I am just trying to implement the scaled Baum-Welch algorithm and I have run into a problem where my backward variables, after scaling, are over the value of 1. Is this normal? After all, probabilities shouldn't be over 1.
I am using the scale factor I obtained from the forward variables:
$$
c_t = 1 / sum_{sin S}alpha_t(s)\
$$
where c_t is the scaling factor for time t, alpha is the forward variable, s are the states in the hmm.
For the backward algorithm I implemented it in java below:
public double[][] backwardAlgo(){ int time = eSequence.size(); double beta[][] = new double[2][time]; // Intialize beta for current time for(int i = 0; i < 2; i++){ beta[i][time-1] = scaler[time-1]; } // Use recursive method to calculate beta double tempBeta = 0; for(int t = time-2; t >= 0; t--){ for(int i = 0; i < 2; i++){ for(int j = 0; j < 2; j++){ tempBeta = tempBeta + (stateTransitionMatrix[i][j] * emissionMatrix[j][eSequence.get(t+1)] * beta[j][t+1]); } beta[i][t] = tempBeta; beta[i][t] = scaler[t] * beta[i][t]; tempBeta = 0; } } return beta; } The scales are stored in the array called scaler. There are 2 states in this hmm.
I should also note that the scale factors I am getting are over 1 as well.
Best Answer
I don't this in itself indicates any problem. $sum_{s in S} alpha_t(s)$ is the probability that the observed output sequence up to $t$ was $t_0, t_1, dots,$ eSequence.get(t). Thus, it's fine for $c_t$ to be greater than one. Also, for instance for $beta_{mathrm{time}-1}$, it's $sum_{s in S}beta_{mathrm{time}-1}(s) = |S|c_{mathrm{t}-1}$, which can very well be over one.