# boostrap particle filter marginal likelihood

I want to calculate the marginal likelihood $$p(y|\Theta)$$ of the parameters of a Markov state space model with unknown parameters $$\Theta$$ that I am trying to estimate the marginal likelihood (marginal over latent states) using a bootstrap particle filter.

More specifically, lets say I know the initial state of my system $$x_0$$ and have a transition kernel $$p(x_t|x_{t-1},\theta)$$ that depends on some parameters $$\theta$$ that are unknown. Further, I have a measurement model $$p(y_t|x_t,\phi)$$ that depends on some other parameters $$\phi$$, which I'm also trying to estimate.

Then, with $$\Theta = \{\theta,\phi\}$$, my current understanding is that to calculate $$p(y|\Theta)$$, I iterate over all time points, the following operations:

1. Sample from $$p(x_t|x_{t-1},\theta)$$
2. Evaluate $$p(y|x_t,\phi)$$ for each of these proposed $$x_t$$ from step 1.
3. Weight each particle according to $$p(y_t|x_t,\phi)$$, normalise weights, and resample them (yielding a sample with uniform weights from $$p(x_{t}|y_{1:t})$$ ).

Then, lets say my bootstrap particle filter has $$P$$ particles, my understanding is that

$$p(y|\Theta) \approx \prod_{t=1}^{T} \frac{1}{P}\sum_{p=1}^{P}p(y_t|x^p_t,\phi)$$

where the inner sum is over the particles proposed (at step 1) using $$p(x_t|x_{t-1},\theta)$$

is this correct?

• Yes, that is the correct expression for the likelihood estimate. I would say, though, that after resampling, you have samples approximately distributed according to the filtering distribution. – Taylor May 2 '19 at 3:08
• yep I agree. This was a typo! Thanks! – user3235916 May 2 '19 at 12:12