I struggle to implement a particle filter to evaluate the likelihood of a textbook example. I got the following process:
$x_t = \alpha + \beta x_{t-1}/(1+x_{t-1}^2) + w_t$ where $w_t \sim \mathcal{N}(0, \sigma^2)$ and
$y_t = \delta x_t + v_t$ where $v_t \sim \mathop{t}(2)$.
I want to know the likelihood $p(y|\alpha, \beta, \sigma, \delta)$ given some observations $y$ and a known $x_0 = 1$. To achieve that, I draw $N$ particles $x_t^j$ for every $x_{t}$ and calculate the density of the residuals $\hat{v}_t^j = y_t - x_t^j$. After that, I resample the particles with stratified resampling. Now $\sum_{t=1}^n\ln \left(\sum_{j=1}^N \mathop{p}(\hat{v}_t^j)/N\right)$ gives me my approximated log-likelihood.
The problem is: It works very poorly. If I deviate a little from the true parameters, the likelihood gets, as expected, smaller. However, there are always alternative constellations which give me a higher likelihood.
Let's suppose e.g. I have $n=1000$, $N = 10000$, $\alpha = 0.5$, $\beta = 0.3$, $\sigma = 1.0$ and $\delta = 1.0$. Then my likelihood is significantly lower for $\delta = 1.1$, but the overall estimate can be extremely poor (e.g. all paramters deviate by $0.5$ from their true state).
Does someone have an idea, what goes wrong in my example?
