Question

Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t, we assume the Y to be independent. Imagine we observe the following data: t1-1 1 outbreaks t2 2 3 outbreaks t3 4 10 outbreaks We want to produce a maximum likelihood estimator for (a, b). To this end, write down the log likelihood l (a, b) of the model for the provided three observations at tı. t2. and ts (plug in their values) What is its gradient? Enter your answer as a pair of derivatives. Oat d, 1 a,l (a, b In order to find the maximum likelihood estimator, we have to solve the nonlinear equation Ve (a,b) 0, which in general does not have a closed solution. Assume that we can reasonably estimate the likelihood estimator using numerical methods, and we obtain a -0.43, b 0.69 Given these results, what would be the predicted expected number of outbreaks for t - 3? Round your answer to the nearest 0.001 We want to model the rate of infection with an infectious disease depending on the day after outbreak t. Denote the recorded number of outbreaks at day t by kt We are going to model the distribution of kt as a Poisson distribution with a time-varying parameter At, which is a common assumption when handling count data First, recall the likelihood of a Poisson distributed random variable Y in terms of the parameter λ, Rewrite this in terms of an exponential family. In other words, write it in the form P (Y-k h(k exp n (A)T (k) - B(A) Since this representation is only unique up to re-scaling by constants, take the convention that T(k) -k. We can write this in canonical form, e.g. as P (Y-k)-h (k) exp[m-b (η)] What is b (n)? Recall that the mean of a Poisson A) distribution is λ. What is the canonical link function ga μ-E [Y] ? Write your answer in terms of . associated with this exponential family here

Answer 1