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BAYESIAN STATISTICS
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Assignment Bayesian statistics
Assume a negative binomial regression model as follows:
1. Explain an algorithm for extracting of β
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probability distribution of a zero-truncated negative binomial probability,
Pr(
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Where
g(
) = Pr(Y=
, ) =
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Negative binomial component includes t (exposure time) and k regressor values for Xi .
Thus,
= exp
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Then calculate MLE of
ln(
) = ln(exp
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) (taking natural log on both sides)
But ln(exp(ln(t)) is a constant
Therefore,
ln(exp
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) =
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Hence (
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2. Proof
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BAYESIAN STATISTICS Student Name Class name Date Assignment – Bayesian statistics Assume a negative binomial regression model as follows: 1. Explain an algorithm for extracting of β = (𝛽1 , 𝛽2 , 𝛽3 )and 𝜅. To obtain the algorithm of extracting β = (𝛽1 , 𝛽2 , 𝛽3 )and 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) (1 − 𝜋)𝑔(𝑦1 ) 𝑖𝑓 𝑗 = 0 𝑖𝑓 𝑗 > 0 Where g(𝑦𝑖 ) = Pr(Y=𝑦𝑖 | 𝜇𝑖 , 𝛼 ) = г(𝑦𝑖 +𝛼−1 ) 𝛼−1 1 ( ) г(𝛼−1 )г(𝑦 +1) 1+𝛼𝜇 𝑖 𝑖 𝛼𝜇 (1+𝛼𝜇𝑖 ) 𝑦𝑖 𝑖 Negative binomial component includes t (exposure time) and k regressor values for Xi . Thus, 𝜇𝑖 = exp{ln(𝑡𝑖 ) + 𝛽1 𝑥1𝑖 + 𝛽2 𝑥2𝑖 + ⋯ . +𝛽𝑘 ? ...
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