probability comparisons

[Nature on] simulations driving the world’s response to COVID-19

Nature of 02 April 2020 has a special section on simulation methods used to assess and predict the pandemic evolution. Calling for caution as the models used therein, like the standard ODE S(E)IR models, which rely on assumptions on the spread of the data and very rarely on data, especially in the early stages of the pandemic. One epidemiologist is quote stating “We’re building simplified representations of reality” but this is not dire enough, as “simplified” evokes “less precise” rather than “possibly grossly misleading”. (The graph above is unrelated to the Nature cover and appears to me as particularly appalling in mixing different types of data, time-scale, population at risk, discontinuous updates, and essentially returning no information whatsoever.)

“[the model] requires information that can be only loosely estimated at the start of an epidemic, such as the proportion of infected people who die, and the basic reproduction number (…) rough estimates by epidemiologists who tried to piece together the virus’s basic properties from incomplete information in different countries during the pandemic’s early stages. Some parameters, meanwhile, must be entirely assumed.”

The report mentions that the team at Imperial College, which predictions impacted the UK Government decisions, also used an agent-based model, with more variability or stochasticity in individual actions, which require even more assumptions or much more refined, representative, and trustworthy data.

“Unfortunately, during a pandemic it is hard to get data — such as on infection rates — against which to judge a model’s projections.”

Unfortunately, the paper was written in the early days of the rise of cases in the UK, which means predictions were not much opposed to actual numbers of deaths and hospitalisations. The following quote shows how far off they can fall from reality:

“the British response, Ferguson said on 25 March, makes him “reasonably confident” that total deaths in the United Kingdom will be held below 20,000.”

since the total number as of April 29 is above 21,000 24,000 29,750 and showing no sign of quickly slowing down… A quite useful general public article, nonetheless.

automatic adaptation of MCMC algorithms

“A typical adaptive MCMC sampler will approximately optimize performance given the kind of sampler chosen in the first place, but it will not optimize among the variety of samplers that could have been chosen.”

Last February (2018), Dao Nguyen and five co-authors arXived a paper that I missed. On a new version of adaptive MCMC that aims at selecting a wider range of proposal kernels. Still requiring a by-hand selection of this collection of kernels… Among the points addressed, beyond the theoretical guarantees that the adaptive scheme does not jeopardize convergence to the proper target, are a meta-exploration of the set of combinations of samplers and integration of the computational speed in the assessment of each sampler. Including the very difficulty of assessing mixing. One could deem the index of the proposal as an extra (cyber-)parameter to its generic parameter (like the scale in the random walk), but the discreteness of this index makes the extension more delicate than expected. And justifies the distinction between internal and external parameters. The notion of a worst-mixing dimension is quite appealing and connects with the long-term hope that one could spend the maximum fraction of the sampler runtime over the directions that are poorly mixing, while still keeping the target as should be. The adaptive scheme is illustrated on several realistic models with rather convincing gains in efficiency and time.

The convergence tools are inspired from Roberts and Rosenthal (2007), with an assumption of uniform ergodicity over all kernels considered therein which is both strong and delicate to assess in practical settings. Efficiency is rather unfortunately defined in terms of effective sample size, which is a measure of correlation or lack thereof, but which does not relate to the speed of escape from the basin of attraction of the starting point. I also wonder at the pertinence of the estimation of the effective sample size when the chain is based on different successive kernels, since the lack of correlation may be due to another kernel. Another calibration issue is the internal clock that relates to the average number of iterations required to tune properly a specific kernel, which again may be difficult to assess in a realistic situation. A last query is whether or not this scheme could be compared with an asynchronous (and valid) MCMC approach that exploits parallel capacities of the computer.

calibrating approximate credible sets

Earlier this week, Jeong Eun Lee, Geoff Nicholls, and Robin Ryder arXived a paper on the calibration of approximate Bayesian credible intervals. (Warning: all three authors are good friends of mine!) They start from the core observation that dates back to Monahan and Boos (1992) of exchangeability between θ being generated from the prior and φ being generated from the posterior associated with one observation generated from the prior predictive. (There is no name for this distribution, other than the prior, that is!) A setting amenable to ABC considerations! Actually, Prangle et al. (2014) relies on this property for assessing the ABC error, while pointing out that the test for exchangeability is not fool-proof since it works equally for two generations from the prior.

“The diagnostic tools we have described cannot be “fooled” in quite the same way checks based on the exchangeability can be.”

The paper thus proposes methods for computing the coverage [under the true posterior] of a credible set computed using an approximate posterior. (I had to fire up a few neurons to realise this was the right perspective, rather than the reverse!) A first solution to approximate the exact coverage of the approximate credible set is to use logistic regression, instead of the exact coverage, based on some summary statistics [not necessarily in an ABC framework]. And a simulation outcome that the parameter [simulated from the prior] at the source of the simulated data is within the credible set. Another approach is to use importance sampling when simulating from the pseudo-posterior. However this sounds dangerously close to resorting to an harmonic mean estimate, since the importance weight is the inverse of the approximate likelihood function. Not that anything unseemly transpires from the simulations.

 

what is a large Kullback-Leibler divergence?

A question that came up on X validated is about scaling a Kullback-Leibler divergence. A fairly interesting question in my opinion since this pseudo-distance is neither naturally nor universally scaled. Take for instance the divergence between two Gaussian

which is scaled by the standard deviation of the second Normal. There is no absolute bound in this distance for which it can be seen as large. Bypassing the coding analogy from signal processing, which has never been clear to me, he only calibration I can think of is statistical, namely to figure out a value extreme for two samples from the same distribution. In the sense of the Kullback between the corresponding estimated distributions. The above is an illustration, providing the distribution of the Kullback-Leibler divergences from samples from a Gamma distribution, for sample sizes n=15 and n=150. The sample size obviously matters.