importance sampling strategies

Question

I am trying to approximate the expectation of the "complete-data likelihood" with respect to the distribution of some missing data, and I am having some trouble. This expectation can be written as $$ \int p(\mathbf{y} \mid \mathbf{x}, \mathbf{\beta})f(\mathbf{x}) d\mathbf{x}_{1:n} = \int \prod_{i=1}^np(y_i \mid \mathbf{x}_i, \mathbf{\beta})f(\mathbf{x}_i) d\mathbf{x}_{1:n} $$ where

1. $p(y_i \mid \mathbf{x}_i, \mathbf{\beta}) = \sigma(\mathbf{x}_i^T\beta)^{y_i} [1-\sigma(\mathbf{x}_i^T\beta)]^{1-y_i}$
2. $f(\mathbf{x}_i)$ is chosen, so let's not make any assumptions about what it is yet. The $\sigma$ function is the inverse logit, or sigmoid function: $\sigma(r) = e^r/(1+e^r)$.

You can use importance sampling by picking an importance density $q(\mathbf{x}^s \mid \mathbf{y})$, simulating $N_s$ times from it, and calculating: $$ \sum_{s = 1}^{N_s} \frac{p(\mathbf{y} \mid \mathbf{x}^s, \mathbf{\beta})f(\mathbf{x}^s)}{q(\mathbf{x}^s \mid \mathbf{y})} \bigg/ N_s. $$

I am interested in reducing relative variance, and not getting all of these $-\infty$s. Are there well-known or popular strategies to accomplish this? Maybe I can simulate $\mathbf{x}^s$ so that all of these $\mathbf{x}_i^T\beta$s are constrained to be close to $0$.

Edit: perhaps I should be more clear about this: this is a logistic regression model with missing data, not a state space model. The phrase “complete data likelihood” could’ve been misleading as it means two different things in these two contexts. Also, it could’ve been misleading that I simplified notation for $f$, as it doesn’t have anything denoting missing and observed subvectors.

Answer 1

Importance sampling is notorious for falling afoul of dimensionality, the so called curse of dimension(ality). This means that a direct frontwise importance sampling attack in this latent variable setting cannot work.

The standard approach is to use instead sequential importance sampling, also called sequential Monte Carlo or particle filtering, where an importance sample is created for each $\mathbf{x}_i$ in the sequence $\mathbf{x}_{1:n}$, targeting the density proportional to $$p(y_i \mid \mathbf{x}_i, \mathbf{\beta})f(\mathbf{x}_i |\mathbf{x}_{1:{i-1}})$$and estimating its normalising constant from that sample.