Which are the default links for the 'Beta' family?

[LucianoDeBenedictis]

#1751 details the same confusion I was having, checking the documentation in brmsfamily and the output of my model. That issue mentions beta_binomial and zero_inflated_beta_binomial, but it seems the same applies also to Beta.

The documentation in brmsfamily tells that the default link for phi is 'log', but this is not shown when printing the model output. The mentioned issue seems to suggest that the link is instead 'identity' if there are no predictors, which led me to some test.

My point is this: even if there are no predictors for phi, it could still be useful to know which link is being used for defining priors. In particular, the default prior gamma(0.01, 0.01) when using the defaults for link seems makes sense with 'identity' more than with log, and in the same way, one would think that changing this prior would be with the log-link described in the documentation.

Currently, print doesn't show the link for phi, which is a consequence of the mentioned issue. My confusion is increased when checking family inside the fit object. Despite the above, it says 'log' for link_phi there, not 'identity'. This can be checked with this toy example:

library(brms)

data <- data.frame(y = 0.5,
                   x = 0)

model <- brm(y ~ x,
           data = data,
           family = Beta,
           prior = prior(normal(0, 1), class = "b"),
           sample_prior = "only",
           chains = 1,
           iter = 10)

I have also checked the above with beta_binomial and appropriate data. If you add an argument phi ~ x inside the brm call, the print method of the fit still doesn't show the link function for phi (anyway it's log inside the fit object).
Finally, forcing the identity link with Beta(link_phi = "identity") for family shows the expected link inside the fit object (with no apparent change in the print).

I think the current, implicit link function is unsatisfactory for people that don't go digging deep into the documentation and then decide to provide every argument. It seems to me that defaults are widely used with brms, and I am not sure if the default `gamma(0.01, 0.01) is supposed to be used with log or identity link (I checked that this is the default intercept prior whether there are predictors or not).
I hope I have provided enough information to improve this great package in a small way.

[paul-buerkner]

Thank you for opening this issue. What concretely do you suggest?

[LucianoDeBenedictis]

Thank you for opening this issue. What concretely do you suggest?

In short, it would be:

1. Make the link function explicit in all cases, also for distributional parameters. I understand that this would partially roll back what's been decided in beta_binomial default link doc vs summary mismatch #1751, but I still think it's worth considering for the reasons mentioned above, both in the print.brmsfit() method and maybe during prior specification.
2. Continuing on the latter, currently the only way to know for sure on which scale the priors are specified before fitting the model is checking the documentation in brmsfamily(), which is fine assuming the link for phi is always 'log', which seems to be the case unlike in beta_binomial default link doc vs summary mismatch #1751. Could it be nice to add some information about links in the print method for get_prior/default_prior?
3. Check if the default gamma(0.01, 0.01) is supposed to be used with the identity or the log link. Currently it's log, but I feel like identity makes more sense. Sorry if I'm wrong.

Finally, thanks to you for all your work. Whether this ends up being useful feedback or not, I'm grateful anyway.

[lunafazio]

The problem I see is that, effectively, every distributional parameter has two defaults: identity when no formula is explicitly specified for it, and whatever brmsfamily states when a formula is added, even if it's just an intercept. I think adding one line clarifying this point in the brmsfamily documentation would be the simplest way to address this source of confusion.

[LucianoDeBenedictis]

The problem I see is that, effectively, every distributional parameter has two defaults: identity when no formula is explicitly specified for it, and whatever brmsfamily states when a formula is added, even if it's just an intercept. I think adding one line clarifying this point in the brmsfamily documentation would be the simplest way to address this source of confusion.

This seemed to be the case according to #1751, but according to my tests above it seems like the default is currently always log, even with no explicit formula. I'm fine with either one, as long as it's clear.

[paul-buerkner]

Luna is right with what the default behavior is. Adding some more doc could be helpful but somehow I thought I had added it there already.

[LucianoDeBenedictis]

Luna is right with what the default behavior is. Adding some more doc could be helpful but somehow I thought I had added it there already.

Is it? Maybe I'm doing something wrong, but running the mini-model above (which doesn't have a distributional sub-model) and then checking:

> model[["family"]][["link_phi"]]
[1] "log"

I checked this with both family = Beta and family = beta_binomial, after adding some lines for trials.

[paul-buerkner]

These links are only used if there is a formula specified.

[LucianoDeBenedictis]

These links are only used if there is a formula specified.

Ok, so that means even an implicit formula like phi ~ 1 as opposed to fixing phi to a value, because I didn't provide a formula for phi, which should mean it's treated as phi ~ 1.