# likelihood of latent state space model

Im trying to calculate the likelihood function of my latent state space model.

My model has Poisson observations

$$p(y_t|\beta_t;x_t) \sim \mathcal{Poiss}(z)$$.

where $$z$$ is the rate of the poisson process, $$z = exp(\beta_t \cdot x_t)$$ and $$x_t$$ are observed covariates on which the rate may depend

and evolves as a linear dynamical system.

$$p(\beta_t|\beta_{t-1}) \sim \mathcal{N}(A\cdot \beta,Q)$$

I'm trying to compare this model with a simpler model with fixed parameters (i.e. just a simple poisson regression). It is my understanding that to compare these models I need to calculate $$p(y|\theta)$$ in both cases but am unsure how to calculate this for the latent variable model.

In order to fit the latent model I've used the EM algorithm outlined here, making a laplace approximation to adjust for the nonlinear observation process: http://mlg.eng.cam.ac.uk/zoubin/course04/tr-96-2.pdf.

The authors provide a closed form expression of the joint likelihood $$p(x,y)$$ of data and latent variables but I'm unsure how to calculate the likelihood.

My current thinking is that I need to integrate out the latent variables $$x$$; i.e.

$$p(y|\theta) = \int p(x,y|\theta) \ dx$$

The only way I can think of doing this, since there is no closed form equation for the $$p(x|y)$$ is some kind of (markov chain) monte carlo approach. Is this correct or am I just way off?

• How does $z_t$ relate to $\beta_t$ and $x_t$? – jbowman Apr 10 '19 at 18:58
• Clarified, z is the rate of the poisson process: 𝑧=𝑒𝑥𝑝(𝛽𝑡⋅𝑥𝑡) – user3235916 Apr 10 '19 at 19:03

$$log \ p(Y|\theta) \approx log \ p(Y|\hat{\beta_{\theta}},\theta) + log \ p(\hat{\beta_{\theta}}|\theta) - \frac{1}{2} \cdot log \ | -H_{\theta} | + K$$
where K is constant independent of $$\theta$$, $$\hat{\beta_{\theta}}$$ is the MAP estimate of $$\beta$$ condition on $$\theta$$.