Question

Suppose observations X1, X2,.. are recorded. We assume these to be conditionally independent and exponen- tially distributed given a parameter θ: Xi ~' Exponential(θ), for all i 1, . . . , n. The exponential distribution is controlled by one rate parameter θ > 0, and its density is for r ER+ 1. Plot the graph of p(x:0) for θ 1 in the interval x E [0,4] 2. What is the visual representation of the likelihood of individual data points? Draw it on the graph above for the samples in a toy dataset χ-: {1,2,4} and θ 1 3. Would a higher rate (e.g. 2) increase or decrease the likelihood of each sample in this toy data set? We introduce a prior distribution q(9) for the parameter. Our objective is to compute the posterior. In general, that requires computation of the evidence as the integral zn)=h (Ilp( r. θ))g@jde p(zī, pT1,...,Tn) - i=1 We will not have to compute the integral in the following, since we choose a prior that is conjugate to the exponential The natural conjugate prior for the exponential distribution is the gamma distribution Γ(a) for θ > 0 and α, β > 0. We have already encountered this distribution in an earlier homework problem (where we computed its maximum likelihood estimator), and you will notice that we are using a different parametrization of the gamma density here

Answer 1

ρ haり 、na hus.ct ee ryegahaam na Mar Henelite, an Sur ned VM