Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation

Approximate Bayesian computation has emerged as a standard computational tool when dealing with the increasingly common scenario of completely intractable likelihood functions in Bayesian inference. We show that many common Markov chain Monte Carlo kernels used to facilitate inference in this setting can fail to be variance bounding, and hence geometrically ergodic, which can have consequences for the reliability of estimates in practice. This phenomenon is typically independent of the choice of tolerance in the approximation. We then prove that a recently introduced Markov kernel in this setting can inherit variance bounding and geometric ergodicity from its intractable Metropolis--Hastings counterpart, under reasonably weak and manageable conditions. We show that the computational cost of this alternative kernel is bounded whenever the prior is proper, and present indicative results on an example where spectral gaps and asymptotic variances can be computed, as well as an example involving inference for a partially and discretely observed, time-homogeneous, pure jump Markov process. We also supply two general theorems, one of which provides a simple sufficient condition for lack of variance bounding for reversible kernels and the other provides a positive result concerning inheritance of variance bounding and geometric ergodicity for mixtures of reversible kernels.Comment: 26 pages, 10 figure

Investing under model uncertainty: decision based evaluation of exchange rate forecasts in the US, UK and Japan

We evaluate the forecast performance of a range of theory-based and atheoretical models explaining exchange rates in the US, UK and Japan. A decision-making environment is fully described for an investor who optimally allocates portfolio shares to domestic and foreign assets. Methods necessary to compute and use forecasts in this context are proposed, including the means of combining density forecasts to deal with model uncertainty. An out-of-sample forecast evaluation exercise is described using both statistical criteria and decision-based criteria. The theory-based models are found to perform relatively well when their forecasts are judged by their economic value

Which ergodic averages have finite asymptotic variance?

We show that the class of $L^2$ functions for which ergodic averages of a reversible Markov chain have finite asymptotic variance is determined by the class of $L^2$ functions for which ergodic averages of its associated jump chain have finite asymptotic variance. This allows us to characterize completely which ergodic averages have finite asymptotic variance when the Markov chain is an independence sampler. In addition, we obtain a simple sufficient condition for all ergodic averages of $L^2$ functions of the primary variable in a pseudo-marginal Markov chain to have finite asymptotic variance

Twisted particle filters

We investigate sampling laws for particle algorithms and the influence of these laws on the efficiency of particle approximations of marginal likelihoods in hidden Markov models. Among a broad class of candidates we characterize the essentially unique family of particle system transition kernels which is optimal with respect to an asymptotic-in-time variance growth rate criterion. The sampling structure of the algorithm defined by these optimal transitions turns out to be only subtly different from standard algorithms and yet the fluctuation properties of the estimates it provides can be dramatically different. The structure of the optimal transition suggests a new class of algorithms, which we term "twisted" particle filters and which we validate with asymptotic analysis of a more traditional nature, in the regime where the number of particles tends to infinity.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1167 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Experience gained in operation of the VLF ATD lightning location system

The United Kingdom (UK) Meteorological Office's Very Low Frequency (VLF) Arrival Time Difference (ATD) System for long-range location of lightning flashes started automatic international issue of lightning-location products on 17 Jun. 1988. Data from before and after this formal start-date were carefully scrutinized to judge performance. Techniques for estimating location accuracy include internal consistency and comparisons against other systems. Other areas studied were range (up to several thousand km); detection efficiency, saturation effects in active situations, and communication difficulties (for this redundant system); and spurious fix rate. Care was taken to assess the potential of the system, in addition to identifying the operational difficulties of the present implementation

Measuring the natural output gap using actual and expected output data

An output gap measure is suggested based on the Beveridge-Nelson decomposition of output using a vector-autoregressive model that includes data on actual output and on expected output obtained from surveys. The paper explains the advantages of using survey data in business cycle analysis and the gap is provided economic meaning by relating it to the natural level of output defined in Dynamic Stochastic General Equilibrium models. The measure is applied to quarterly US data over the period 1970q1-2007q4 and the resultant gap estimates are shown to have sensible statistical properties and perform well in explaining inflation in estimates of New Keynesian Phillips curves

The iterated auxiliary particle filter

We present an offline, iterated particle filter to facilitate statistical inference in general state space hidden Markov models. Given a model and a sequence of observations, the associated marginal likelihood L is central to likelihood-based inference for unknown statistical parameters. We define a class of "twisted" models: each member is specified by a sequence of positive functions psi and has an associated psi-auxiliary particle filter that provides unbiased estimates of L. We identify a sequence psi* that is optimal in the sense that the psi*-auxiliary particle filter's estimate of L has zero variance. In practical applications, psi* is unknown so the psi*-auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate psi*, and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the bootstrap particle filter in challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm