Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models

Abstract

A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set of sufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach -including a computationally efficient version based on the auxiliary score -achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multi-parameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the paper.

Recalibration: A post-processing method for approximate Bayesian computation

“…This use of the auxiliary model is different to some previous usages where the MAP or MLE of the auxiliary model defined summary statistics that were then used for a standard ABC analysis (e.g. Gleim and Pigorsch 2013;Drovandi et al 2015;Martin et al 2017).…”

Multivariate Count Data Models for Time Series Forecasting

“…where η t ∼ N(0, Σ η ). Equation (24) shows that the observations have Poisson distribution with mean λ t defined through the Equation ( 25), and λ t nonlinearly depends on the latent process h t which is defined through Equation (26). Note that the latent process is defined through a VAR(p) process, and hence corresponding theory applies.…”

“…As is discussed in [22], pMCMC outperforms other methods (variational Bayes [23], integrated nested Laplace approximation [24] and Riemann manifold Hamiltonian Monte Carlo [25]) in terms of parameter estimation. There are other recent methods for the estimation of the state-space models such as auxiliary likelihood-based approximate Bayesian computation [26] and variational Sequential Monte Carlo [27], but their performance has to be investigated further, which is outside of the scope of this paper.…”

Likelihood-free approximate Gibbs sampling

“…State space models provide a flexible and well-structured framework to probabilistically describe an extensive array of applied problems (West and Harrison 1997;Petris 2010 but increasing efforts to tackle this issue are being made (Jasra et al 2012;Dean et al 2014;Martin et al 2014;Calvet and Czellar 2012;Yildirim et al 2013;Picchini and Samson 2018;Martin et al 2016). Our approach extends the method given by Peters et al (2016).…”