I'm updating cluster assignments in the context of a non-parametric Bayesian mixture model. When computing the probability of starting a new cluster, in the absence of cluster parameters (and using a conjugate model), we would evaluate the likelihood integrating over all cluster parameters.
In my particular case, I have data points $x$ and covariates $y$, which I model as follows: $$x_i \vert \Theta, z_i \sim \text{Normal}(\theta_{z_i} \cdot y_i, \sigma^2)$$ where $x_i$ is a datum $i$, $y_i$ are covariates, $z_i$ the cluster assignment, $\Theta$ all the cluster parameters. The mean for a data point is the dot product of cluster parameters and the point's covariates.
The prior over cluster parameters is multivariate normal: $$\theta \sim \text{Multivariate-Normal}(\mu, \Sigma)$$
What I wish to compute basically is $$P(z_i = z \vert \dots) \propto P(z) \int \!\! P(x_i \vert \theta)p(\theta)d\theta$$ where $z$ represents a new cluster.
Is there any hope to get an analytic solution? If not, how would you recommend to approach this? Monte Carlo integration?
