ABSTRACT
In this chapter, we look at Markov chain Monte Carlo (MCMC) methods for a class of time series models called state-space models. The idea of state-space models is that there is an unobserved state of interest the evolves through time, and that partial observations of the state are made at successive time points. We will denote the state byX and observations by Y, and assume that our state-space model has the following structure:
Xt | {x1:t−1, y1:t−1} ∼ p(xt | xt−1, θ), (21.1) Yt | {x1:t, y1:t−1} ∼ p(yt | xt, θ). (21.2)
Here, and throughout, we use the notation x1:t = (x1, . . ., xt), and write p(· | ·) for a generic conditional probability density ormass function (with the argumentsmaking it clearwhich conditional distribution it relates to). To fully define the distribution of the hidden state we further specify an initial distribution p(x1 | θ). We have made explicit the dependence of the model on an unknown parameter θ, which may be multidimensional. The assumptions in this model are that, conditional on the parameter θ, the state model is Markov, and that we have a conditional independence property for the observations: observation Yt only depends on the state at that time, Xt. For concreteness we give three examples of state-space models:
Example 21.1: Stochastic Volatility
Example 21.3: Change-Point Model
Our aim is to performBayesian inference for a state-spacemodel givendata y1:n.We assume a prior for the parameters, p(θ), has been specified, and we wish to obtain the posterior of the parameters p(θ | y1:n), or in some cases we may be interested in the joint distribution of the state and the parameters p(θ, x1:n | y1:n). How can we design an MCMC algorithm to sample from either of these posterior distri-
butions? In both cases, this can be achieved using data augmentation (see Chapter 10, this volume). That is, we design a Markov chain whose state is (θ,X1:n), and whose stationary distribution is p(θ, x1:n | y1:n) (samples from themarginal posterior p(θ | y1:n) can be obtained from samples from p(θ, x1:n | y1:n) just by discarding the x1:n component of each sample). The reason for designing anMCMC algorithm on this state space is that, for state-spacemodels of the form Equations 21.1 through 21.2, we can write down the stationary distribution of the MCMC algorithm up to proportionality:
p(θ, x1:n | y1:n) ∝ p(θ)p(x1 | θ) ( n∏ t=2
p(xt | xt−1, θ) )( n∏
t=1 p(yt | xt, θ)
) . (21.5)
Hence, it is straightforward to use standard moves within our MCMC algorithm. In most applications it is straightforward to implement an MCMC algorithm with
Equation 21.5 as its stationary distribution. A common approach is to design moves that update θ conditional on the current values of X1:n and then update X1:n conditional on θ. We will describe various approaches within this framework. We first focus on the problem of updating the state; and to evaluate different methods we will consider models where θ is known. Then we will consider moves to update the parameters.
