Recursive bayesian prediction, which model to use?

Let's say that I have a set of random variables $X=\{X_1,..X_t,..X_T\}$ ($t$ is a time index). I know that every one of these random variables $X_t$ generate a multivariate Gaussian Distribution and are related somehow (for every $i$, $X_i$ and $X_{i+1}$ obe a certain relationship: $X_{i+1}=X_i+D_i$). I want to do a recursive inference and I have no idea what kind of model I should use.

So having a full knowledge about the distributions of $X_1,..X_t,..X_T$ (all Gaussians) and knowing the relationship between every two indexed consecutive random variables, I want to be able to predict the last output at time $T$, $Y_T$ for a certain input observation $Y_t$. I googled for nearly an hour and got Gaussian processes (to model time-indexed Gaussian distributions), recursive Bayesian models and Bayesian networks.

Can anyone please show me the right direction to start from?

• Is $t$ continuous or discrete? – jlimahaverford Oct 7 '15 at 15:01
• $t$ is discrete, I have a fixed set of possible values for $t$. – vphenix Oct 7 '15 at 15:09
• just to save you the trouble in advance, definitely not Bayesian networks. – Zhubarb Oct 7 '15 at 16:13
• Two questions for clarification: How are $Y_T$ and $Y_t$ related to the $X_i$? Given that $X_{i+1}-X_i=D_i$ is true for some $D_i$ for any random variables, this seems rather vacuous. Do you know something specific about this "certain relationship"? – g g Oct 7 '15 at 20:56
• And I forgot: What is exactly the data you would like to base your inference on? iid observations of $(X_1,\ldots,X_{T-1})$ or something else? – g g Oct 7 '15 at 21:07