Homework  12/7/2016  COSC 681

Note, these questions are "plug and chug" and intended to make sure you
understand the notation. There are 6 questions.

PART A.

Answer the following questions using the dynamic model and the recursive
prediction equation from the youtube lecture https://www.youtube.com/watch?v=jY2E6ExLxaw

The recursion for prediction is given at approximately time 44:20.

The dynamic model:

initial distribution
P( X(0) = s ) = 0.2        P( X(0) = h ) = 0.8

transition probabilities P( X(t)  (table form)

                    X(t)
              |   s     h
              ----------------
         s    |  0.99   0.01 |
X(t-1)        |              |   
         h    |  0.1    0.9  |
              ----------------
         
        
observation model (emission probabilities)   P(Y | X)

                      Y
              w     s     c     f
            -------------------------
        s |  0.1   0.3   0.5   0.1  |
P(Y|X)    |                         |
        h |  0.4   0.4   0.2   0    |
            ------------------------


1.  What is X2, given Y1 = c   (predict X2  
    (need probabilities of X2 = s and  X2 = h) )

2. What is X3, given Y1, Y2 = c c   (predict X3 --
     Again, need the probabilities (two values) )

3. What is X4, given Y1, Y2, Y3 = c c w  (predict X4)


PART B

The video continues to give the formula (recursive) that obtains
P( X(t) | Y1, Y2, ... Yt )

at approximately 48:00.

For the same model as in PART A, answer the following:

NOTE!  Part B #1 was solved in Part A #2. Just restate the answer.
1. What is X2 given Y1, Y2 = c c ?

Note! Part B #2 was solved in Part A #3. Just restate the answer.
2. What is X3 given Y1, Y2, Y3 = c c w ?

3. What is X4 given Y1, Y2, Y3, Y4 = c c w w ?