Adaptive Metropolis algorithm using variational Bayesian adaptive Kalman filter

Abstract

Markov chain Monte Carlo (MCMC) methods are powerful computational tools for analysis of complex statistical problems. However, their computational efficiency is highly dependent on the chosen proposal distribution, which is generally difficult to find. One way to solve this problem is to use adaptive MCMC algorithms which automatically tune the statistics of a proposal distribution during the MCMC run. A new adaptive MCMC algorithm, called the variational Bayesian adaptive Metropolis (VBAM) algorithm, is developed. The VBAM algorithm updates the proposal covariance matrix using the variational Bayesian adaptive Kalman filter (VB-AKF). A strong law of large numbers for the VBAM algorithm is proven. The empirical convergence results for three simulated examples and for two real data examples are also provided.

A Variational Bayesian Multiple Particle Filtering Scheme for Large-Dimensional Systems

“…Applying the unsupervised scheme to a real-world problem, in which some (or all) of the hyper-parameters are not known, and thus need to be estimated along with the state, would then be of great interest. (21), (29), (30) We present here the main steps for computing the computational complexity of the sampling and weighting steps of the proposed VBMPF in nonlinear system (17) with Q n diagonal; (32), (33), (36), (37), (39) and (40) can be obtained similarly.…”

Fast Kalman-Like Filtering for Large-Dimensional Linear and Gaussian State-Space Models

“…4, we can easily verify that q(x k n+1 |y 0:n ) is Gaussian with parameters given by (50) and (51), and that q(x k n |y 0:n ) is Gaussian with a variance v k n|n given by (48) and a mean equal to…”

“…The VB approach has been already used in the context of the KF considering (reasonable) low-dimensional state-space systems (see for instance [40] [43] [44] [45] [48] and references therein). More precisely, the work in [40] aimed at estimating the system state, x n , and the measurement noise covariance, R n .…”

“…A similar algorithm has then been derived in [45] in the context of backward state-space systems, i.e., systems that evolve in decreasing time direction, and for which a robust VB-like fixed-interval Kalman smoother has been also proposed. More recently, an adaptive VB-like KF has been introduced in [48] to update the covariance of the (assumed Gaussian) proposal distribution of the Markov Chain Monte Carlo-based Metropolis algorithm. Unlike these works, we focus here on large-dimensional state-space systems and we adopt the idea of splitting the system state using the VB approach, which, in the best of our knowledge, is original.…”

Locating a two-wheeled robot using extended Kalman filter

“…In addition, the Monte Carlo method solves problems which cannot be solved by conventional techniques by generating random numbers. In combination of this method and the Adaptive Kalman Filter, one of the most important conditions for the convergence rate and the required time is to select the proposed distribution [7]. Another application of Kalman filter is calibration of satellite positioning components like a gyroscope.…”