APPENDIX: CO M PUTING THE M ARG INAL LIK ELIHO O D USING THE G ELFAND AND DEY M ETHO D

8. Appendix: Computing the marginal likelihood using the Gelfand and Dey method

Gelfand and Dey (1994) introduce a method for computing the marginal likelihood that is particularly convenient to use when employing the Metropolis Hastings algorithm. This method is based on the following result.

 ∙  (Φ)  ( \Φ) ×  (Φ)\ ¸ = 1  ( ) (8.1)

where  ( \Φ) denotes the likelihood function,  (Φ) is the prior distribution,  ( ) is the marginal likelihood and  (Φ) is any pdf with support Θ defined within the region of the posterior. The proof of equation 8.1 can be obtained by noting that h_{ (\Φ)× (Φ)} (Φ) _\i =

Z

 (Φ)

 (\Φ)× (Φ) ×  (Φ\ ) Φ where  (Φ\ ) is the posterior distribution.

Note that  (Φ\ ) =  (\Φ)× (Φ) ( ) and the density  (Φ) integrates to 1 leaving us with the right hand side in

equation 8.1.

We can approximate the marginal likelihood as 1 



X

=1

 (Φ)

 (\Φ)× (Φ) where Φ denotes draws of the parameters

from Metropolis Hastings algorithm and  ( \Φ) ×  (Φ) is the posterior evaluated at each draw. Geweke (1998)

recommends using a truncated normal distribution for  (Φ). This distribution is truncated at the tails to ensure that  (Φ) is bounded from above, a requirement in Gelfand and Dey (1994). In particular, Geweke (1998) suggest using  (Φ) = 1  (2)2 ¯ ¯ ¯ˆΣ¯¯_¯−12exp ∙ −05³Φ− ˆΦ ´ ˆ Σ−1³Φ− ˆΦ ´0¸ × ³Φ∈ ˆΘ ´ (8.2) where ˆΦ is the posterior mean, ˆΣ is the posterior covariance and k is the number of parameters. The indicator function ³Φ∈ ˆΘ ´ takes a value of 1 if ∙³ Φ − ˆΦ´Σˆ−1³_{Φ − ˆ}Φ´0 ¸ ≤ 21−() where 2

1−() is the inverse 2 cumulative distribution function with degrees of freedom  and probability  Thus

2

1−() denotes the value that exceeds 1 − % of the samples from a 2distribution with  degrees of freedom. The

indicator function ³Φ ∈ ˆΘ

´

therefore removes ‘extreme’ values of Φ For more details, see Koop (2003) page 104.

In figures 30 and 31 we estimate the marginal likelihood for a linear regression model via the Gelfand and Dey method. The model is exactly used in the appendix to Chapter 1 and is based on artifical data. A simple random walk Metropolis Hastings algorithm is used to approximate the posterior on lines 35 to 62 and we save the log posterior evaluated at each draw and each draw of the parameters. Lines 65 and 66 calculate the posterior mean and variance. We set 1 −  = 01 on line 68. In practice, different value of 1 −  can be tried to check robustness of the estimate. On line 70 we evaluate the inverse 2_{CDF. Line 71 to 78, loop through the saved draws of the parameters. On 73 we}

calculate ³_{Φ − ˆ}Φ´Σˆ−1³_{Φ − ˆΦ}´0_{ If this is less than or equal to }2

1−() we evaluate

 (Φ)

 (\Φ)× (Φ) in logs, adding

8. APPENDIX: CO M PUTING THE M ARG INAL LIK ELIHO O D USING THE G ELFAND AND DEY M ETHO D 135

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