4) Maximum Likelihood Estimation (15 pts). Consider a random variable X (possibly a vector) whose distribution (density function or mass function) belongs to a parametric family. The density or mass function may be written f(x;0), where O is called the parameter, and can be either a scalar or vector. For example, in the Gaussian family, e can be a two-dimensional vector consisting of the mean and variance. Suppose the parametric family is known, but the value of the parameter is unknown. It is often of interest to estimate this parameter from observations of X. Marimum likelihood estimation is one of the most important parameter estimation techniques. Let X1, ..., X.. be iid (independent and identically distributed) random variables distributed according to f(x;0). By independence, the joint distribution of the observations is the product II s( X;0) (1) Viewed as a function of , this quantity is called the likelihood of 0. It is often more convenient to work with the log-likelihood, Clog / ( X1;0) (2) A maximum likelihood estimate (MLE) of 0 is any parameter E arg max > log f (X,; 0) (3) 1=1 2 EECS 545 - Machine Learning - Homework #1 Due 11:00pm 01/25/2016 where "arg max" denotes the set of all values achieving the maximum. If there is a unique maximizer, it is called the maximum likelihood estimate. Let X1, ..., X, be iid Poisson random variables with intensity parameter A. Determine the maximum likelihood estimator of 1