Problem
I have a system that is measured at regular intervals. The state of the system at those times is given by the vector $\vec x=(x_0, x_1, x_2,\cdots)$. In between each measurement, a random variable $z$, drawn from a Gaussian distribution with zero mean and standard deviation $\sigma_z$ is added onto the state and there is some decay, i.e., $x_i=(1-\gamma)x_{i-1}+z_i$. (The underlying model is a damped harmonic oscillator in a thermal environment, probed at multiples of its period.)
My measurements are imperfect, i.e., I measure $\vec y=(y_0,y_1,y_2,\cdots)$, where $y_i=x_i+d_i$. Again, the $d$ are drawn from a Gaussian distribution of zero mean, with standard deviation $\sigma_d$. Note that this setup is a close variant of an example on the Wikipedia page for the Kalman filter. But as far as I understand, the Kalman filter does not take the whole measurement record into account, but provides an update rule for the best guess. In particular, it's estimate for $x_0$, which I am particularly interested in, is bad.
I would like to find $\vec x$ given some measurement record $\vec y$ (parameters $\sigma_d, \sigma_z$, $\gamma$ are known). I believe I should use some maximum likelihood analysis, but I have trouble finding the right probability distributions for $P(\vec y|\vec x), P(\vec x), P(\vec y)$ that I need to determine $P(\vec x|\vec y)$.
Attempt
I tried guessing what the right probability distributions should be. This one is fairly easy: $$P(\vec y|\vec x) = \prod_{i=0}^{N-1}\frac{1}{\sqrt{2\pi\sigma_z^2}}\exp\left(-\frac{(y_i-x_i)^2}{2\sigma_z^2}\right).$$
Now for $P(\vec x)$, I'm already slightly unsure. I start at equilibrium of the above update rule for $x_i$, i.e., when the covariance $\sigma_{x_i}^2\to(1-\gamma)\sigma_{x_{i-1}}^2+\sigma_d^2$ has reached a steady state, i.e., $\sigma_{x_0}=\sigma_d/\sqrt{\gamma}$. This is my expectation for $x_0$. Thus I think I should write $$ P(\vec x) = \frac{1}{\sqrt{2\pi\sigma_{x_0}^2}}\exp\left(-\frac{x_0^2}{2\sigma_{x_0}^2}\right)\times\prod_{i=1}^{N-1}\frac{1}{\sqrt{2\pi\sigma_d^2}}\exp\left(-\frac{(x_i-x_{i-1})^2}{2\sigma_d^2}\right). $$
However, the real problem is with $P(\vec y)$, and I think this is due to my limited experience with this method. Naively, without thinking about Bayesian updates, my guess for $P(\vec x|\vec y)$ would have looked like the product of $P(\vec x)$ and $P(\vec y|\vec x)$. I think my expectation for $\vec y$ should be $$ P(\vec y) = \int d\vec x\, P(\vec x)P(\vec y|\vec x). $$ It is possible to do all the Gaussian integrals, but it doesn't seem to give a simple result.
Is there some simpler assumption I can make? And how is it justified? Is there a mistake somewhere else?
Attempt #2
I'm still not entirely sure about $P(\vec y)$ (I think it's correct though), but one can get by without explicitly calculating it.
One mistake above is calling this approach "maximum likelihood". It's not. MLE would mean that I chose $\vec x$ such that $P(\vec y|\vec x)$ above becomes maximal for a given set of measurement results $\vec y$. This is a bad guess, as it neglects what we know about how the state evolves (note that $\sigma_d$ does not crop up in $P(\vec y|\vec x)$). Instead, what I am doing is Bayesian inference, i.e., finding the conditional probability $P(\vec x|\vec y)$.
To do this, we start with the joint distribution $P(\vec x, \vec y)=P(\vec y|\vec x)P(\vec x)$. This is a normal distribution $$ \begin{pmatrix}\vec x \\ \vec y\end{pmatrix} \sim N\left[\begin{pmatrix}\vec \mu_x \\ \vec \mu_y\end{pmatrix}, \begin{pmatrix}\Sigma_{xx} & \Sigma_{xy} \\ \Sigma_{xy} & \Sigma_{yy}\end{pmatrix}\right],$$ with $\mu_x,\mu_y=0$, $\Sigma_{yy} = \Sigma_{xy} = (1/2\sigma_z^2)\mathbb 1$, and $$ [\Sigma_{xx}]_{ij} = \left( \frac{1}{2\sigma_z^2} + \frac{1}{\sigma_d^2} \right)\delta_{i,j} + \left( \frac{1}{2\sigma_{x_0}^2} - \frac{1}{2\sigma_d^2} \right)\delta_{i,0}\delta_{j,0}+\frac{1}{2\sigma_d^2}(\delta_{i,j+1}+\delta_{i,j-1}).$$ NB: $\Sigma_{xx}$ is tri-diagonal apart from the first element, which differs (indexing from 0).
It's actually straightforward to derive the conditional distribution for $\vec x$ from the joint distribution [2]. It is given through $$ \vec x \sim N(\mu_x+\Sigma_{xy}\Sigma_{yy}^{-1}(\vec y-\vec\mu_y), \Sigma_{xx}-\Sigma_{xy}\Sigma_{yy}^{-1}\Sigma_{xy}). $$ However, in our case, this reduces to $$ \vec x\sim N( \vec y, \Sigma_{xx}-1/2\sigma_z^2 ),$$ which is not at all what I am expecting. All values of $\vec y$ should be used to determine a specific $x_i$. The problem lies in the fact that $\Sigma_{yy}$ is diagonal. What am I missing?
