# bayessox - Bayesian Inference Inference Regarding a...

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Bayesian Inference – Inference Regarding a Proportion Probability Red Sox Beat Yankees in a Game Parameter: π = Probability that Boston would beat NY in a baseball game (0 π 1) Prior (to season) belief regarding the value of π : No information (anywhere between 0 and 1, equally likely) Values closer to ½ are most likely with no “bias” away from ½. “Expect” that π ≈ 2/3. Family of “prior distributions” with nice properties and very flexible: Beta Distribution: Support is from 0 to 1, indexed by 2 parameters: α,β Density Function: ) 1 ( ) ( ) ( ) ( )! 1 ( ) ( 0 , , 1 0 ) 1 ( ) ( ) ( ) ( ) ( 2 1 1 + + + = + = - = Γ - Γ Γ + Γ = - - β α αβ π V E n n p Case 1: Non-informative Prior ( α = β = 1 29 p( π )=1 0 2 y 0 .2 .4 .6 .8 1 x

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Case 2: Informative Prior with mean 0.5 ( α = β = 2 29 p( π )=3 π(1-π29 Case 3: Informative Prior with mean 0.67 ( α = 2 , β = 1 29 p( π )=2 π 0 .5 1 1.5 y 0 .2 .4 .6 .8 1 x 2 0 .2 .4 .6 .8 1 x
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bayessox - Bayesian Inference Inference Regarding a...

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