Posterior Model Odds using Monte Carlo Integration
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I have a model $M_1$ with three parameters and a model $M_2$ with two parameters and data $y_t$. I want to compute the quantity $$frac{p(M_1|y)}{p(M_2|y)}= frac{p(y|M_1)}{p(y|M_2)}$$ for equal priors $p(M_1)=p(M_2)=1/2$. The numerator for example can be written as $$int int int p(y| mu, omega, rho, M_1) p( mu|M_1 ) p( omega | M_1 ) p( rho M_1) d mu d omega drho$$ or as $$ int int int p(y| mu, omega, rho, M_1) p(mu, omega ,rho |M_1) d mu domega drho$$ if we assume independent priors. The full conditionals are easily computed but I think that it is impossible to simulate using Gibbs algorithm from the posterior distribution of the parameters and take the mean on the likelihood for two reasons. The one is that the right posterior must be conditioned on my data as well so $ p( mu|M_1 ) p( omega | M_1 ) p( rho M_1)$ is more likely the prior. But if I simulated from the priors I think that it would be wrong again as the likelihood is a function of y here and parameters are considered given for the likelihood so I can't use my data. Am I right ? So this is a more difficult problem than I think?
bayesian
asked Dec 31 '18 at 17:59
Manos LouManos Lou
62
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add a comment |
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I have a model $M_1$ with three parameters and a model $M_2$ with two parameters and data $y_t$. I want to compute the quantity $$frac{p(M_1|y)}{p(M_2|y)}= frac{p(y|M_1)}{p(y|M_2)}$$ for equal priors $p(M_1)=p(M_2)=1/2$. The numerator for example can be written as $$int int int p(y| mu, omega, rho, M_1) p( mu|M_1 ) p( omega | M_1 ) p( rho M_1) d mu d omega drho$$ or as $$ int int int p(y| mu, omega, rho, M_1) p(mu, omega ,rho |M_1) d mu domega drho$$ if we assume independent priors. The full conditionals are easily computed but I think that it is impossible to simulate using Gibbs algorithm from the posterior distribution of the parameters and take the mean on the likelihood for two reasons. The one is that the right posterior must be conditioned on my data as well so $ p( mu|M_1 ) p( omega | M_1 ) p( rho M_1)$ is more likely the prior. But if I simulated from the priors I think that it would be wrong again as the likelihood is a function of y here and parameters are considered given for the likelihood so I can't use my data. Am I right ? So this is a more difficult problem than I think?
bayesian
asked Dec 31 '18 at 17:59
Manos LouManos Lou
62
$endgroup$
add a comment |
$begingroup$
I have a model $M_1$ with three parameters and a model $M_2$ with two parameters and data $y_t$. I want to compute the quantity $$frac{p(M_1|y)}{p(M_2|y)}= frac{p(y|M_1)}{p(y|M_2)}$$ for equal priors $p(M_1)=p(M_2)=1/2$. The numerator for example can be written as $$int int int p(y| mu, omega, rho, M_1) p( mu|M_1 ) p( omega | M_1 ) p( rho M_1) d mu d omega drho$$ or as $$ int int int p(y| mu, omega, rho, M_1) p(mu, omega ,rho |M_1) d mu domega drho$$ if we assume independent priors. The full conditionals are easily computed but I think that it is impossible to simulate using Gibbs algorithm from the posterior distribution of the parameters and take the mean on the likelihood for two reasons. The one is that the right posterior must be conditioned on my data as well so $ p( mu|M_1 ) p( omega | M_1 ) p( rho M_1)$ is more likely the prior. But if I simulated from the priors I think that it would be wrong again as the likelihood is a function of y here and parameters are considered given for the likelihood so I can't use my data. Am I right ? So this is a more difficult problem than I think?
bayesian
asked Dec 31 '18 at 17:59
Manos LouManos Lou
62
$endgroup$
I have a model $M_1$ with three parameters and a model $M_2$ with two parameters and data $y_t$. I want to compute the quantity $$frac{p(M_1|y)}{p(M_2|y)}= frac{p(y|M_1)}{p(y|M_2)}$$ for equal priors $p(M_1)=p(M_2)=1/2$. The numerator for example can be written as $$int int int p(y| mu, omega, rho, M_1) p( mu|M_1 ) p( omega | M_1 ) p( rho M_1) d mu d omega drho$$ or as $$ int int int p(y| mu, omega, rho, M_1) p(mu, omega ,rho |M_1) d mu domega drho$$ if we assume independent priors. The full conditionals are easily computed but I think that it is impossible to simulate using Gibbs algorithm from the posterior distribution of the parameters and take the mean on the likelihood for two reasons. The one is that the right posterior must be conditioned on my data as well so $ p( mu|M_1 ) p( omega | M_1 ) p( rho M_1)$ is more likely the prior. But if I simulated from the priors I think that it would be wrong again as the likelihood is a function of y here and parameters are considered given for the likelihood so I can't use my data. Am I right ? So this is a more difficult problem than I think?
bayesian
bayesian
asked Dec 31 '18 at 17:59
Manos LouManos Lou
62
asked Dec 31 '18 at 17:59
Manos LouManos Lou
62
asked Dec 31 '18 at 17:59
Manos LouManos Lou
62
asked Dec 31 '18 at 17:59
Manos LouManos Lou
62
asked Dec 31 '18 at 17:59
Manos LouManos Lou
62
62
add a comment |
add a comment |
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