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Unformatted text preview: Bayesian Estimation Confidence Intervals: Lecture XXII Charles B. Moss October 21, 2010 Charles B. Moss () Bayesian Confidence Interval October 21, 2010 1 / 15 1 Bayesian Estimation 2 Bayesian Confidence Intervals Charles B. Moss () Bayesian Confidence Interval October 21, 2010 2 / 15 Bayesian Estimation Implicitly in our previous discussions about estimation, we adopted a classical viewpoint. I We had some process generating random observations. I This random process was a function of fixed, but unknown. I We then designed procedures to estimate these unknown parameters based on observed data. Charles B. Moss () Bayesian Confidence Interval October 21, 2010 3 / 15 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. I We can estimate the parameters of this distribution using maximum likelihood. I 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 X i =1 ( X i − μ ) 2 # (1) I Our estimates of μ and σ 2 are then based on the value of each parameter that maximizes the likelihood of drawing that sample. Charles B. Moss () Bayesian Confidence Interval October 21, 2010 4 / 15 Turning this process around slightly, Bayesian analysis assumes that we can make some kind of probability statement about parameters before we start. The sample is then used to update our prior distribution. I 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.parameter we are interested in estimating....
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 Spring '10
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 Normal Distribution, Probability theory, Estimation theory, Likelihood function, Bayesian confidence interval, Charles B. Moss

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