MMEmaxlikelihood - 2011 Steven Tschantz Maximum-likelihood...

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© 2011 - Steven Tschantz Maximum-likelihood estimation Steven Tschantz 3/22/11 Problem A mathematical model of a real world situation will often be defined with one or more free parameters to be calibrated to observed data. Rarely do we expect a model to fit the observed data exactly however. Measurements are subject to some error, not all factors determining an outcome will be known or observable, and randomness is an essential part of real world phenomena. A statistical model includes random factors in its specification. But observed samples will generally not deter- mine parameters of a statistical model uniquely. The problem is how to estimate the parameters for a statistical model. Method While there are many methods for estimating the parameters of a statistical model, and many criteria for judging such methods, this notebook will concentrate on a particular method applicable to many problems which is based on a simple heuristic. For given parameters, a statistical model should predict the probability of a particular observation. While a single observation may give a highly unusual result, one with a low probability, repeated trials should reflect the probabilities predicted by the model. One set of values of parameters might thus be considered a better estimate of the true parameters if the model with those parameter values predicts a higher probability for the actual observations. Maximum-likelihood estimation takes for parameter estimates the values of the parameters in the model that make the observed data the most likely. For details see wikipedia, here are the basics. Let Θ be a vector of parameters of a statistical model, and x be a vector of observed quantities in the model. The model should assign a probability density f H x ; Θ L to the observation of x . Suppose we make n observations x 1 , x 2 , . .., x n , assumed to be identically and independently distributed according to the probability density function f H x ; Θ L for some "true" but unknown values of the parameters Θ . We define the likelihood of the parameters being Θ given the observations x 1 , x 2 , . .. to be the probability of these observations, L H Θ ; x 1 , x 2 , ..., x n L = ± i = 1 n f H x i ; Θ L . A maximum-likelihood estimator for the model is a function that determines, as a function of the observations x 1 , x 2 , . .., a set of parameters Θ that maximize this likelihood (often uniquely determined). The Θ that maximizes the likelihood will maximize the average log-likelihood { H Θ ; x 1 , x 2 , ..., x n L = 1 n log H L H Θ ; x 1 , x 2 , ..., x n LL = 1 n ² i = 1 n log H f H x i ; Θ LL and this will often be more convenient to work with. Adjusting L by a constant multiple, or adding a constant to { will not change the values of Θ giving the maximum.
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Example - Binomial distribution For many simple models, the maximum-likelihood estimator can be simply understood. Suppose we model the flip of a coin as coming up heads with some probability
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MMEmaxlikelihood - 2011 Steven Tschantz Maximum-likelihood...

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