Inference in state space models

Question

I have the following state space model $$x_{n}=x_{n-1}+cos(1.2n)$$ $$y_{n}=x_{n}^{2}+w_{n}$$ $$w_{n}\thicksim N(0,σ_{w}^{2})$$ For the observation pdf, we have $y_{n}\thicksim N(x_{n},{x_{n}σ_{w}}^{2})$ so we can use this to compute the observation pdf. On the other hand, the transition equation dosent't have process noise. In this case, how am I supposed to compute the transition pdf $p(x_{n+1}|x_{n})$.

Answer 1

Your equations are $$\begin{split}x_n&=x_{n-1}+\cos(1.2n)\\ y_n&=x_n^2+w_n\\ w_n&\sim N(0, \sigma^2)\end{split}$$

It is claimed this is a hidden markov model. Indeed, for a hidden markov model, we have the joint pdf

$$f(\textbf x|\textbf y)=p(x_1)p(y_1|x_1)\prod_{j=2}^n p(x_j|x_{j-1})p(y_j|x_j)$$

But here, the $x_j$'s are not random (except maybe the first one). So you get a recursive relation. That is, you plug in the $x_n$ into the equation for $y_n$.

$$\begin{split}y_1|x_1&\sim N(x_1^2,\sigma^2)\\ y_2&\sim N([x_1+\cos(2.4)]^2,\sigma^2)\\ &\vdots\\ y_n&\sim N\left((((x_1+\cos(2.4))^2+\cos(3.6))^2+\dots)^2,\sigma^2\right)\end{split}$$

And from there you can get the pdf. If $x_1$ or $x_0$ is random, you multiply those by the prior pdf of $x_1$ or $x_0$ to get $p(\textbf y, x_0)=p(x_0)\prod_{i=1}^np(y_i|x_0)$. If it not, you just leave it out and are left with $p(\textbf y)=\prod_{i=1}^np(y_i)$.