HW3, CMPS 290C, S04, Warmuth
Handed out May 18
Due May 27 in class

1) Develop a recurrence for the following quantity of HMMs
using first principles: 
      xi(q_t,q_t+1) = p(q_t,q_t+1|y)
(See Section 12.6 for a discussion of these variables)

2) Derive the equations 13.28-13.31

3) Show that the following distributions
are in the exponential family:

In each case give the
- natural parameter
- sufficient statistic
- log normalization (cumulant function)
- reference density
- expectation parameter
- transformation from natural to expectation parameter
  and back
- density i.t.o. the expectation parameter
- function dual to the cumulant function
- Bregmann divergence i.t.o. the natural parameters
- Bregmann divergence i.t.o. the expectation parameters


a) Poisson distribution on the non-negative integers

 p(x) = lambda^x e^(-lambda) / x!

b) Dirichlet distribution on the probability simplex

 p(x) = 

   gamma(sum_i=1^k alpha_i) 
  -------------------------  *   prod_i=1^k x_i^(alpha_i-1)
   prod_i=1^k gamma(alpha_i)

(It seems like it is not possible to give a closed
form of the natural parameter i.t.o. the expectation
parameter. Any progress on this one will be EXTRA CREDIT
Do the below one instead!)

c) Exponenential distribution on the non-negative reals

p(x) = lambda exp(-lambda x),  where x >=0, lambda >=0