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As far as Estimation theory goes, for the start I would suggest Steven Kay's two volumes on Statistical Signal Processing. In my opinion that is the most gentle introduction to the subject. After reading those two books and solving as many of problems as you can, you can hop on to any of the following standard (my personal favorite) texts: Parameter ...

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To answer your first question, note that $X_t$ depends only on $X_{t-1}$ and $V_t$ through the evolution equation $X_t = f(X_{t-1}, V_t).$ Hence given that we know the values of $X_{t-1}$ and $V_t,$ $X_t$ is not random anymore and hence its pdf is given by a Dirac delta function. Hence $X_t$ occurs with probability one at $f(X_{t-1}, V_t)$ (the Dirac delta ...

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The prior distribution is $UNIF(0,1) \equiv BETA(1,1)$ with density function $p(\theta) = \theta^{\alpha_0 - 1}(1 - \theta)^{\beta_0 - 1},$ where $0 \le \theta \le 1$ and $\alpha_0 = \beta_0 = 1.$ The likelihood function is proportional to $\theta^2(1-\theta)^{n-2},$ where $n = 5.$ The posterior distribution is found as Posterior \propto Prior \times ...

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