Particle-based likelihood inference in partially observed diffusion processes using generalised Poisson estimators

This paper concerns the use of the expectation-maximisation (EM) algorithm for inference in partially observed diffusion processes. In this context, a well known problem is that all except a few diffusion processes lack closed-form expressions of the transition densities. Thus, in order to estimate efficiently the EM intermediate quantity we construct, using novel techniques for unbiased estimation of diffusion transition densities, a random weight fixed-lag auxiliary particle smoother, which avoids the well known problem of particle trajectory degeneracy in the smoothing mode. The estimator is justified theoretically and demonstrated on a simulated example

On the auxiliary particle filter

In this article we study asymptotic properties of weighted samples produced by the auxiliary particle filter (APF) proposed by pitt and shephard (1999). Besides establishing a central limit theorem (CLT) for smoothed particle estimates, we also derive bounds on the Lp error and bias of the same for a finite particle sample size. By examining the recursive formula for the asymptotic variance of the CLT we identify first-stage importance weights for which the increase of asymptotic variance at a single iteration of the algorithm is minimal. In the light of these findings, we discuss and demonstrate on several examples how the APF algorithm can be improved.Comment: 26 page

Long-term stability of sequential Monte Carlo methods under verifiable conditions

This paper discusses particle filtering in general hidden Markov models (HMMs) and presents novel theoretical results on the long-term stability of bootstrap-type particle filters. More specifically, we establish that the asymptotic variance of the Monte Carlo estimates produced by the bootstrap filter is uniformly bounded in time. On the contrary to most previous results of this type, which in general presuppose that the state space of the hidden state process is compact (an assumption that is rarely satisfied in practice), our very mild assumptions are satisfied for a large class of HMMs with possibly noncompact state space. In addition, we derive a similar time uniform bound on the asymptotic $\mathsf{L}^p$ error. Importantly, our results hold for misspecified models; that is, we do not at all assume that the data entering into the particle filter originate from the model governing the dynamics of the particles or not even from an HMM.Comment: Published in at http://dx.doi.org/10.1214/13-AAP962 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sequential sampling of junction trees for decomposable graphs

The junction-tree representation provides an attractive structural property for organizing a decomposable graph. In this study, we present a novel stochastic algorithm, which we call the junction-tree expander, for sequential sampling of junction trees for decomposable graphs. We show that recursive application of the junction-tree expander, expanding incrementally the underlying graph with one vertex at a time, has full support on the space of junction trees with any given number of underlying vertices. A direct application of our suggested algorithm is demonstrated in a sequential Monte Carlo setting designed for sampling from distributions on spaces of decomposable graphs, where the junction-tree expander can be effectively employed as proposal kernel; see the companion paper Olsson et al. 2019 [16]. A numerical study illustrates the utility of our approach by two examples: in the first one, how the junction-tree expander can be incorporated successfully into a particle Gibbs sampler for Bayesian structure learning in decomposable graphical models; in the second one, we provide an unbiased estimator of the number of decomposable graphs for a given number of vertices. All the methods proposed in the paper are implemented in the Python library trilearn.Comment: 31 pages, 7 figure

“Tax Simplification”—Grave Threat to the Charitable Contribution Deduction: The Problem and a Proposed Solution

The present National Administration has continued to support proposed legislative changes aimed at substantially reducing the number of income tax returns in which deductions are itemized. The author contends that these tax simplification proposals are incompatible with the preservation of the charitable contribution deduction and would undermine the position of voluntary charitable organizations by reducing the incentives for giving. He proposes a solution to this dilemma by promoting the charitable contribution deduction, with certain limitations, to the position of a deduction from gross income, rather than a deduction from adjusted gross income