package aima.core.probability.hmm.exact;
import java.util.ArrayList;
import java.util.List;
import aima.core.probability.CategoricalDistribution;
import aima.core.probability.hmm.HiddenMarkovModel;
import aima.core.probability.proposition.AssignmentProposition;
import aima.core.util.math.Matrix;
/**
* Artificial Intelligence A Modern Approach (3rd Edition): page 579.<br>
* <br>
*
* Smoothing for any time slice <em>k</em> requires the simultaneous presence of
* both the forward and backward messages, <b>f</b><sub>1:k</sub> and
* <b>b</b><sub>k+1:t</sub>, according to Equation (15.8). The forward-backward
* algorithm achieves this by storing the <b>f</b>s computed on the forward pass
* so that they are available during the backward pass. Another way to achieve
* this is with a single pass that propagates both <b>f</b> and <b>b</b> in the
* same direction. For example, the "forward" message <b>f</b> can be propagated
* backward if we manipulate Equation (15.12) to work in the other direction:<br>
*
* <pre>
* <b>f</b><sub>1:t</sub> = α<sup>'</sup>(<b>T</b><sup>T</sup>)<sup>-1</sup><b>O</b><sup>-1</sup><sub>t+1</sub><b>f</b><sub>1:t+1</sub>
* </pre>
*
* The modified smoothing algorithm works by first running the standard forward
* pass to compute <b>f</b><sub>t:t</sub> (forgetting all intermediate results)
* and then running the backward pass for both <b>b</b> and <b>f</b> together,
* using them to compute the smoothed estimate at each step. Since only one copy
* of each message is needed, the storage requirements are constant (i.e.
* independent of t, the length of the sequence). There are two significant
* restrictions on the algorithm: it requires that the transition matrix be
* invertible and that the sensor model have no zeroes - that is, that every
* observation be possible in every state.
*
* @author Ciaran O'Reilly
*/
public class HMMForwardBackwardConstantSpace extends HMMForwardBackward {
public HMMForwardBackwardConstantSpace(HiddenMarkovModel hmm) {
super(hmm);
}
//
// START-ForwardBackwardInference
@Override
public List<CategoricalDistribution> forwardBackward(
List<List<AssignmentProposition>> ev, CategoricalDistribution prior) {
// local variables: f, the forward message <- prior
Matrix f = hmm.convert(prior);
// b, a representation of the backward message, initially all 1s
Matrix b = hmm.createUnitMessage();
// sv, a vector of smoothed estimates for steps 1,...,t
List<Matrix> sv = new ArrayList<Matrix>(ev.size());
// for i = 1 to t do
for (int i = 0; i < ev.size(); i++) {
// fv[i] <- FORWARD(fv[i-1], ev[i])
f = forward(f, hmm.getEvidence(ev.get(i)));
}
// for i = t downto 1 do
for (int i = ev.size() - 1; i >= 0; i--) {
// sv[i] <- NORMALIZE(fv[i] * b)
sv.add(0, hmm.normalize(f.arrayTimes(b)));
Matrix e = hmm.getEvidence(ev.get(i));
// b <- BACKWARD(b, ev[i])
b = backward(b, e);
// f1:t <-
// NORMALIZE((T<sup>T<sup>)<sup>-1</sup>O<sup>-1</sup><sub>t+1</sub>f<sub>1:t+1</sub>)
f = forwardRecover(e, f);
}
// return sv
return hmm.convert(sv);
}
// END-ForwardBackwardInference
//
/**
* Calculate:
*
* <pre>
* <b>f</b><sub>1:t</sub> = α<sup>'</sup>(<b>T</b><sup>T</sup>)<sup>-1</sup><b>O</b><sup>-1</sup><sub>t+1</sub><b>f</b><sub>1:t+1</sub>
* </pre>
*
* @param O_tp1
* <b>O</b><sub>t+1</sub>
* @param f1_tp1
* <b>f</b><sub>1:t+1</sub>
* @return <b>f</b><sub>1:t</sub>
*/
public Matrix forwardRecover(Matrix O_tp1, Matrix f1_tp1) {
return hmm.normalize(hmm.getTransitionModel().transpose().inverse()
.times(O_tp1.inverse()).times(f1_tp1));
}
}