The asymptotic behaviour of (Yn, n e N) is of fundamental importance in probability theory. Indeed, if the Xj have common mean fi and variance a, then by taking each an = n/u and b„ = n a, the celebrated Central Limit Theorem tells us that (Yn) converges in distribution to a standard normal. The class of random variables which appear as limits in distribution of (Yn) for more general (an) and (b„) are called the stable random variables and they have a number of elegant properties, such as the preservation of their distribution functions under convolution. A stochastic process is said to be stable if all its finite-dimensional distributions are themselves stable and it is the systematic study of such processes which is the aim of the present volume.

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Sequential Monte Carlo (SMC) methods are a class of techniques to sample approximately from any sequence of probability distributions using a combination of importance sampling and resampling steps. This paper is concerned with the convergence analysis of a class of SMC methods where the times at which resampling occurs are computed online using criteria such as the effective sample size. This is a popular approach amongst practitioners but there are very few convergence results available for these methods. By combining semigroup techniques with an original coupling argument, we obtain functional central limit theorems and uniform exponential concentration estimates for these algorithms.

On adaptive resampling strategies for sequential Monte Carlo methods

Approximate Bayesian Computation (ABC) can be viewed as an analytic approximation of an intractable likelihood coupled with an elementary simulation step. Such a view, combined with a suitable instrumental prior distribution permits maximum-likelihood (or maximum-a-posteriori) inference to be conducted, approximately, using essentially the same techniques. An elementary approach to this problem which simply obtains a nonparametric approximation of the likelihood surface which is then maximised is developed here and the convergence of this class of algorithms is characterised theoretically. The use of non-sufficient summary statistics in this context is considered. Applying the proposed method to four problems demonstrates good performance. The proposed approach provides an alternative for approximating the maximum likelihood estimator (MLE) in complex scenarios.

/pdf/a-simple-approach-to-maximum-intractable-likelihood-3ntp2mrwra.pdf

A simple approach to maximum intractable likelihood estimation

Approximate Bayesian Computation (ABC) methods have become a “mainstream” statistical technique in the past decade, following the realisation by statisticians that they are a special type of non-parametric inference. In this survey of ABC methods, we focus on the recent literature, building on the previous survey of Marin et al. Stat Comput 21(2):279–291, 2011, [39]. Given the importance of model choice in the applications of ABC, and the associated difficulties in its implementation, we also give emphasis to this aspect of ABC techniques.

Approximate Bayesian Computation: A Survey on Recent Results

Making decisions and predictions from noisy observations are two important and challenging problems in many areas of society. Some examples of applications are recommendation systems for online sho ...

/pdf/accelerating-monte-carlo-methods-for-bayesian-inference-in-4cy9suknih.pdf

Accelerating Monte Carlo methods for Bayesian inference in dynamical models

In this article we focus on Maximum Likelihood estimation (MLE) for the static model parameters of hidden Markov models (HMMs). We will consider the case where one cannot or does not want to compute the conditional likelihood density of the observation given the hidden state because of increased computational complexity or analytical intractability. Instead we will assume that one may obtain samples from this conditional likelihood and hence use approximate Bayesian computation (ABC) approximations of the original HMM. Although these ABC approximations will induce a bias, this can be controlled to arbitrary precision via a positive parameter ϵ, so that the bias decreases with decreasing ϵ. We first establish that when using an ABC approximation of the HMM for a fixed batch of data, then the bias of the resulting log- marginal likelihood and its gradient is no worse than $\mathcal{O}(n\epsilon)$, where n is the total number of data-points. Therefore, when using gradient methods to perform MLE for the ABC approximation of the HMM, one may expect parameter estimates of reasonable accuracy. To compute an estimate of the unknown and fixed model parameters, we propose a gradient approach based on simultaneous perturbation stochastic approximation (SPSA) and Sequential Monte Carlo (SMC) for the ABC approximation of the HMM. The performance of this method is illustrated using two numerical examples.

/pdf/gradient-free-parameter-estimation-for-hidden-markov-models-56a3fk136i.pdf

Gradient Free Parameter Estimation for Hidden Markov Models with Intractable Likelihoods

In this article, we consider approximate Bayesian parameter inference for observation-driven time series models. Such statistical models appear in a wide variety of applications, including econometrics and applied mathematics. This article considers the scenario where the likelihood function cannot be evaluated pointwise; in such cases, one cannot perform exact statistical inference, including parameter estimation, which often requires advanced computational algorithms, such as Markov Chain Monte Carlo (MCMC). We introduce a new approximation based upon Approximate Bayesian Computation (ABC). Under some conditions, we show that as n → ∞, with n the length of the time series, the ABC posterior has, almost surely, a Maximum A Posteriori (MAP) estimator of the parameters that is often different from the true parameter. However, a noisy ABC MAP, which perturbs the original data, asymptotically converges to the true parameter, almost surely. In order to draw statistical inference, for the ABC approximation adopted, standard MCMC algorithms can have acceptance probabilities that fall at an exponential rate in n and slightly more advanced algorithms can mix poorly. We develop a new and improved MCMC kernel, which is based upon an exact approximation of a marginal algorithm, whose cost per iteration is random, but the expected cost, for good performance, is shown to be O(n2) per iteration. We implement our new MCMC kernel for parameter inference from models in econometrics.

/pdf/approximate-inference-for-observation-driven-time-series-zf5f8hyftd.pdf

Approximate Inference for Observation-Driven Time Series Models with Intractable Likelihoods

In the following article we consider approximate Bayesian parameter inference for observation driven time series models. Such statistical models appear in a wide variety of applications, including econometrics and applied mathematics. This article considers the scenario where the likelihood function cannot be evaluated point-wise; in such cases, one cannot perform exact statistical inference, including parameter estimation, which often requires advanced computational algorithms, such as Markov chain Monte Carlo (MCMC). We introduce a new approximation based upon approximate Bayesian computation (ABC). Under some conditions, we show that as $n\rightarrow\infty$, with $n$ the length of the time series, the ABC posterior has, almost surely, a maximum \emph{a posteriori} (MAP) estimator of the parameters which is different from the true parameter. However, a noisy ABC MAP, which perturbs the original data, asymptotically converges to the true parameter, almost surely. In order to draw statistical inference, for the ABC approximation adopted, standard MCMC algorithms can have acceptance probabilities that fall at an exponential rate in $n$ and slightly more advanced algorithms can mix poorly. We develop a new and improved MCMC kernel, which is based upon an exact approximation of a marginal algorithm, whose cost per-iteration is random but the expected cost, for good performance, is shown to be $\mathcal{O}(n^2)$ per-iteration.

Approximate Inference for Observation Driven Time Series Models with Intractable Likelihoods

In this article we focus on Maximum Likelihood estimation (MLE) for the static parameters of hidden Markov models (HMMs). We will consider the case where one cannot or does not want to compute the conditional likelihood density of the observation given the hidden state because of increased computational complexity or analytical intractability. Instead we will assume that one may obtain samples from this conditional likelihood and hence use approximate Bayesian computation (ABC) approximations of the original HMM. ABC approximations are biased, but the bias can be controlled to arbitrary precision via a parameter \epsilon>0; the bias typically goes to zero as \epsilon \searrow 0. We first establish that the bias in the log-likelihood and gradient of the log-likelihood of the ABC approximation, for a fixed batch of data, is no worse than \mathcal{O}(n\epsilon), n being the number of data; hence, for computational reasons, one might expect reasonable parameter estimates using such an ABC approximation. Turning to the computational problem of estimating $\theta$, we propose, using the ABC-sequential Monte Carlo (SMC) algorithm in Jasra et al. (2012), an approach based upon simultaneous perturbation stochastic approximation (SPSA). Our method is investigated on two numerical examples

/pdf/static-parameter-estimation-for-abc-approximations-of-hidden-oim2lb1ztu.pdf

Static Parameter Estimation for ABC Approximations of Hidden Markov Models

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Gradient Free Parameter Estimation for Hidden Markov Models with Intractable Likelihoods

Approximate Inference for Observation-Driven Time Series Models with Intractable Likelihoods

Approximate Inference for Observation Driven Time Series Models with Intractable Likelihoods

Static Parameter Estimation for ABC Approximations of Hidden Markov Models