Beta Binomial in Bayesian linear regression

XuhuiZhang

By using the default setting of beta binomial (i.e., alpha and beta are all 1), the P(M) should be 1/number of models. But the result seems not like below:

XuhuiZhang

According to BAS package on which JASP relies, the function beta.binomial looks different from the definition of beta binomial distribution or other R packages (e.g., rmutil). Here is the description of beta.binomial:

The beta-binomial distribution on model size is obtained by assigning each variable inclusion indicator independent Bernoulli distributions with probability w, and then giving w a beta(alpha,beta) distribution. Marginalizing over w leads to the distribution on model size having the beta-binomial distribution. The default hyperparameters lead to a uniform distribution over model size. 

As for the definition or other packages, e.g., rmutil:

https://forum.cogsci.nl/uploads/410/ODEZKZDVKASB.png

I guess this is why the output of P(M) differ among different models. And it seems that the beta.binomial function JASP uses is not as intuitive as others. That's to say, the default setting (i.e., alpha and beta are both 1), leads to model prior probabilities not so uninformative. Maybe someone could help clarify this question?

Thanks!

XuhuiZhang

According to BAS package on which JASP relies, the function beta.binomial looks different from the definition of beta binomial distribution or other packages (e.g., rmutil). Here is the description of beta.binomial:

The beta-binomial distribution on model size is obtained by assigning each variable inclusion indicator independent Bernoulli distributions with probability w, and then giving w a beta(alpha,beta) distribution. Marginalizing over w leads to the distribution on model size having the beta-binomial distribution. The default hyperparameters lead to a uniform distribution over model size. 

As for the definition or other packages, e.g. rmutil:

https://forum.cogsci.nl/uploads/410/ODEZKZDVKASB.png

I guess this is why the output of P(M) differ among different models. But different model prior probabilities generated by JASP do not look like noninformative with the default setting of beta binomial distribution (i.e., alpha and beta are both 1). Maybe someone could help clarify?

Thanks!

EJ

Dear XuhuiZhang,

The beta-binomial formulation is based on an idea by Harold Jeffreys that was elaborated upon by Scott & Berger (2006,2010) -- references are hopefully in the help file. The idea is that a beta(1,1) prior assigns equal prior probability to all of the model *classes* (e.g., the class of models with no predictors, with one predictor, with two predictors, etc) and then distributes it evenly across all members of the class. If the model probability would be distributed evenly across all models this would effectively bias the analysis towards models that contain about half of the predictors.

Cheers,

EJ

XuhuiZhang

Dear EJ,

Thanks for your reply! The ref you mentioned helps a lot.