We now introduce a poisson intensity parameter for every time point and denote the parameter ( ) that gives the canonical exponential family representation as above by . we choose to employ a linear model connecting the time points with the canonical parameter theta of the poisson distribution above, i. e., =+ in other words, we choose a generalized linear model with poisson distribution and its canonical link function. that also means that conditioned on , we assume the to be independent. imagine we observe the following data: =1 1 outbreaks =2 3 outbreaks =4 10 outbreaks we want to produce a maximum likelihood estimator for . to this end, write down the log likelihood ℓ of the model for the provided three observations at , , and (plug in their values).