Page 54 - Applied Probability

2. Counting Methods and the EM Algorithm
12. Chun Li has derived an extension of Problem 10 for hidden multino-
mial trials. Let N denote the number of hidden trials, θ i the proba-
bility of outcome i of k possible outcomes, and L(θ) the loglikelihood
of the observed data Y . Derive the EM update
  37
k
θ n ∂ n n ∂
i
n
n+1
θ = θ +  L(θ ) − θ L(θ ) .
n 
i i n j
E(N | Y, θ ) ∂θ i ∂θ j
j=1
Here the superscripts indicate iteration number.
13. In the spirit of Problem 10, formulate models for hidden Poisson and
exponential trials [16]. If the number of trials is N and the mean per
trial is θ, then show that the EM update in the Poisson case is
θ n d
= θ n + L(θ n )
θ n+1
E(N | Y, θ n ) dθ
and in the exponential case is
θ 2 n d
= θ n + L(θ n ),
θ n+1
E(N | Y, θ n ) dθ
where L(θ) is the loglikelihood of the observed data Y .
14. Suppose light bulbs have an exponential lifetime with mean θ.Two
experiments are conducted. In the first, the lifetimes y 1 ,...,y n of n
independent bulbs are observed. In the second, p independent bulbs
are observed to burn out before time t, and q independent bulbs are
observed to burn out after time t. In other words, the lifetimes in the
second experiment are both left and right censored. Construct an EM
algorithm for finding the maximum likelihood estimate of θ [7].
15. A palindromic DNA string such as ggatcc equals its reverse comple-
ment. Amend the EM algorithm of Section 2.7 so that it handles
palindromic binding domain patterns. What restrictions does this
imply on the domain probabilities p(m, b)?

2.9 References
[1] Chakraborty R, Srinivasan MR, Daiger SP (1993) Evaluation of stan-
dard error and confidence interval of estimated multilocus genotype
probabilities, and their applications in DNA forensics. Amer J Hum
Genet 52:60–70
[2] Clarke CA, Price-Evans DA, McConnell RB, Sheppard PM (1959)
Secretion of blood group antigens and peptic ulcers. Brit Med J 1:603–
607