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Likelihood_and_Bayesian

# Likelihood_and_Bayesian - Examples on Maximum Likelihood...

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Examples on Maximum Likelihood Estimation and Bayesian Inference November 13, 2009 1 Estimating a Normal Population Mean 1.1 Unknown μ , Known σ 2 Let X i ∼ N ( μ, σ 2 ) where σ 2 is known and μ is to be estimated based on the observed n samples, x 1 , ..., x n . To be able to find the maximum likelihood estimator of μ or to obtain a posterior distribution over μ using the Bayesian route, we first need to state the likelihood. Recall that the likelihood is obtained from the data generating machinery and in this case it is a normal distribution. Our observations x i are generated by a normal distribution with mean μ which we would like to learn about. Thus the likelihood of μ (or the joint density of the observed x i ) stated as the product of normal densities evaluated at each x i . L ( μ ) = p ( x 1 , ..., x n | μ ) = n Y i =1 1 2 πσ exp ( - 1 2 x i - μ σ 2 ) The maximum likelihood estimator, ˆ μ , is arg max μ log L ( μ ), i.e. the μ value that maximizes the logarithm of the likelihood function given above. If we take the log of the likelihood function we obtain log L ( μ ) = ( μ ) = - n 2 log(2 π ) - n 2 log σ 2 - 1 2 σ 2 n X i =1 ( x i - μ ) 2 . If we take the derivative with respect to μ , and set it equal to zero evaluated at the maximum ˆ μ , we obtain ˆ μ = i x i /n = ¯ x . If desired confidence intervals may be obtained on μ . Our goal is to obtain Bayesian inferences on μ and likelihood is a central element to this procedure.

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Likelihood_and_Bayesian - Examples on Maximum Likelihood...

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