We have independent observations Xi, for i = 1, . . . , n, from a mixture of m Poisson distributions with component probabilities dc and rates lc, for c = 1, . . . ,m. We decide to reparameterise with

hc = log lc, c = 1, . . . ,m

tc = log dc - log d1, c = 2, . . . ,m.

(a) What is the purpose of the reparameterisation?

(b) Derive expressions for lc and dc, for c = 1 and for c > 1, in terms of the new parameters.

(c) What would be the consequence of defining tc = log dc for c = 1, . . . ,m, instead of (?) above?

(d) Write down the reparameterised likelihood for m = 2.