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I got a an exercise problem which should be seen as a HMM scenario and argument some statements. However I'm quite confused about how to properly solve and argument my solutions.

Problem tells:

Imagine you want to determine the annual temperature centuries of years ago, when of course there wasn't any thermometer or records. So, nature as an evidence is a worth to try resource, we may achieve it by watching at tree's inside rings. There's reliable evidence suggesting that there's a relation among the rings inside trees and temperature. There will be 2 different temperature states, WARM (W) and COLD (C) and three discretized tree rings sizes: SMALL (S), MEDIUM (M) and LARGE (L). Some researchers have provided two matrixes:

$\begin{bmatrix}.7 & .3\\.4 & .6\end{bmatrix}$

As transition matrix, so the probability of remaining in COLD state if COLD is present is $.6$ and the probability of passing from COLD to WARM is $.7$.

Also, a second matrix with the relation among the ring's size and the temperature over the year:

$\begin{bmatrix}.1 & .4 & .5\\.7 & .2 & .1\end{bmatrix}$

So, problem asks what I should do to calculate the chance of a sequence (for example):

SSSMMLLL

To happen. I considered multiplying the 4 distinct cases and so generate a Markov matrix with transitions among trees ring sizes. However I never got a matrix whose rows sum 1 as it should.

How could I solve this?

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    $\begingroup$ What research & self-study have you done? What have you tried? Where specifically did you get stuck? If you're not sure how to approach this, that probably means you need to go back and spend some more time studying HMMs. There is lots written about HMMs and how to work with them; there would be little point in us repeating that here. Can you articulate a specific conceptual question? (This is not a site where you can paste your exercise and have us solve it for you. We want to help you, but as it is not clear what the issue is, it is hard to know how to do that.) $\endgroup$ – D.W. Dec 20 '14 at 22:08
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The Viterbi algorithm solves the problem. It calculates the most probable state sequence for an HMM given an observation sequence. In your example, ring sizes are observations and temperatures are hidden states. I suggest to refer to the Wikipedia article about the Viterbi algorithm (http://en.wikipedia.org/wiki/Viterbi_algorithm). It contains an example of an HMM with 2 hidden states and observations with 3 discrete probabilities.

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I think since likelihood of observation sequence was asked in the question you should use the Forward Backward probabilities. This uses a technique of dynamic programming.

Reference to wiki page

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