HMM-forward backward algorithm

Let $x_t, \, t=1, \dots ,T$ be a time series and suppose that $x_t | \xi_t \sim N(\mu_{\xi_t},\sigma^2_{\xi_t})$, where $\xi_t \in [1, \dots,K]$ is a group indicator (or regime or state), the parameter of the normal distribution depends on the value of $\xi_t$ and we suppose that $P(\xi_t=k|\xi_{t-1}=j) = \pi_{jk}$. I want to estimates this in a Bayesian framework and then the prior are $\pi_{j1},\pi_{j2},\dots, \pi_{jK} \sim Dirichlet(\alpha_1,\dots, \alpha_k)$, $\mu_{k} \sim N(M,V)$ and $\sigma_{k}^2 \sim InvGamma(a,b)$. I suppose $\xi_0=1$.

I am trying to understand how sample from the full conditional of $\xi_t, t=1,\dots,T$. Can someone explain me if I have to use the forward, the backward or the Viterbi algorithm or something else? and how to use it? I am really confuse about all these stuff :-). A link where it is well explained is accepted as well.

• My favorite HMM tutorial is Rabiner's HMM tutorial – bdeonovic Aug 1 '14 at 17:55
• I don't see where it describe how to find the full conditional distribution – niandra82 Aug 1 '14 at 18:32
• You may want to simplify the question by removing the priors on everything but the group indicators $\xi$ since the full conditional is equivalent to knowing all $\pi$, $\mu$, and $\sigma^2$. – jaradniemi Aug 5 '14 at 20:32