Answer Happy

1. Consider yi to be observed counts coming from the Poisson distribution. Consider the following model with parameter 1: p(yil) = Yiel yi! 1>0, Yi E N. (a) Write down the likelihood function appropriate for ni.i.d. observations y1, ..., Yn. [2 MARKS] (b) Explain the Bayesian concept of a conjugate prior. Show that the Ga(a,b) dis- tribution is conjugate for i.i.d. data Y1, ..., Yn using the likelihood derived in part (la). [2 MARKS] (c) Derive the Jeffreys' prior p^) for the parameter 1 in the Poisson model for n observations above. [4 MARKS] (d) Is the obtained Jeffreys' prior for proper? Demonstrate this. [2 MARKS] (e) Show that the posterior distribution resulting from using the Jeffreys' prior is equivalent to Ga (1-1 Yi + 1/2,n), and demonstrate whether it is proper. [2 MARKS] (f) Assume that yi, ..., yn have been observed. Using the Jeffreys' prior, derive the posterior predictive distribution for Yn+1. [6 MARKS] (g) Using a Gamma(2, 2) prior on 1, and having observed counts 3, 2, 1, and 2; what is the maximum a posteriori (MAP) estimate of \? (The mode of a Gamma distribution Gala,B) is a ?). [2 MARKS]