public RandomVariable predict(RandomVariable aBelief, String action) {
    RandomVariable newBelief = aBelief.duplicate();

    Matrix beliefMatrix = aBelief.asMatrix();
    Matrix transitionMatrix = transitionModel.asMatrix(action);
    Matrix predicted = transitionMatrix.transpose().times(beliefMatrix);
    newBelief.updateFrom(predicted);
    return newBelief;
  }
  public List<RandomVariable> forward_backward(List<String> perceptions) {
    RandomVariable forwardMessages[] = new RandomVariable[perceptions.size() + 1];
    RandomVariable backwardMessage = priorDistribution.createUnitBelief();
    RandomVariable smoothedBeliefs[] = new RandomVariable[perceptions.size() + 1];

    forwardMessages[0] = priorDistribution;
    smoothedBeliefs[0] = null;

    // populate forward messages
    for (int i = 0; i < perceptions.size(); i++) { // N.B i starts at 1,
      // not zero
      forwardMessages[i + 1] = forward(forwardMessages[i], perceptions.get(i));
    }
    for (int i = perceptions.size(); i > 0; i--) {
      RandomVariable smoothed = priorDistribution.duplicate();
      smoothed.updateFrom(forwardMessages[i].asMatrix().arrayTimes(backwardMessage.asMatrix()));
      smoothed.normalize();
      smoothedBeliefs[i] = smoothed;
      backwardMessage =
          calculate_next_backward_message(
              forwardMessages[i], backwardMessage, perceptions.get(i - 1));
    }

    return Arrays.asList(smoothedBeliefs);
  }
  public RandomVariable perceptionUpdate(RandomVariable aBelief, String perception) {
    RandomVariable newBelief = aBelief.duplicate();

    // one way - use matrices
    Matrix beliefMatrix = aBelief.asMatrix();
    Matrix o_matrix = sensorModel.asMatrix(perception);
    Matrix updated = o_matrix.times(beliefMatrix);
    newBelief.updateFrom(updated);
    newBelief.normalize();
    return newBelief;

    // alternate way of doing this. clearer in intent.
    // for (String state : aBelief.states()){
    // double probabilityOfPerception= sensorModel.get(state,perception);
    // newBelief.setProbabilityOf(state,probabilityOfPerception *
    // aBelief.getProbabilityOf(state));
    // }
  }
 public RandomVariable calculate_next_backward_message(
     RandomVariable forwardBelief, RandomVariable present_backward_message, String perception) {
   RandomVariable result = present_backward_message.duplicate();
   // System.out.println("fb :-calculating new backward message");
   // System.out.println("fb :-diagonal matrix from sens model = ");
   Matrix oMatrix = sensorModel.asMatrix(perception);
   // System.out.println(oMatrix);
   Matrix transitionMatrix = transitionModel.asMatrix(); // action
   // should
   // be
   // passed
   // in
   // here?
   // System.out.println("fb :-present backward message = "
   // +present_backward_message);
   Matrix backwardMatrix =
       transitionMatrix.times(oMatrix.times(present_backward_message.asMatrix()));
   Matrix resultMatrix = backwardMatrix.arrayTimes(forwardBelief.asMatrix());
   result.updateFrom(resultMatrix);
   result.normalize();
   // System.out.println("fb :-normalized new backward message = "
   // +result);
   return result;
 }