Archive for birth-and-death process

Naturally amazed at non-identifiability

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on May 27, 2020 by xi'an

A Nature paper by Stilianos Louca and Matthew W. Pennell,  Extant time trees are consistent with a myriad of diversification histories, comes to the extraordinary conclusion that birth-&-death evolutionary models cannot distinguish between several scenarios given the available data! Namely, stem ages and daughter lineage ages cannot identify the speciation rate function λ(.), the extinction rate function μ(.)  and the sampling fraction ρ inherently defining the deterministic ODE leading to the number of species predicted at any point τ in time, N(τ). The Nature paper does not seem to make a point beyond the obvious and I am rather perplexed at why it got published [and even highlighted]. A while ago, under the leadership of Steve, PNAS decided to include statistician reviewers for papers relying on statistical arguments. It could time for Nature to move there as well.

“We thus conclude that two birth-death models are congruent if and only if they have the same rp and the same λp at some time point in the present or past.” [S.1.1, p.4]

Or, stated otherwise, that a tree structured dataset made of branch lengths are not enough to identify two functions that parameterise the model. The likelihood looks like

\frac{\rho^{n-1}\Psi(\tau_1,\tau_0)}{1-E(\tau)}\prod_{i=1}^n \lambda(\tau_i)\Psi(s_{i,1},\tau_i)\Psi(s_{i,2},\tau_i)$

where E(.) is the probability to survive to the present and ψ(s,t) the probability to survive and be sampled between times s and t. Sort of. Both functions depending on functions λ(.) and  μ(.). (When the stem age is unknown, the likelihood changes a wee bit, but with no changes in the qualitative conclusions. Another way to write this likelihood is in term of the speciation rate λp

e^{-\Lambda_p(\tau_0)}\prod_{i=1}^n\lambda_p(\tau_I)e^{-\Lambda_p(\tau_i)}

where Λp is the integrated rate, but which shares the same characteristic of being unable to identify the functions λ(.) and μ(.). While this sounds quite obvious the paper (or rather the supplementary material) goes into fairly extensive mode, including “abstract” algebra to define congruence.

 

“…we explain why model selection methods based on parsimony or “Occam’s razor”, such as the Akaike Information Criterion and the Bayesian Information Criterion that penalize excessive parameters, generally cannot resolve the identifiability issue…” [S.2, p15]

As illustrated by the above quote, the supplementary material also includes a section about statistical model selections techniques failing to capture the issue, section that seems superfluous or even absurd once the fact that the likelihood is constant across a congruence class has been stated.

non-reversibility in discrete spaces

Posted in Books, Statistics, University life with tags , , , , , , , , , on January 3, 2020 by xi'an

Following a recent JASA paper by Giacomo Zanella (which I have not yet read but is discussed on this blog), Sam Power and Jacob Goldman have recently arXived a paper on Accelerated sampling on discrete spaces with non-reversible Markov processes, where they use continuous-time, non-reversible algorithms à la PDMP, even though differential equations do not exist on discrete spaces. More specifically, they devise discrete versions of the coordinate sampler and of the Zig-Zag sampler, using Markov jump processes instead of differential equations, with detailed balance on the jump rate rather than the Markov kernel. A use of jump processes originating at least from Peskun (1973) and connected with MCMC algorithms in Matthew Stephens‘ 1999 PhD thesis. A neat thing about discrete settings is that the jump process can be implemented with no discretisation! However, as we noticed when working on birth-and-death processes with Olivier Cappé and Tobias Rydèn, there is a potential for disastrous implementation if an infinite sequence of instantaneous moves (out of zero probability states) is proposed.

The authors make the further assumption(s) that the discrete space is endowed with a graphical structure with a group G acting upon this graph, with an involution keeping the target (or a completion of the original target) invariant. In this framework, reversibility amounts to repeatedly using (group) generators þ with a low order (as in Bayesian variable selection, binary spin systems, where þ.þ=id, and other permutation problems), since they bring the chain back to its starting point. Their first sampler is called a Tabu sampler for avoiding such behaviour, forcing the next step to use other generators þ in the generator set Þ thanks to a binary auxiliary variable that partitions Þ into forward vs backward moves. For high order generators, the discrete coordinate and Zig-Zag samplers are instead repeatedly using the same generator (although it is unclear to me why this is beneficial, given that neither graph nor generator is not necessarily linked with the target). With the coordinate sampler being again much cheaper since it only looks at one direction in the generator group.

The paper contains a range of comparisons with (only) Zanella’s sampler, some presenting heavy gains in terms of ESS. Including one on hundreds of sensors in a football stadium. As I am not particularly familiar with these examples, except for the Bayesian variable selection one, I found it rather hard to determine whether or not the compared samplers were indeed exploring the entirety of the (highly complex and highly dimensional) target. The collection of examples is however quite rich and support the use of such non-reversible schemes. It may also be that the discrete nature of the target could facilitate the theoretical study of their convergence properties.

Big Bayes goes South

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , , on December 5, 2018 by xi'an

At the Big [Data] Bayes conference this week [which I found quite exciting despite a few last minute cancellations by speakers] there were a lot of clustering talks including the ones by Amy Herring (Duke), using a notion of centering that should soon appear on arXiv. By Peter Müller (UT, Austin) towards handling large datasets. Based on a predictive recursion that takes one value at a time, unsurprisingly similar to the update of Dirichlet process mixtures. (Inspired by a 1998 paper by Michael Newton and co-authors.) The recursion doubles in size at each observation, requiring culling of negligible components. Order matters? Links with Malsiner-Walli et al. (2017) mixtures of mixtures. Also talks by Antonio Lijoi and Igor Pruenster (Boconni Milano) on completely random measures that are used in creating clusters. And by Sylvia Frühwirth-Schnatter (WU Wien) on creating clusters for the Austrian labor market of the impact of company closure. And by Gregor Kastner (WU Wien) on multivariate factor stochastic models, with a video of a large covariance matrix evolving over time and catching economic crises. And by David Dunson (Duke) on distance clustering. Reflecting like myself on the definitely ill-defined nature of the [clustering] object. As the sample size increases, spurious clusters appear. (Which reminded me of a disagreement I had had with David McKay at an ICMS conference on mixtures twenty years ago.) Making me realise I missed the recent JASA paper by Miller and Dunson on that perspective.

Some further snapshots (with short comments visible by hovering on the picture) of a very high quality meeting [says one of the organisers!]. Following suggestions from several participants, it would be great to hold another meeting at CIRM in a near future. Continue reading

normalising constants of G-Wishart densities

Posted in Books, Statistics with tags , , , , , , on June 28, 2017 by xi'an

Abdolreza Mohammadi, Hélène Massam, and Gérard Letac arXived last week a paper on a new approximation of the ratio of two normalising constants associated with two G-Wishart densities associated with different graphs G. The G-Wishart is the generalisation of the Wishart distribution by Alberto Roverato to the case when some entries of the matrix are equal to zero, which locations are associated with the graph G. While enjoying the same shape as the Wishart density, this generalisation does not enjoy a closed form normalising constant. Which leads to an intractable ratio of normalising constants when doing Bayesian model selection across different graphs.

Atay-Kayis and Massam (2005) expressed the ratio as a ratio of two expectations, and the current paper shows that this leads to an approximation of the ratio of normalising constants for a graph G against the graph G augmented by the edge e, equal to

Γ(½{δ+d}) / 2 √π Γ(½{δ+d+1})

where δ is the degree of freedom of the G-Wishart and d is the number of minimal paths of length 2 linking the two end points of e. This is remarkably concise and provides a fast approximation. (The proof is quite involved, by comparison.) Which can then be used in reversible jump MCMC. The difficulty is obviously in evaluating the impact of the approximation on the target density, as there is no manageable available alternative to calibrate the approximation. In a simulation example where such an alternative is available, the error is negligible though.

SMC on a sequence of increasing dimension targets

Posted in Statistics with tags , , , , , , , , , on February 15, 2017 by xi'an

mixdirRichard Everitt and co-authors have arXived a preliminary version of a paper entitled Sequential Bayesian inference for mixture models and the coalescent using sequential Monte Carlo samplers with transformations. The central notion is an SMC version of the Carlin & Chib (1995) completion in the comparison of models in different dimensions. Namely to create auxiliary variables for each model in such a way that the dimension of the completed models are all the same. (Reversible jump MCMC à la Peter Green (1995) can also be interpreted this way, even though only relevant bits of the completion are used in the transitions.) I find the paper and the topic most interesting if only because it relates to earlier papers of us on population Monte Carlo. It also brought to my awareness the paper by Karagiannis and Andrieu (2013) on annealed reversible jump MCMC that I had missed at the time it appeared. The current paper exploits this annealed expansion in the devising of the moves. (Sequential Monte Carlo on a sequence of models with increasing dimension has been studied in the past.)

The way the SMC is described in the paper, namely, reweight-subsample-move, does not strike me as the most efficient as I would try to instead move-reweight-subsample, using a relevant move that incorporate the new model and hence enhance the chances of not rejecting.

One central application of the paper is mixture models with an unknown number of components. The SMC approach applied to this problem means creating a new component at each iteration t and moving the existing particles after adding the parameters of the new component. Since using the prior for this new part is unlikely to be at all efficient, a split move as in Richardson and Green (1997) can be considered, which brings back the dreaded Jacobian of RJMCMC into the picture! Here comes an interesting caveat of the method, namely that the split move forces a choice of the split component of the mixture. However, this does not appear as a strong difficulty, solved in the paper by auxiliary [index] variables, but possibly better solved by a mixture representation of the proposal, as in our PMC [population Monte Carlo] papers. Which also develop a family of SMC algorithms, incidentally. We found there that using a mixture representation of the proposal achieves a provable variance reduction.

“This puts a requirement on TSMC that the single transition it makes must be successful.”

As pointed by the authors, the transformation SMC they develop faces the drawback that a given model is only explored once in the algorithm, when moving to the next model. On principle, there would be nothing wrong in including regret steps, retracing earlier models in the light of the current one, since each step is an importance sampling step valid on its own right. But SMC also offers a natural albeit potentially high-varianced approximation to the marginal likelihood, which is quite appealing when comparing with an MCMC outcome. However, it would have been nice to see a comparison with alternative estimates of the marginal in the case of mixtures of distributions. I also wonder at the comparative performances of a dual approach that would be sequential in the number of observations as well, as in Chopin (2004) or our first population Monte Carlo paper (Cappé et al., 2005), since subsamples lead to tempered versions of the target and hence facilitate moves between models, being associated with flatter likelihoods.