## Archive for ODEs

## consistency of ABC

Posted in pictures, Statistics, Travel, University life with tags ABC, consistency, convergence diagnostics, Ian Potter collection, identifiability, indirect inference, MA(p) model, Melbourne, Monash University, ODEs on August 25, 2015 by xi'an**A**long with David Frazier and Gael Martin from Monash University, Melbourne, we have just completed (and arXived) a paper on the (Bayesian) consistency of ABC methods, producing sufficient conditions on the summary statistics to ensure consistency of the ABC posterior. Consistency in the sense of the prior concentrating at the true value of the parameter when the sample size and the inverse tolerance (intolerance?!) go to infinity. The conditions are essentially that the summary statistics concentrates around its mean and that this mean identifies the parameter. They are thus weaker conditions than those found earlier consistency results where the authors considered convergence to the genuine posterior distribution (given the summary), as for instance in Biau et al. (2014) or Li and Fearnhead (2015). We do not require here a specific rate of decrease to zero for the tolerance ε. But still they do not hold all the time, as shown for the MA(2) example and its first two autocorrelation summaries, example we started using in the Marin et al. (2011) survey. We further propose a consistency assessment based on the main consistency theorem, namely that the ABC-based estimates of the marginal posterior densities for the parameters should vary little when adding extra components to the summary statistic, densities estimated from simulated data. And that the mean of the resulting summary statistic is indeed one-to-one. This may sound somewhat similar to the stepwise search algorithm of Joyce and Marjoram (2008), but those authors aim at obtaining a vector of summary statistics that is as informative as possible. We also examine the consistency conditions when using an auxiliary model as in indirect inference. For instance, when using an AR(2) auxiliary model for estimating an MA(2) model. And ODEs.

## hypothesis testing for MCMC

Posted in Books, Statistics, University life with tags Markov chain Monte Carlo algorithm, Markov chains, MCMC, ODEs, sequential testing on October 6, 2014 by xi'an**A** recent arXival by Benjamin Gyori and Daniel Paulin considers sequential testing based on MCMC simulation. The test is about an expectation under the target and stationary distribution of the Markov chain (i.e., the posterior in a Bayesian setting). Hence testing whether or not the posterior expectation is below a certain bound is not directly relevant from a Bayesian perspective. One would test instead whether or not *the parameter itself* is below the bound… The paper is then more a study of sequential tests when the data is a Markov chain than in any clear connection with MCMC topics. Despite the paper including an example of a Metropolis-Hastings scheme for approximating the posterior on the parameters of an ODE. I am a bit puzzled by the purpose of the test, as I was rather expecting tests connected with the convergence of the Markov chain or of the empirical mean. (But, given the current hour, I may also have missed a crucial point!)