Corrections to the slides "Automatic Speech Recognition: 
From Theory to Practice" by Prof. Bryan Pellom.


Lecture 4:

* slide 32: Definition of the backward probabilities should be:

  beta_t(i) = P(o_{t+1} o_{t+2} ... o_T | q_t = i, lambda)

  not

  beta_t(i) = P(o_{t+1} o_{t+2} ... o_T , q_t = i | lambda)

  In other words, backward probabilities are defined as
  the probabiltiy of observation sequence o_{t+1} to o_T
  GIVEN state i at time t and HMM parameters lambda.

* slide 31: disregard the example of forward calculation, the initial
  conditions are strange. In our implementation, we always start at time 1
  in state 1. Thus, the initialization of the forward algorithm is
  alpha_1(1) = 1*b_1(1) and alpha_1(i) = 0 for i > 1.

* slides 56-57: In calculating mu_jk and Sigma_jk,
  the summation in the denominator should be only over t
  and not over k. That is, delete "sum_{k=1}^M" from
  the denominator of the two formulas.
  The formula for c_jk on page 56 is correct: there are two
  summations in the denominator.