arXiv subjects

Jordan Franks

Publications and source records attributed to Jordan Franks.

6 recordsLinked to original sources

arXiv2019

Markov chain Monte Carlo importance samplers for Bayesian models with intractable likelihoods

We consider the efficient use of an approximation within Markov chain Monte Carlo (MCMC), with subsequent importance sampling (IS) correction of the Markov chain inexact output, leading to asymptotically exact inference. We detail convergence and central limit theorems for the resulting MCMC-IS estimators. We also consider the case where the approximate Markov chain is pseudo-marginal, requiring unbiased estimators for its approximate marginal target. Convergence results with asymptotic variance formulae are shown for this case, and for the case where the IS weights based on unbiased estimators are only calculated for distinct output samples of the so-called `jump' chain, which, with a suitable reweighting, allows for improved efficiency. As the IS type weights may assume negative values, extended classes of unbiased estimators may be used for the IS type correction, such as those obtained from randomised multilevel Monte Carlo. Using Euler approximations and coupling of particle filters, we apply the resulting estimator using randomised weights to the problem of parameter inference for partially observed It\^{o} diffusions. Convergence of the estimator is verified to hold under regularity assumptions which do not require that the diffusion can be simulated exactly. In the context of approximate Bayesian computation (ABC), we suggest an adaptive MCMC approach to deal with the selection of a suitably large tolerance, with IS correction possible to finer tolerance, and with provided approximate confidence intervals. A prominent question is the efficiency of MCMC-IS compared to standard direct MCMC, such as pseudo-marginal, delayed acceptance, and ABC-MCMC. We provide a comparison criterion which generalises the covariance ordering to the IS setting.

stat.CO↗

arXiv2019

On the use of approximate Bayesian computation Markov chain Monte Carlo with inflated tolerance and post-correction

Approximate Bayesian computation allows for inference of complicated probabilistic models with intractable likelihoods using model simulations. The Markov chain Monte Carlo implementation of approximate Bayesian computation is often sensitive to the tolerance parameter: low tolerance leads to poor mixing and large tolerance entails excess bias. We consider an approach using a relatively large tolerance for the Markov chain Monte Carlo sampler to ensure its sufficient mixing, and post-processing the output leading to estimators for a range of finer tolerances. We introduce an approximate confidence interval for the related post-corrected estimators, and propose an adaptive approximate Bayesian computation Markov chain Monte Carlo, which finds a `balanced' tolerance level automatically, based on acceptance rate optimisation. Our experiments show that post-processing based estimators can perform better than direct Markov chain targetting a fine tolerance, that our confidence intervals are reliable, and that our adaptive algorithm leads to reliable inference with little user specification.

stat.CO↗

arXiv2018

Unbiased inference for discretely observed hidden Markov model diffusions

We develop a Bayesian inference method for diffusions observed discretely and with noise, which is free of discretisation bias. Unlike existing unbiased inference methods, our method does not rely on exact simulation techniques. Instead, our method uses standard time-discretised approximations of diffusions, such as the Euler--Maruyama scheme. Our approach is based on particle marginal Metropolis--Hastings, a particle filter, randomised multilevel Monte Carlo, and importance sampling type correction of approximate Markov chain Monte Carlo. The resulting estimator leads to inference without a bias from the time-discretisation as the number of Markov chain iterations increases. We give convergence results and recommend allocations for algorithm inputs. Our method admits a straightforward parallelisation, and can be computationally efficient. The user-friendly approach is illustrated on three examples, where the underlying diffusion is an Ornstein--Uhlenbeck process, a geometric Brownian motion, and a 2d non-reversible Langevin equation.

stat.ME↗

arXiv2017

Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance

We establish an ordering criterion for the asymptotic variances of two consistent Markov chain Monte Carlo (MCMC) estimators: an importance sampling (IS) estimator, based on an approximate reversible chain and subsequent IS weighting, and a standard MCMC estimator, based on an exact reversible chain. Essentially, we relax the criterion of the Peskun type covariance ordering by considering two different invariant probabilities, and obtain, in place of a strict ordering of asymptotic variances, a bound of the asymptotic variance of IS by that of the direct MCMC. Simple examples show that IS can have arbitrarily better or worse asymptotic variance than Metropolis-Hastings and delayed-acceptance (DA) MCMC. Our ordering implies that IS is guaranteed to be competitive up to a factor depending on the supremum of the (marginal) IS weight. We elaborate upon the criterion in case of unbiased estimators as part of an auxiliary variable framework. We show how the criterion implies asymptotic variance guarantees for IS in terms of pseudo-marginal (PM) and DA corrections, essentially if the ratio of exact and approximate likelihoods is bounded. We also show that convergence of the IS chain can be less affected by unbounded high-variance unbiased estimators than PM and DA chains.

stat.CO↗

arXiv2016

Importance sampling type estimators based on approximate marginal MCMC

We consider importance sampling (IS) type weighted estimators based on Markov chain Monte Carlo (MCMC) targeting an approximate marginal of the target distribution. In the context of Bayesian latent variable models, the MCMC typically operates on the hyperparameters, and the subsequent weighting may be based on IS or sequential Monte Carlo (SMC), but allows for multilevel techniques as well. The IS approach provides a natural alternative to delayed acceptance (DA) pseudo-marginal/particle MCMC, and has many advantages over DA, including a straightforward parallelisation and additional flexibility in MCMC implementation. We detail minimal conditions which ensure strong consistency of the suggested estimators, and provide central limit theorems with expressions for asymptotic variances. We demonstrate how our method can make use of SMC in the state space models context, using Laplace approximations and time-discretised diffusions. Our experimental results are promising and show that the IS type approach can provide substantial gains relative to an analogous DA scheme, and is often competitive even without parallelisation.

stat.CO↗

arXiv2013

On 1-cocycles induced by a positive definite function on a locally compact abelian group

For $φ$ a normalized positive definite function on a locally compact abelian group $G$, we consider on the one hand the unitary representation $π_φ$ associated to $φ$ by the GNS construction, on the other hand the probability measure $μ_φ$ on the Pontryagin dual $\hat{G}$ provided by Bochner's theorem. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,π_φ)$ and reduced 1-cohomology $\bar{H}^1(G,π_φ)$. For example, $\bar{H}^1(G,π_φ)=0$ if and only if either $Hom(G,\mathbb{C})=0$ or $μ_φ(1_G)=0$, where $1_G$ is the trivial character of $G$.

math.RT↗