the penalty method

Posted in Statistics, University life with tags , , , , , , , , , , on July 7, 2016 by xi'an

“In this paper we will make conceptually simple generalization of Metropolis algorithm, by adjusting the acceptance ratio formula so that the transition probabilities are unaffected by the fluctuations in the estimate of [the acceptance ratio]…”

Last Friday, in Paris-Dauphine, my PhD student Changye Wu showed me a paper of Ceperley and Dewing entitled the penalty method for random walks with uncertain energies. Of which I was unaware of (and which alas pre-dated a recent advance made by Changye).  Despite its physics connections, the paper is actually about estimating a Metropolis-Hastings acceptance ratio and correcting the Metropolis-Hastings move for this estimation. While there is no generic solution to this problem, assuming that the logarithm of the acceptance ratio estimate is Gaussian around the true log acceptance ratio (and hence unbiased) leads to a log-normal correction for the acceptance probability.

“Unfortunately there is a serious complication: the variance needed in the noise penalty is also unknown.”

Even when the Gaussian assumption is acceptable, there is a further issue with this approach, namely that it also depends on an unknown variance term. And replacing it with an estimate induces further bias. So it may be that this method has not met with many followers because of those two penalising factors. Despite precluding the pseudo-marginal approach of Mark Beaumont (2003) by a few years, with the later estimating separately numerator and denominator in the Metropolis-Hastings acceptance ratio. And hence being applicable in a much wider collection of cases. Although I wonder if some generic approaches like path sampling or the exchange algorithm could be applied on a generic basis… [I just realised the title could be confusing in relation with the current football competition!]

commentaries in financial econometrics

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on April 27, 2016 by xi'an

My comment(arie)s on the moment approach to Bayesian inference by Ron Gallant have appeared, along with other comment(arie)s:

Invited Article
Reflections on the Probability Space Induced by Moment Conditions with
Implications for Bayesian Inference
A. Ronald Gallant . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Commentaries
Dante Amengual and Enrique Sentana .. . . . . . . . . . 248
John Geweke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253
Jae-Young Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Oliver Linton and Ruochen Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261
Christian P. Robert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Christopher A. Sims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Wei Wei and Asger Lunde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . .278
Author Response
A. Ronald Gallant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284

While commenting on commentaries is formally bound to induce an infinite loop [or l∞p], I remain puzzled by the main point of the paper, which is that setting a structural distribution on a moment function Z(x,θ) plus a prior p(θ) induces a distribution on the pair (x,θ) in a possibly weaker σ-algebra. (The two distributions may actually be incompatible.) Handling this framework requires checking that a posterior exists, which sounds rather unnatural (even though we also have to check properness of the posterior). And the meaning of such a posterior remains unclear, as for instance in this assertion that (4) above is a likelihood, when it does not define a density in x but on the object inside the exponential.

“…it is typically difficult to determine whether there exists a p(x|θ) such that the implied distribution of m(x,θ) is the one stated, and if not, what damage is done thereby” J. Geweke (p.254)

estimating constants [impression soleil levant]

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on April 25, 2016 by xi'an

The CRiSM workshop on estimating constants which took place here in Warwick from April 20 till April 22 was quite enjoyable [says most objectively one of the organisers!], with all speakers present to deliver their talks  (!) and around sixty participants, including 17 posters. It remains a exciting aspect of the field that so many and so different perspectives are available on the “doubly intractable” problem of estimating a normalising constant. Several talks and posters concentrated on Ising models, which always sound a bit artificial to me, but also are perfect testing grounds for approximations to classical algorithms.

On top of [clearly interesting!] talks associated with papers I had already read [and commented here], I had not previously heard about Pierre Jacob’s coupling SMC sequence, which paper is not yet out [no spoiler then!]. Or about Michael Betancourt’s adiabatic Monte Carlo and its connection with the normalising constant. Nicolas Chopin talked about the unnormalised Poisson process I discussed a while ago, with this feature that the normalising constant itself becomes an additional parameter. And that integration can be replaced with (likelihood) maximisation. The approach, which is based on a reference distribution (and an artificial logistic regression à la Geyer), reminded me of bridge sampling. And indirectly of path sampling, esp. when Merrilee Hurn gave us a very cool introduction to power posteriors in the following talk. Also mentioning the controlled thermodynamic integration of Chris Oates and co-authors I discussed a while ago. (Too bad that Chris Oates could not make it to this workshop!) And also pointing out that thermodynamic integration could be a feasible alternative to nested sampling.

Another novel aspect was found in Yves Atchadé’s talk about sparse high-dimension matrices with priors made of mutually exclusive measures and quasi-likelihood approximations. A simplified version of the talk being in having a non-identified non-constrained matrix later projected onto one of those measure supports. While I was aware of his noise-contrastive estimation of normalising constants, I had not previously heard Michael Gutmann give a talk on that approach (linking to Geyer’s 1994 mythical paper!). And I do remain nonplussed at the possibility of including the normalising constant as an additional parameter [in a computational and statistical sense]..! Both Chris Sherlock and Christophe Andrieu talked about novel aspects on pseudo-marginal techniques, Chris on the lack of variance reduction brought by averaging unbiased estimators of the likelihood and Christophe on the case of large datasets, recovering better performances in latent variable models by estimating the ratio rather than taking a ratio of estimators. (With Christophe pointing out that this was an exceptional case when harmonic mean estimators could be considered!)

Bayesian model averaging in astrophysics [guest post]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , on August 12, 2015 by xi'an

.[Following my posting of a misfiled 2013 blog, Ewan Cameron told me of the impact of this paper in starting his own blog and I asked him for a guest post, resulting in this analysis, much deeper than mine. No warning necessary this time!]

Back in February 2013 when “Bayesian Model Averaging in Astrophysics: A Review” by Parkinson & Liddle (hereafter PL13) first appeared on the arXiv I was a keen, young(ish) postdoc eager to get stuck into debates about anything and everything ‘astro-statistical’. And with its seemingly glaring flaws, PL13 was more grist to the mill. However, despite my best efforts on various forums I couldn’t get a decent fight started over the right way to do model averaging (BMA) in astronomy, so out of sheer frustration two months later I made my own soapbox to shout from at Another Astrostatistics Blog. Having seen PL13 reviewed recently here on Xian’s Og it feels like the right time to revisit the subject and reflect on where BMA in astronomy is today.

As pointed out to me back in 2013 by Tom Loredo, the act of Bayesian model averaging has been around much longer than its name; indeed an early astronomical example appears in Gregory & Loredo (1992) in which the posterior mean representation of an unknown signal is constructed for an astronomical “light-curve”, averaging over a set of constant and periodic candidate models. Nevertheless the wider popularisation of model averaging in astronomy has only recently taken place through a variety of applications in cosmology: e.g. Liddle, Mukherjee, Parkinson & Wang (2006) and Vardanyan, Trotta & Silk (2011).

In contrast to earlier studies like Gregory & Loredo (1992)—or the classic review on BMA by Hoeting et al. (1999)—in which the target of model averaging is typically either a utility function, a set of future observations, or a latent parameter of the observational process (e.g. the unknown “light-curve” shape) shared naturally by all competing models, the proposal of cosmological BMA studies is to produce a model-averaged version of the posterior for a given ‘shared’ parameter: a so-called “model-averaged PDF”. This proposal didn’t sit well with me back in 2013, and it still doesn’t sit well with me today. Philosophically: without a model a parameter has no meaning, so why should we want to seek meaning in the marginalised distribution of a parameter over an entire set of models? And, practically: to put it another way, without knowing the model ‘label’ to which a given mass of model-averaged parameter probability belongs there’s nothing much useful we can do with this ‘PDF’: nothing much we can say about the data we’ve just analysed and nothing much we can say about future experiments. Whereas the space of the observed data is shared automatically by all competing models, it seems to me to be somehow “un-Bayesian” to place the further restriction that the parameters of separate models share the same scale and topology. I say “un-Bayesian” since this mode of model averaging suggests a formulation of the parameter space + prior pairing stronger than the statement of one’s prior beliefs for the distribution of observable data given the model. But I would be happy to hear arguments from the other side in the comments box below … ! Continue reading

scalable Bayesian inference for the inverse temperature of a hidden Potts model

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , on April 7, 2015 by xi'an

Matt Moores, Tony Pettitt, and Kerrie Mengersen arXived a paper yesterday comparing different computational approaches to the processing of hidden Potts models and of the intractable normalising constant in the Potts model. This is a very interesting paper, first because it provides a comprehensive survey of the main methods used in handling this annoying normalising constant Z(β), namely pseudo-likelihood, the exchange algorithm, path sampling (a.k.a., thermal integration), and ABC. A massive simulation experiment with individual simulation times up to 400 hours leads to select path sampling (what else?!) as the (XL) method of choice. Thanks to a pre-computation of the expectation of the sufficient statistic E[S(Z)|β].  I just wonder why the same was not done for ABC, as in the recent Statistics and Computing paper we wrote with Matt and Kerrie. As it happens, I was actually discussing yesterday in Columbia of potential if huge improvements in processing Ising and Potts models by approximating first the distribution of S(X) for some or all β before launching ABC or the exchange algorithm. (In fact, this is a more generic desiderata for all ABC methods that simulating directly if approximately the summary statistics would being huge gains in computing time, thus possible in final precision.) Simulating the distribution of the summary and sufficient Potts statistic S(X) reduces to simulating this distribution with a null correlation, as exploited in Cucala and Marin (2013, JCGS, Special ICMS issue). However, there does not seem to be an efficient way to do so, i.e. without reverting to simulating the entire grid X…

computational methods for statistical mechanics [day #3]

Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on June 6, 2014 by xi'an

The third day [morn] at our ICMS workshop was dedicated to path sampling. And rare events. Much more into [my taste] Monte Carlo territory. The first talk by Rosalind Allen looked at reweighting trajectories that are not in an equilibrium or are missing the Boltzmann [normalizing] constant. Although the derivation against a calibration parameter looked like the primary goal rather than the tool for constant estimation. Again papers in J. Chem. Phys.! And a potential link with ABC raised by Antonietta Mira… Then Jonathan Weare discussed stratification. With a nice trick of expressing the normalising constants of the different terms in the partition as solution(s) of a Markov system

$v\mathbf{M}=v$

Because the stochastic matrix M is easier (?) to approximate. Valleau’s and Torrie’s umbrella sampling was a constant reference in this morning of talks. Arnaud Guyader’s talk was in the continuation of Toni Lelièvre’s introduction, which helped a lot in my better understanding of the concepts. Rephrasing things in more statistical terms. Like the distinction between equilibrium and paths. Or bias being importance sampling. Frédéric Cérou actually gave a sort of second part to Arnaud’s talk, using importance splitting algorithms. Presenting an algorithm for simulating rare events that sounded like an opposite nested sampling, where the goal is to get down the target, rather than up. Pushing particles away from a current level of the target function with probability ½. Michela Ottobre completed the series with an entry into diffusion limits in the Roberts-Gelman-Gilks spirit when the Markov chain is not yet stationary. In the transient phase thus.

controlled thermodynamic integral for Bayesian model comparison [reply]

Posted in Books, pictures, Running, Statistics, University life with tags , , , , , , , , , , , , on April 30, 2014 by xi'an

Chris Oates wrotes the following reply to my Icelandic comments on his paper with Theodore Papamarkou, and Mark Girolami, reply that is detailed enough to deserve a post on its own:

Thank you Christian for your discussion of our work on the Og, and also for your helpful thoughts in the early days of this project! It might be interesting to speculate on some aspects of this procedure:

(i) Quadrature error is present in all estimates of evidence that are based on thermodynamic integration. It remains unknown how to exactly compute the optimal (variance minimising) temperature ladder “on-the-fly”; indeed this may be impossible, since the optimum is defined via a boundary value problem rather than an initial value problem. Other proposals for approximating this optimum are compatible with control variates (e.g. Grosse et al, NIPS 2013, Friel and Wyse, 2014). In empirical experiments we have found that the second order quadrature rule proposed by Friel and Wyse 2014 leads to substantially reduced bias, regardless of the specific choice of ladder.

(ii) Our experiments considered first and second degree polynomials as ZV control variates. In fact, intuition specifically motivates the use of second degree polynomials: Let us presume a linear expansion of the log-likelihood in θ. Then the implied score function is constant, not depending on θ. The quadratic ZV control variates are, in effect, obtained by multiplying the score function by θ. Thus control variates can be chosen to perfectly correlate with the log-likelihood, leading to zero-variance estimators. Of course, there is an empirical question of whether higher-order polynomials are useful when this Taylor approximation is inappropriate, but they would require the estimation of many more coefficients and in practice may be less stable.

(iii) We require that the control variates are stored along the chain and that their sample covariance is computed after the MCMC has terminated. For the specific examples in the paper such additional computation is a negligible fraction of the total computational, so that we did not provide specific timings. When non-diffegeometric MCMC is used to obtain samples, or when the score is unavailable in closed-form and must be estimated, the computational cost of the procedure would necessarily increase.

For the wide class of statistical models with tractable likelihoods, employed in almost all areas of statistical application, the CTI we propose should provide state-of-the-art estimation performance with negligible increase in computational costs.