# mh1 - MC methods MH Ch 6 Hw Ch 6 Metropolis-Hasting...

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Unformatted text preview: MC methods: MH. Ch 6 Hw Ch 6. Metropolis-Hasting Algorithm Explain clearly what you do. When describing the MH algorithm write which is your target distribution f and the proposal q . Ex 1. The negative binomial density can be parametrized in terms of its mean μ and overdis- persion parameter α , p ( k | α,μ ) = 1 B ( k + 1 ,α- 1 )( k + α- 1 ) αμ αμ + 1 k 1 αμ + 1 1 /α , k = 0 , 1 ,..., (1) where B ( x,y ) = Γ( x )Γ( y ) / Γ( x + y ) is the beta function. Under this parameterization E ( k ) = μ and var ( k ) = μ + αμ 2 . The marginal density of α is obtained by integrating μ out of the density. To do this we assume a conditional density for μ , assuming that μ | α ∼ F( ν 1 ,ν 2 ), the F-distribution with ν 1 and ν 2 degrees of freedom. This is a conjugate prior distribution for μ , and allows us to calculate the marginal distribution of α . In practice, we take ν 1 = 2 a μ and ν 2 = 2 a μ /α , where a μ > 0 is a chosen constant. Then, denoting the conditional density by p ( μ | α ), we have p ( μ | α ) = 1 B ( a μ ,a μ /α ) α a μ μ a μ- 1 (1 + αμ ) a μ + a μ /α , and the integrated likelihood function is given by p ( k | α ) = Z ∞ p ( k | α,μ ) p ( μ | α ) dμ (2) = " Y j 1 B ( k j + 1 ,α- 1 )( k j + α- 1 ) # B ( a μ + ∑ j k j , ( J + a μ ) /α ) B ( a μ ,a μ /α ) , where k denotes the sample k = ( k 1 ,...,k J ) ∼ NegBin( μ,α ), and p ( k | α,μ ) = Q j p...
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mh1 - MC methods MH Ch 6 Hw Ch 6 Metropolis-Hasting...

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