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
- $p(y_i \mid \mathbf{x}_i, \mathbf{\beta}) = \sigma(\mathbf{x}_i^T\beta)^{y_i} [1-\sigma(\mathbf{x}_i^T\beta)]^{1-y_i}$
- $f(\mathbf{x}_i)$ 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.