Difference between revisions of "Bayes Factor"
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(New page: The ''Bayes Factor'' of model class M<sub>1</sub> to model class M<sub>2</sub> for some set of observations ''X'' is the ratio of their associated marginal class likelihoods. The marginal ...) |
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| − | The ''Bayes Factor'' of model class M<sub>1</sub> to model class M<sub>2</sub> for some set of observations ''X'' is the ratio of their associated marginal class likelihoods. The marginal class likelihood of some set of observations is the [[marginal likelihood]] of the observations for the model class obtained by marginalizing the [[joint probability distribution]] of the observations and the model class parameters by treating the model class parameters as [[nuisance parameters]]. | + | The ''Bayes Factor'' of model class M<sub>1</sub> to model class M<sub>2</sub> for some set of observations ''X'' is the ratio of their associated marginal class likelihoods. The marginal class likelihood of some set of observations is the [[marginal likelihood]] of the observations for the model class obtained by marginalizing the [[joint probability distribution]] of the observations and the model class parameters by treating the model class parameters <math>\theta</math> as [[nuisance parameters]], i.e., |
::<math>m_{\theta}(X|M) = \int_\theta p(X,\theta|M) \, d\theta = \int_\theta p(X|\theta , M) \, p(\theta|M) \, d\theta </math> | ::<math>m_{\theta}(X|M) = \int_\theta p(X,\theta|M) \, d\theta = \int_\theta p(X|\theta , M) \, p(\theta|M) \, d\theta </math> | ||
| − | showing that it may be calculated directly using the [[joint probability distribution]] or the [[conditional probability distribution]] where the second form of the integral was obtained | + | |
| + | showing that it may be calculated directly using the [[joint probability distribution]] or from the [[conditional probability distribution]] where the second form of the integral was obtained using the [[Bayesian Product Rule]]. | ||
[[Category:Probability]] | [[Category:Probability]] | ||
Revision as of 03:46, December 28, 2007
The Bayes Factor of model class M1 to model class M2 for some set of observations X is the ratio of their associated marginal class likelihoods. The marginal class likelihood of some set of observations is the marginal likelihood of the observations for the model class obtained by marginalizing the joint probability distribution of the observations and the model class parameters by treating the model class parameters <math>\theta</math> as nuisance parameters, i.e.,
- <math>m_{\theta}(X|M) = \int_\theta p(X,\theta|M) \, d\theta = \int_\theta p(X|\theta , M) \, p(\theta|M) \, d\theta </math>
showing that it may be calculated directly using the joint probability distribution or from the conditional probability distribution where the second form of the integral was obtained using the Bayesian Product Rule.