Solving a hidden markov model.

You sit down at a table at the unfair casino. The dealer rolls a 6-sided die 10 times, and records the sequence of numbers. Between each roll, the dealer secretly switches between 3 different dice. To win, determine the most likely die used at each draw during this sequence.

The 3 dice:

F: The "Fair" die has an equal chance of rolling {1,2,3,4,5,6}.

O: The "Ones" loaded die has the sides {1,1,1,1,1,6}.

S: The "Sixes" loaded die has the sides {1,6,6,6,6,6}

At the start, the dealer chooses one of the three die at random. At each turn, the dealer has a 50% chance of staying with the same die, and a 25% chance to switching to either of the other ones.

1 (4 pts) Draw an HMM model depicting this game, including hidden states and arrows labeled with transition probabilities between states. Also draw tables for matrix pi (initial probabilities of each state), matrix A (transition probabilities between states), and matrix E (emission probabilities for each state).

2 (1 pt) What specific aspect of this game makes it suited for an HMM?

3 (5 pts ) The dealer rolls the sequence 1 1 1 6 1 6 6 1 2 3.

Determine the most probable sequence of dice {F,O,S} that would have generated these numbers.

Show all of your work.

Hint: Perform calculations in Log10 space. This can be solved using R/Matlab/python or creative use of Excel.