`bas.predict` error when n<p when using `estimator= BMA` and `se.fit =T`

[petersen-f]

Describe the bug
Great package! A noticed a bug however when trying to perform predictions.
When training a model with more predictors than variables (n<p) via method = BAS, the prediction of new data (with se.fit =T and estimator= BMA) fails with the following error: Error in solve.default(qr.R(qr(oldX))) : 'a' (14 x 15) must be square

To Reproduce
Steps to reproduce the behavior:

data("bodyfat")
bas_mod <- bas.lm(Bodyfat ~.,data = bodyfat[1:14,], method = 'BAS')
pred <- predict(bas_mod,newdata = bodyfat[15:20,], se.fit = T, estimator = 'BMA') 

Expected behavior
The function should return predictions with the 95% credible interval.
If this behavior is not a bug and this type of prediction is impossible I would expect a more informative error that se.fit =T is not supported for n>p scenarios via BMA and the BAS method. It seems to work fine if the method is set to MCMC however.

Desktop (please complete the following information):

• OS: Ubuntu 22.04
• R Version: 4.2.2
• BAS version: 1.6.4

[merliseclyde]

Thanks! That is a bug in the n<p case. I am guessing the reason that it does not happen with MCMC as the sampler is not visiting the non-full rank models, while BAS in this case is enumerating them and non-full rank models are part of the model space (but there should have been a warning about that - an additional issue).

[merliseclyde]

error also triggered using method='deterministic' as this also samples all models.

data("bodyfat")
bas_mod <- bas.lm(Bodyfat ~.,data = bodyfat[1:14,], method = 'BAS')
pred <- predict(bas_mod,newdata = bodyfat[15:20,], se.fit = T, estimator = 'BMA') 

This is also y a problem in bas.glm as well.

1. Short-term fix is to remove rank deficient models in predict as an option (post-process).
2. Assign rank-deficient models prior probability 0 in C which would fix this via solution to issue bas.lm and bas.glm ignoring prior model probabilities that are 0  #74
3. Fix code so that models are not saved for sampling methods BAS, deterministic and MCMC+BAS