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Lecture22-2010 - Bayesian Estimation Condence Intervals...

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Bayesian Estimation Confidence Intervals: Lecture XXII Charles B. Moss October 21, 2010 I. Bayesian Estimation A. Implicitly in our previous discussions about estimation, we adopted a classical viewpoint. 1. We had some process generating random observations. 2. This random process was a function of fixed, but unknown. 3. We then designed procedures to estimate these unknown pa- rameters based on observed data. B. Specifically, if we assumed that a random process such as students admitted to the University of Florida, generated heights. This height process can be characterized by a normal distribution. 1. We can estimate the parameters of this distribution using maximum likelihood. 2. The likelihood of a particular sample can be expressed as L X 1 , X 2 , · · · X n | μ, σ 2 = 1 (2 π ) n/ 2 σ n exp 1 2 σ 2 n i =1 ( X i μ ) 2 (1) 3. Our estimates of μ and σ 2 are then based on the value of each parameter that maximizes the likelihood of drawing that sample. 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXII Fall 2010 C. Turning this process around slightly, Bayesian analysis assumes that we can make some kind of probability statement about pa- rameters before we start. The sample is then used to update our prior distribution. 1. First, assume that our prior beliefs about the distribution function can be expressed as a probability density function π ( θ ) where θ is the parameter we are interested in estimating.
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