Suppose you are at a table at a casino and notice that things don’t look quite right. Either the casino is extremely lucky, or things should have averaged out more than they have. You view this as a pattern recognition problem and would like to understand the number of ‘loaded’ dice that the casino is using and how these dice are loaded. To accomplish this you set up a number of Hidden Markov Models, where the loaded die are the latent (hidden) variables, and would like to determine which of these, if any is more likely to be using.
First lets go over a few things.
We will call each roll of the dice an observation. The observations will be stored in variables o1, o2, …, oT, where T is the number of total observations.
To generate a hidden Markov Model (HMM) 
we need to determine 5 parameters:
- The N states of the model, defined by S = {S1, …, SN}
- The M possible output symbols, defined by
= {
1, 2, …, M} - The State transition probability distribution A = {aij}, where aij is the probability that the state at time t+1 is Sj, given that the state at time t is Si.
- The Observation symbol probability distribution B = {bj(k)} where bj(k) is the probability that the symbol k is emitted in state Sj.
- The initial state distribution
= {i}, where i is the probability that the model is in state Si at time t = 0.The HMMs we’ve generated are based on two questions. For each question, you have provided 3 different answers which leads to 9 possible HMMs. Each of these models has its corresponding state transition and emission distributions.
- How often does the casino change dice?
- 0) Dealer Repeatedly Uses Same Dice
- 1) Dealer Uniformly Changes Die
- 2) Dealer Rarely Uses Same Dice
- Which sides on the loaded dice are more likely?
- 0) Larger Numbers Are More Likely
- 1) Numbers Are Randomly Likely
- 2) Smaller Numbers Are More Likely
How often does the casino change dice? Which sides on
the loaded dice
are more likely?(0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0) (2, 1) (2, 2) One of the interesting problems associated with Hidden Markov Models is called the Learning Problem, which asks the question “How can I improve a HMM
so that it would be more likely to have generated the sequence O = o1, o2, …, oT?The Baum-Welch algorithm answers this question using an Expectation-Maximization approach. It creates two auxiliary variables
t(i) and 
t(i, j). The variable t(i) represents the probability of being in state i at time t, given the entire observation sequence. Likewise t(i, j) represents the joint probability of being in state i at time t and of being in state j at time t+1, given the entire observation sequence. They can be calculated byt(i) =( 
t(i) * 
t(i) )j = 1 to N(t(j) * t(j))and
t(i, j) =( t(i) * ai, j * t+1(j) * bj(ot+1) )i’ = 1 to Nj’ = 1 to N(t(i’) * ai’, j’ * t+1(j’) * bj’(ot+1) )As you can see, these are a direct result of calculations of
from the Forward algorithm and from the Backwards algorithm. Once we have calculated these variables, we can update the parameters of the model as follows:
i = 1(i)
i,j =t = t to T-1 (t(i, j))t = 1 to T-1(t(i))// [b bar]_{j, k} = Sigma_{t = 1 to T, o_t = o_k} gamma_{t, j} / Sigma_{t = 1 to T} gamma_{t, j}, 1 <= j <= N, 1 <= k <= M
j(ok) =t = 1 to T-1, ot = ok t(j)t = 1 to T-1 t(j)We can iterate this procedure a finite number of times or until it converges. This will generate a new model,
= {N, , , , }.There is more on this example at LEARNINGlover.com: Hidden Marokv Models: The Baum-Welch Algorithm.
Some further reading on Hidden Markov Models:
- How often does the casino change dice?
t(i) which has the highest probability that the partial observation sequence o1, …, ot can have, given that the current state is i. This variable can be calculated by the following formula:
j 
)2. We can get the average deviation from the mean then by computing the average of these values.
i = 1 to n((vi –