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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Point Estimation 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 29, 2007 Lecture 24 More Bayesian Metrics Today our focus will be on testing in a Bayesian setting. We will see that even in nice, simple cases Bayesian tests have different results than frequentists tests. Our setup is the same as last time. We have: • Likelihood f ( Y T | θ ) • Prior p ( θ ) which give the posterior f ( p ( θ |Y T = Y T | θ ) p ( θ ) ) R f ( Y T | θ ) p ( θ ) dθ In addition, we have a decision theory setup: • A – action space • decision rule δ ( Y T ) ∈ A , mapping observations to the action space • Loss function: L ( δ, θ ) : A ⊗ Θ → < • Bayesian decision rule: δ ( Y T ) = arg min E ( L ( δ, θ ) δ |Y T ) ∈A Most things we might be interested in estimating can be put into a decision theory framework. Point Estimation Our action is to choose an estimate, A = Θ. There are a number of loss functions we could use: 1. If Θ is discrete, the loss function could be L ( δ, θ ) = ( 1 δ = θ δ = θ The optimal decision rule is then δ ( Y T ) = arg max P ( θ = θ θ |Y T ), which is the mode In the continuous case, ∀ ² > 0, L ( δ, θ ; ² ) = 1- 1 { θ ∈N ² ( δ ) } Let δ ² denote the optimal decision rule for this loss function. Then lim ² δ → ² = the mode of the posterior distribution....
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lec24 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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