5102-Lecture-04

5102-Lecture-04 - Lecture 4 Stat 5102-004 26 January 2011...

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Lecture 4 Stat 5102-004 26 January 2011
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Bayesian Paradigm We’d like to I communicate uncertainty about a parameter I (equivalently, the distribution it indexes) I in light of the data we have seen. The Bayesian paradigm accomplishes this by I assigning a probability distribution to the parameter space, I conditional on the data, I via Bayes’ theorem. STAT 5102 (Theory of Statistics) Lecture 4 1 / 40
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Bayesian Paradigm 1. Let the random variable Y take values in the observable space Y . Assume it is in a parametric family; denote its density f ( y | θ ) . 2. Let the random variable Θ take values in the parameter space ϑ . Denote the density of this distribution at θ as π ( θ ) . 3. Bayes’ theorem relates expressions of the density over Y × ϑ . π ( θ | y ) f ( y ) = f ( y | θ ) π ( θ ) 4. The density π ( θ | y ) expresses uncertainty about θ given the data y . STAT 5102 (Theory of Statistics) Lecture 4 2 / 40
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Binomial Example Consider sampling from a Binomial with n fixed. lik( θ | y ) θ y (1 - θ ) n - y θ (0 , 1) y ∈ { 0 , . . . , n } Assume a uniform prior for the success probability θ . π ( θ ) 1 θ (0 , 1) The posterior is a Beta( y + 1 , n - y + 1 ). π ( θ | y ) θ ( y +1) - 1 (1 - θ ) ( n - y +1) - 1 θ (0 , 1) STAT 5102 (Theory of Statistics) Lecture 4 3 / 40
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Binomial: n = 12 , y = 0 θ π ( θ |y 29 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 12 π ( θ | y ) = Γ( n + 2) Γ( y + 1)Γ( n - y + 1) θ ( y +1) - 1 (1 - θ ) ( n - y +1) - 1 STAT 5102 (Theory of Statistics) Lecture 4 4 / 40
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Binomial: n = 12 , y = 1 θ π ( θ |y 29 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 π ( θ | y ) = Γ( n + 2) Γ( y + 1)Γ( n - y + 1) θ ( y +1) - 1 (1 - θ ) ( n - y +1) - 1 STAT 5102 (Theory of Statistics) Lecture 4 5 / 40
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Binomial: n = 12 , y = 2 θ π ( θ |y 29 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 π ( θ | y ) = Γ( n + 2) Γ( y + 1)Γ( n - y + 1) θ ( y +1) - 1 (1 - θ ) ( n - y +1) - 1 STAT 5102 (Theory of Statistics) Lecture 4 6 / 40
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Binomial: n = 12 , y = 3 θ π ( θ |y 29 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 π ( θ | y ) = Γ( n + 2) Γ( y + 1)Γ( n - y + 1) θ ( y +1) - 1 (1 - θ ) ( n - y +1) - 1 STAT 5102 (Theory of Statistics) Lecture 4 7 / 40
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5102-Lecture-04 - Lecture 4 Stat 5102-004 26 January 2011...

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