Chapter24_TAMU.pdf - ECEN 489 Information Theory Inference and Learning Algorithms Chapter 24 Exact Marginalization Dr Chao TIAN Texas A&M University 1

Chapter24_TAMU.pdf - ECEN 489 Information Theory Inference...

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ECEN 489: Information Theory, Inference, and Learning Algorithms Chapter 24: Exact Marginalization Dr. Chao TIAN Texas A&M University 1 / 14
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Inferring the Mean and the Variance of a Gaussian The one dimensional Gaussian distribution: P ( x | μ, σ ) = 1 2 πσ exp - ( x - μ ) 2 2 σ 2 , Normal( x ; μ, σ 2 ) . Given the model, but unknown parameters μ, σ 2 ; Prior distribution on μ, σ 2 : I No prior knowledge: uniform distribution over the ranges; I More convenient choices: conjugate priors. 2 / 14
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FYI: What are the Conjugate Priors for μ and σ ? Conjugate prior: When the posterior distributions P ( θ | x ) are in the same family as the prior probability distribution P ( θ ). For the mean μ of a Gaussian: a Gaussian distribution Normal( μ ; μ 0 , σ 2 μ ) For the standard deviation σ of a Gaussian: a Gamma distribution P ( β ) = Γ( β ; b β , c β ) = 1 Γ( c β ) β c β - 1 b c β β exp - β b β , 0 β ≤ ∞ . where β = 1 2 and Γ( x ) , R 0 u x - 1 e - u du . 3 / 14
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FYI: What are the Conjugate Priors for μ and σ ? Conjugate prior: When the posterior distributions P ( θ | x ) are in the same family as the prior probability distribution P ( θ ). For the mean μ of a Gaussian: a Gaussian distribution Normal( μ ; μ 0 , σ 2 μ ) For the standard deviation σ of a Gaussian: a Gamma distribution P ( β ) = Γ( β ; b β , c β ) = 1 Γ( c β ) β c β - 1 b c β β exp - β b β , 0 β ≤ ∞ . where β = 1 2 and Γ( x ) , R 0 u x - 1 e - u du . 3 / 14
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FYI: What are the Conjugate Priors for μ and σ ? Conjugate prior: When the posterior distributions P ( θ | x ) are in the same family as the prior probability distribution P ( θ ).
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