127
edits
Changes
Jump to navigation
Jump to search
mLine 1:
Line 1: − + Line 5:
Line 5: − 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]]. Thus the ''Bayes factor'' for model class M<sub>1</sub> with parameters <math>\theta_1</math> to model class M<sub>2</sub> with parameters <math>\theta_2</math> is+ Line 11:
Line 11: − In determining which model class has the largest [[posterior odds ratio]] (of which the ''Bayes factor'' is a principle component) when compared against all others, the model class or [[mathematical model]] which best explains the data is determined. It is then left to determine the best inference as to the values of that models parameters via [[parameter estimation]].+
no edit summary
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.,
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.,
illustrating 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]]. Thus the ''Bayes factor'' for model class M<sub>1</sub> with parameters <math>\theta_1</math> to model class M<sub>2</sub> with parameters <math>\theta_2</math> is
By integrating out the model class parameters for comparitive classes of likelihood models, one is effectively comparing the strengths of the predictions of the two model classes concerning the observed data. I.e., in determining which model class has the largest [[posterior odds ratio]] (of which the ''Bayes factor'' is a principle component) when compared against all others, the model class or [[mathematical model]] which best explains the data is determined. It is then left to determine the best inference as to the values of that models parameters via [[parameter estimation]].
[[Category:Probability]]
[[Category:Probability]]