Archive for maximum likelihood estimation

simulation under zero measure constraints [a reply]

Posted in Books, pictures, Statistics, University life with tags , , , , , on November 21, 2016 by xi'an

grahamFollowing my post of last Friday on simulating over zero measure sets, as, e.g., producing a sample with a given maximum likelihood estimator, Dennis Prangle pointed out the recent paper on the topic by Graham and Storkey, and a wee bit later, Matt Graham himself wrote an answer to my X Validated question detailing the resolution of the MLE problem for a Student’s t sample. Including the undoubtedly awesome picture of a 3 observation sample distribution over a non-linear manifold in R³. When reading this description I was then reminded of a discussion I had a few months ago with Gabriel Stolz about his free energy approach that managed the same goal through a Langevin process. Including the book Free Energy Computations he wrote in 2010 with Tony Lelièvre and Mathias Rousset. I now have to dig deeper in these papers, but in the meanwhile let me point out that there is a bounty of 200 points running on this X Validated question for another three days. Offered by Glen B., the #1 contributor to X Validated question for all times.

simulation under zero measure constraints

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , on November 17, 2016 by xi'an

A theme that comes up fairly regularly on X validated is the production of a sample with given moments, either for calibration motives or from a misunderstanding of the difference between a distribution mean and a sample average. Here are some entries on that topic:

In most of those questions, the constraint in on the sum or mean of the sample, which allows for an easy resolution by a change of variables. It however gets somewhat harder when the constraint involves more moments or, worse, an implicit solution to an equation. A good example of the later is the quest for a sample with a given maximum likelihood estimate in the case this MLE cannot be derived analytically. As for instance with a location-scale t sample…

Actually, even when the constraint is solely on the sum, a relevant question is the production of an efficient simulation mechanism. Using a Gibbs sampler that changes one component of the sample at each iteration does not qualify, even though it eventually produces the proper sample. Except for small samples. As in this example

n=3;T=1e4
s0=.5 #fixed average
sampl=matrix(s0,T,n)
for (t in 2:T){
 sampl[t,]=sampl[t-1,]
 for (i in 1:(n-1)){
  sampl[t,i]=runif(1,
  min=max(0,n*s0-sum(sampl[t,c(-i,-n)])-1),
  max=min(1,n*s0-sum(sampl[t,c(-i,-n)])))
 sampl[t,n]=n*s0-sum(sampl[t,-n])}}

For very large samples, I figure that proposing from the unconstrained density can achieve a sufficient efficiency, but the in-between setting remains an interesting problem.

Nature highlights

Posted in Books, Kids, pictures, Statistics with tags , , , , , , , on November 1, 2016 by xi'an

A mostly genetics issue of Nature this week (of October 13), as the journal contains an article on the genomes of 300 individuals from 142 diverse populations across the globe, and another one on the genetic history of Australia Aborigines, plus a third one of 483 individuals from 125 populations drawing genetic space barriers, leading to diverging opinions on the single versus multiple out-of-Africa scenario. As some of these papers are based on likelihood-based techniques, I wish I had more time to explore the statistics behind. Another paper builds a phylogeny of violence in mammals, rising as one nears the primates. I find the paper most interesting but I am not convinced by the genetic explanation of violence, in particular because it seems hard to believe that data about Palaeolithic, Mesolithic, and Neolithic periods can be that informative about the death rate due to intra-species violence. And to conclude on a “pessimistic” note, the paper that argues there is a maximum lifespan for humans, meaning that the 122 years enjoyed (?) by Jeanne Calment from France may remain a limit. However, the argument seems to be that the observed largest, second largest, &tc., ages at death reached a peak in 1997, the year Jeanne Calment died, and is declining since then. That does not sound super-convincing when considering extreme value theory, since 1997 is the extreme event and thus another extreme event of a similar magnitude is not going to happen immediately after.

tractable Bayesian variable selection: beyond normality

Posted in R, Statistics, University life with tags , , , , , , , on October 17, 2016 by xi'an

bird tracks in the first snow of Winter, Feb. 05, 2012 David Rossell and Francisco Rubio (both from Warwick) arXived a month ago a paper on non-normal variable selection. They use two-piece error models that preserve manageable inference and allow for simple computational algorithms, but also characterise the behaviour of the resulting variable selection process under model misspecification. Interestingly, they show that the existence of asymmetries or heavy tails leads to power losses when using the Normal model. The two-piece error distribution is made of two halves of location-scale transforms of the same reference density on the two sides of the common location parameter. In this paper, the density is either Gaussian or Laplace (i.e., exponential?). In both cases the (log-)likelihood has a nice compact expression (although it does not allow for a useful sufficient statistic). One is the L¹ version versus the other which is the L² version. Which is the main reason for using this formalism based on only two families of parametric distributions, I presume. (As mentioned in an earlier post, I do not consider those distributions as mixtures because the component of a given observation can always be identified. And because as shown in the current paper, maximum likelihood estimates can be easily derived.) The prior construction follows the non-local prior principles of Johnson and Rossell (2010, 2012) also discussed in earlier posts. The construction is very detailed and hence highlights how many calibration steps are needed in the process.

“Bayes factor rates are the same as when the correct model is assumed [but] model misspecification often causes a decrease in the power to detect truly active variables.”

When there are too many models to compare at once, the authors propose a random walk on the finite set of models (which does not require advanced measure-theoretic tools like reversible jump MCMC). One interesting aspect is that moving away from the normal to another member of this small family is driven by the density of the data under the marginal densities, which means moving only to interesting alternatives. But also sticking to the normal only for adequate datasets. In a sense this is not extremely surprising given that the marginal likelihoods (model-wise) are available. It is also interesting that on real datasets, one of the four models is heavily favoured against the others, be it Normal (6.3) or Laplace (6.4). And that the four model framework returns almost identical values when compared with a single (most likely) model. Although not immensely surprising when acknowledging that the frequency of the most likely model is 0.998 and 0.998, respectively.

“Our framework represents a middle-ground to add flexibility in a parsimonious manner that remains analytically and computationally tractable, facilitating applications where either p is large or n is too moderate to fit more flexible models accurately.”

Overall, I find the experiment quite conclusive and do not object [much] to this choice of parametric family in that it is always more general and generic than the sempiternal Gaussian model. That we picked in our Bayesian Essentials, following tradition. In a sense, it would be natural to pick the most general possible parametric family that allows for fast computations, if this notion does make any sense…

weak convergence (…) in ABC

Posted in Books, Statistics, University life with tags , , , , , , on January 18, 2016 by xi'an

Samuel Soubeyrand and Eric Haon-Lasportes recently published a paper in Statistics and Probability Letters that has some common features with the ABC consistency paper we wrote a few months ago with David Frazier and Gael Martin. And to the recent Li and Fearnhead paper on the asymptotic normality of the ABC distribution. Their approach is however based on a Bernstein-von Mises [CLT] theorem for the MLE or a pseudo-MLE. They assume that the density of this estimator is asymptotically equivalent to a Normal density, in which case the true posterior conditional on the estimator is also asymptotically equivalent to a Normal density centred at the (p)MLE. Which also makes the ABC distribution normal when both the sample size grows to infinity and the tolerance decreases to zero. Which is not completely unexpected. However, in complex settings, establishing the asymptotic normality of the (p)MLE may prove a formidable or even impossible task.

efficient approximate Bayesian inference for models with intractable likelihood

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , on July 6, 2015 by xi'an

Awalé board on my garden table, March 15, 2013Dalhin, Villani [Mattias, not Cédric] and Schön arXived a paper this week with the above title. The type of intractable likelihood they consider is a non-linear state-space (HMM) model and the SMC-ABC they propose is based on an optimised Laplace approximation. That is, replacing the posterior distribution on the parameter θ with a normal distribution obtained by a Taylor expansion of the log-likelihood. There is no obvious solution for deriving this approximation in the case of intractable likelihood functions and the authors make use of a Bayesian optimisation technique called Gaussian process optimisation (GPO). Meaning that the Laplace approximation is the Laplace approximation of a surrogate log-posterior. GPO is a Bayesian numerical method in the spirit of the probabilistic numerics discussed on the ‘Og a few weeks ago. In the current setting, this means iterating three steps

  1. derive an approximation of the log-posterior ξ at the current θ using SMC-ABC
  2. construct a surrogate log-posterior by a Gaussian process using the past (ξ,θ)’s
  3. determine the next value of θ

In the first step, a standard particle filter cannot be used to approximate the observed log-posterior at θ because the conditional density of observed given latent is intractable. The solution is to use ABC for the HMM model, in the spirit of many papers by Ajay Jasra and co-authors. However, I find the construction of the substitute model allowing for a particle filter very obscure… (A side effect of the heat wave?!) I can spot a noisy ABC feature in equation (7), but am at a loss as to how the reparameterisation by the transform τ is compatible with the observed-given-latent conditional being unavailable: if the pair (x,v) at time t has a closed form expression, so does (x,y), at least on principle, since y is a deterministic transform of (x,v). Another thing I do not catch is why having a particle filter available prevent the use of a pMCMC approximation.

The second step constructs a Gaussian process posterior on the log-likelihood, with Gaussian errors on the ξ’s. The Gaussian process mean is chosen as zero, while the covariance function is a Matérn function. With hyperparameters that are estimated by maximum likelihood estimators (based on the argument that the marginal likelihood is available in closed form). Turning the approach into an empirical Bayes version.

The next design point in the sequence of θ’s is the argument of the maximum of a certain acquisition function, which is chosen here as a sort of maximum regret associated with the posterior predictive associated with the Gaussian process. With possible jittering. At this stage, it reminded me of the Gaussian process approach proposed by Michael Gutmann in his NIPS poster last year.

Overall, the method is just too convoluted for me to assess its worth and efficiency without a practical implementation to… practice upon, for which I do not have time! Hence I would welcome any comment from readers having attempted such implementations. I also wonder at the lack of link with Simon Wood‘s Gaussian approximation that appeared in Nature (2010) and was well-discussed in the Read Paper of Fearnhead and Prangle (2012).

hierarchical models are not Bayesian models

Posted in Books, Kids, Statistics, University life with tags , , , , , , , on February 18, 2015 by xi'an

When preparing my OxWaSP projects a few weeks ago, I came perchance on a set of slides, entitled “Hierarchical models are not Bayesian“, written by Brian Dennis (University of Idaho), where the author argues against Bayesian inference in hierarchical models in ecology, much in relation with the previously discussed paper of Subhash Lele. The argument is the same, namely a possibly major impact of the prior modelling on the resulting inference, in particular when some parameters are hardly identifiable, the more when the model is complex and when there are many parameters. And that “data cloning” being available since 2007, frequentist methods have “caught up” with Bayesian computational abilities.

Let me remind the reader that “data cloning” means constructing a sequence of Bayes estimators corresponding to the data being duplicated (or cloned) once, twice, &tc., until the point estimator stabilises. Since this corresponds to using increasing powers of the likelihood, the posteriors concentrate more and more around the maximum likelihood estimator. And even recover the Hessian matrix. This technique is actually older than 2007 since I proposed it in the early 1990’s under the name of prior feedback, with earlier occurrences in the literature like D’Epifanio (1989) and even the discussion of Aitkin (1991). A more efficient version of this approach is the SAME algorithm we developed in 2002 with Arnaud Doucet and Simon Godsill where the power of the likelihood is increased during iterations in a simulated annealing version (with a preliminary version found in Duflo, 1996).

I completely agree with the author that a hierarchical model does not have to be Bayesian: when the random parameters in the model are analysed as sources of additional variations, as for instance in animal breeding or ecology, and integrated out, the resulting model can be analysed by any statistical method. Even though one may wonder at the motivations for selecting this particular randomness structure in the model. And at an increasing blurring between what is prior modelling and what is sampling modelling as the number of levels in the hierarchy goes up. This rather amusing set of slides somewhat misses a few points, in particular the ability of data cloning to overcome identifiability and multimodality issues. Indeed, as with all simulated annealing techniques, there is a practical difficulty in avoiding the fatal attraction of a local mode using MCMC techniques. There are thus high chances data cloning ends up in the “wrong” mode. Moreover, when the likelihood is multimodal, it is a general issue to decide which of the modes is most relevant for inference. In which sense is the MLE more objective than a Bayes estimate, then? Further, the impact of a prior on some aspects of the posterior distribution can be tested by re-running a Bayesian analysis with different priors, including empirical Bayes versions or, why not?!, data cloning, in order to understand where and why huge discrepancies occur. This is part of model building, in the end.