Applications of Hidden Markov Models

Homework 2: HMMs and Speech Recognition

Transition Probabilities: Above is the hidden Markov model (HMM) regarding how a particular person P may pronounce a word W, going through a sequence of basic stages from A, B, C, to D. Note that a stage may be repeated more than once when the person pronounces the word. The diagram above shows the probability of how the process of pronunciation may transition from one stage to the next.

Probabilities of Acoustic Patterns: There are seven acoustic patterns related to the acoustic signals we may observe in this process through the microStage sampling during a basic unit of time, and we refer to these seven opatterns as X1, X2, X3, X4, X5, X6, X7. The table below tells us the following probabilities of for each stage (each row in the table) how likely we may observe a particular acoustic pattern in that particular stage:

X1 X2 X3 X4 X5 X6 X7

Stage A: 0.5 0.2 0.3

Stage B: 0.2 0.7 0.1 Stage C: 0.1 0.5 0.4

Stage D: 0.3 0.7

The patterns observed: During speech recognition, we observe the following sequence of acoustic patterns X1, X3, X4, X6, X6, and X6 from the microphone. How likely the perosn P was pronouncing the word

1 W? Probabilistic Inference with the Hidden Markov Model

We need to determine the probability of observing the acoustic patterns X1, X3, X4, X6, X6, and X6 while going through 6 underlying stages (obviously with repetition in some of stages A, B, C, or D) and ending in Stage D. You should follow the approach below and complete all the 6 steps below to determine the probability. This probability is also the probability pr(W|P) that person P was pronouncing the word W.

Approach: Incrementally determine the probability of ending in Stage A, or Stage B, or Stage C, or Stage D after visiting one, two, three, four, five, and six stages respectively.

Step 1: Calculate the probabilities of starting and going through one pass at stage A and seeing X1.

X1 X3 X4 X6 X6 X6 Stage A 1*0.5=0.5 Stage B 0 Stage C 0 Stage D 0

Step 2: Based on step 1, calculate the probabilities of starting at stage A and going through two stages, seeing X1, X3, and ending in stage A, or B, or C, or D respectively.

X1 X3 X4 X6 X6 X6 Stage A 0.5 0.5*0.3*0.3 = 0.045 (extended from A) Stage B 0 0.5*0.7*0.2 = 0.070 (extended from A) Stage C 0 0 Stage D 0 0

Stage 3: Based on step 2, calculate the probabilities of starting at stage A and going through three stages, seeing X1, X3, X4 and ending in A, or B, or C, or D respectively.

2 X1 X3 X4 X6 X6 X6 Stage A 0.5 0.045 0.045*0.3*0 = 0 (extended from A) Stage B 0 0.070 Sum( 0.045*0.7*0.7 if from A 0.070*0.9*0.7 if from B ) = 0.06615 (extended from A or B) Stage C 0 0 0.070*0.1*0.1 = 0.0007 (extended from B) Stage D 0 0 0

Step 4: Based on step 3, calculate the probabilities of starting at stage A and going through four stages, seeing X1, X3, X4, X6 and ending in A, or B, or C, or D respectively.

X1 X3 X4 X6 X6 X6 Stage A 0.5 0.045 0 0 Stage B 0 0.070 0.06615 0.06615*0.9*0 = 0 Stage C 0 0 0.0007 Sum( 0.06615*0.1*0.4 if from B 0.0007 *0.4*0.4 if from C ) = 0.0.002758 (extended from B or C) Stage D 0 0 0 0.0007*0.6*0.3=0.000126

Step 5: Based on step 4, calculate the probabilities of starting at stage A and going through five stages, seeing X1, X3, X4, X6, X6 and ending in A, or B, or C, or D respectively.

Step 6: Based on step 5, calculate the probabilities of starting at stage A and going through six stages, seeing X1, X3, X4, X6, X6, X6 and ending in A, or B, or C, or D respectively.

Assuming that any of A, B, C, and D could be the final stage of the process of pronouncing W, the probability pr(W|P) is simply the sum of the probabilities you have calculated in Step 6 (i.e. the probabilities appear in the very last column of the table).