# Integral of normal likelihood and multivariate normal prior

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?

• Sosa et al give an example of clustering with conditionally-conjugate priors: arxiv.org/abs/2110.10565 Apr 11 at 19:18