.e. (a) how does one subsample pixels and (b)what are the validity guarantees?

A test that would be of interest is to start from a standard ABC solution and use this solution as the reference estimator of θ, then proceeding to apply BCbl for that estimator. Given that the reference table would have to be produced only once, this would not necessarily increase the computational cost by a large amount…

model selection by likelihood-free Bayesian methods

Posted in Books, pictures, Running, Statistics, University life with tags , , , , , , on May 29, 2014 by xi'an

Just glanced at the introduction of this arXived paper over breakfast, back from my morning run: the exact title is “Model Selection for Likelihood-free Bayesian Methods Based on Moment Conditions: Theory and Numerical Examples” by Cheng Li and Wenxin Jiang. (The paper is 81 pages long.) I selected the paper for its title as it connected with an interrogation of ours on the manner to extend our empirical likelihood [A]BC work to model choice. We looked at this issue with Kerrie Mengersen and Judith Rousseau the last time Kerrie visited Paris but could not spot a satisfying entry… The current paper is of a theoretical nature, considering a moment defined model

$\mathbb{E}[g(D,\theta)]=0,$

where D denotes the data, as the dimension p of the parameter θ grows with n, the sample size. The approximate model is derived from a prior on the parameter θ and of a Gaussian quasi-likelihood on the moment estimating function g(D,θ). Examples include single index longitudinal data, quantile regression and partial correlation selection. The model selection setting is one of variable selection, resulting in 2p models to compare, with p growing to infinity… Which makes the practical implementation rather delicate to conceive. And the probability one of hitting the right model a fairly asymptotic concept. (At least after a cursory read from my breakfast table!)