April1_MaximumLikelihood

April1_MaximumLikelihood - Integrative Biology...

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1 Integrative Biology 200A “PRINCIPLES OF PHYLOGENETICS” Spring 2008 University of California, Berkeley Kipling Will- 3 April* Maximum Likelihood: Maximum likelihood is a general statistical method for estimating unknown parameters of a probability model. A parameter is some descriptor of the model. A familiar model might be the normal distribution with two parameters: the mean and variance. In phylogenetics there are many parameters, including rates, differential transformation costs, and, most important, the tree itself Likelihood is defined to be a quantity proportional to the probability of observing the data given the model, P(D|M). Thus, if we have a model (i.e. the tree and parameters), we can calculate the probability the observations would have actually been observed as a function of the model. We then examine this likelihood function to see where it is greatest, and the value of the parameter of interests (usually the tree and/or branch lengths) at that point is the maximum likelihood estimate of the parameter. Simple Coin Flip example: The likelihood for heads probability p for a series of 11 tosses assumed to be independent- HHTTHTHHTTT 5 heads ( p ), 6 tails (1- p ) Assuming a fair coin what is the likelihood of this series results? L=(n!/k!(n-k)!)p k (1-p) n-k = 0.22559 Where n is the number of tosses, k is the number coming up heads, p is the probability of heads. L is maximized (0.23609) when p= 0.45454, intuitively 5/11. Thinking of L as a function dependent on the number of tosses ( n ), number of heads observed ( k ) and the true probability ( p ) of getting heads in a single toss. Since n and k are observed you can try various values for p to find the one that maximizes L . In other words, this can be
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