Bayesian Statistics: Algorithm for Extracting Beta and Proof of Negative Binomial Regression Model
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Added on 2023/05/28
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This text explains the algorithm for extracting β=( β 1 , β 2 , β3 ) ∧κ in Bayesian statistics and provides a proof of the negative binomial regression model. It covers the probability distribution of a zero-truncated negative binomial probability and the calculation of MLE of μi.
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Assignment – Bayesian statistics Assume a negative binomial regression model as follows: 1.Explain an algorithm for extracting ofβ=(β1,β2,β3)∧κ. To obtain the algorithm of extractingβ=(β1,β2,β3)∧κ, we first we write down the probability distribution of a zero-truncated negative binomial probability, Pr(yi=j) ={π+(1−π)g(y1=0)ifj=0 (1−π)g(y1)ifj>0 Where g(yi) = Pr(Y=yi∨¿μi,α) =г(yi+α−1) г(α−1)г(yi+1)(1 1+αμi) α−1 (αμi 1+αμi)yi Negative binomial component includes t (exposure time) and k regressor values for Xi . Thus, μi= exp{ln(ti)+β1x1i+β2x2i+….+βkxki} Then calculate MLE ofμi ln(μi) = ln(exp{ln(ti)+β1x1i+β2x2i+….+βkxki}) (taking natural log on both sides) But ln(exp(ln(t)) is a constant Therefore, ln(exp{ln(ti)+β1x1i+β2x2i+….+βkxki}) =β1x1i+β2x2i+….+βkxki Hence (β1,β2,β3) 2.Proof yi|λiPoisson(λi),λi|xiGamma(κ,κμi),
log(μi)=β1+β2xi+β3xi 2 The pdf of P(X=x) =−e−λ ∑ k=0 nλk k! ⌈(a,b) =∫ b ∞ tα−1e−tdt (where is⌈(a,b) denotes incomplete gamma function ) Then ⌈(n,b) = (n-1)!−e−b ∑ k=0 n−1bk k!given that n>0 P(X=x) =1 ⌈(β)∫ 0 λ tα−1e−tdt But μi= exp{ln(ti)+β1x1i+β2x2i+….+βkxki} Then calculate MLE ofμi ln(μi) = ln(exp{ln(ti)+β1x1i+β2x2i+….+βkxki}) (taking natural log on both sides) But ln(exp(ln(t)) is a constant Therefore, ln(exp{ln(ti)+β1x1i+β2x2i+….+βkxki}) =β1x1i+β2x2i+….+βkxki