Augmentation schemes for particle MCMC

Abstract

Particle MCMC involves using a particle filter within an MCMC algorithm. For inference of a model which involves an unobserved stochastic process, the standard implementation uses the particle filter to propose new values for the stochastic process, and MCMC moves to propose new values for the parameters. We show how particle MCMC can be generalised beyond this. Our key idea is to introduce new latent variables. We then use the MCMC moves to update the latent variables, and the particle filter to propose new values for the parameters and stochastic process given the latent variables. A generic way of defining these latent variables is to model them as pseudo-observations of the parameters or of the stochastic process. By choosing the amount of information these latent variables have about the parameters and the stochastic process we can often improve the mixing of the particle MCMC algorithm by trading off the Monte Carlo error of the particle filter and the mixing of the MCMC moves. We show that using pseudo-observations within particle MCMC can improve its efficiency in certain scenarios: dealing with initialisation problems of the particle filter; speeding up the mixing of particle Gibbs when there is strong dependence between the parameters and the stochastic process; and enabling further MCMC steps to be used within the particle filter.

Appendices

Appendix 1: Calculations for the stochastic volatility model

First consider $Z_{\beta _X}$. Standard calculations give

$$\begin{aligned} p(\beta _X|z_{\beta _X})\propto & {} p(\beta _X)p(z_{\beta _X}|\beta _X) \\\propto & {} \beta _x^{a_x-1}\exp \{-b_x\beta _x\}\left( \beta _X^{n_x}\exp \{-\beta _x z_{\beta _X}\}\right) . \end{aligned}$$

This gives that the conditional distribution of $\beta _X$ given $z_{\beta _X}$ is gamma with parameters $n_x+a_x$ and $b_x+z_{\beta _X}$. Furthermore the marginal distribution for $Z_{\beta _X}$ is

$$\begin{aligned} p(z_{\beta _X})= & {} \int p(\beta _X)p(z_{\beta _X}|\beta _X)\text{ d }\beta _X \\= & {} \frac{b_x^{a_x}z_{\beta _x}^{n_x-1}}{\Gamma (a_x)\Gamma (n_x)}\int \beta _x^{a_x+n_x-1}\exp \{-(b_x+z_{\beta _X}\beta _x\} \text{ d }\beta _X \\= & {} \left( \frac{\Gamma (a_x+n_x)b_x^{a_x}}{\Gamma (a_x)\Gamma (n_x)} \right) \left( \frac{z_{\beta _x}^{n_x-1}}{(z_{\beta _x}+b_x)^{n_x+a_x}} \right) . \end{aligned}$$

The calculations for $Z_{\beta _Y}$ are identical.

Calculations for $Z_X$ and $Z_\gamma$ are as for the linear Gaussian model (see Sect. 3).

Appendix 2: PMMH algorithm with particle learning

Appendix 3: Calculations for the Dirichlet process mixture model

The conditional distribution of $Z_x$ given $x_{1:n}$ can be split into (i) the marginal distribution for v, p(v); (ii) the conditional distribution of the sampled individuals, $i_1,\ldots ,i_v$, given v. Given $i_1,\ldots ,i_v$, the clustering of these individuals is deterministic, being defined by the clustering $(x_{i_1},\ldots ,x_{i_v})$.

The marginal distribution of $Z_x$ thus can be written as

$$\begin{aligned} p(Z_x)=p(v)p(i_1,\ldots ,i_v|v)p(x_{i_1},\ldots ,x_{i_v}). \end{aligned}$$

Where we that, due to uniform sampling of the individuals,

$$\begin{aligned} p(i_1,\ldots ,i_v|v)=\left( \begin{array}{c} n \\ v \end{array} \right) . \end{aligned}$$

Finally, $p(x_{i_1},\ldots ,x_{i_v})$ is given by the Dirichlet process prior. If we relabel the populations so that $x_{i_1}=1$, population 2 is the population of the first individual in $i_1,\ldots ,i_v$ that is not in population 1, and so on; then for $v>1$,

$$\begin{aligned} p(x_{i_1},\ldots ,x_{i_v})=\prod _{j=2}^v p(x_{i_j}|x_{i_1},\ldots ,x_{i_{j-1}}), \end{aligned}$$

with $p(x_{i_j}|x_{i_1},\ldots ,x_{i_{j-1}})$ defined by (4).

Within the PMMH we use a proposal for $Z_x$ given $X_{1:n}$ that is its full conditional

$$\begin{aligned} q(Z_x|x_{1:n})=p(Z_x|x_{1:n})=p(v)p(i_1,\ldots ,i_v|v). \end{aligned}$$

In practice we take the distribution of v to be a Poisson distribution with mean 5, truncated to take values less than n. (Similar results were observed as we varied both the distribution and the mean value.)