# Fractional logit model in Stan

I was reading about methods that people have used for fractional regression models (regression models where the outcome is between [0,1], and one popular method is the fractional logit model. This is a quasi-bernoulli likelihood specified as:

f(y_{i}|x_{i}, \beta) = (G(x_{i}\beta))^{y_{i}} (1-G(x_{i}\beta))^{1-y_{i}}

and a typical choice for G(x_{i}\beta) is

G(x_{i}\beta) = \frac{exp(x_{i}\beta)}{1+x_{i}\beta}

If we were to put a prior on \beta, I was wondering what the community’s thoughts are on implementing this model in Stan is, or more generally, if we can use these quasi-likelihoods in Bayesian inference.

If p = G(x_i \beta), then Wolfram alpha says

\int_0^1 p^y (1 - p)^{1 - y} dy = \frac{1 - 2 p}{2 \text{tan}^{-1}(1 - 2 p)}

So since we can normalize the distribution over y, then it’s just a regular likelihood and it will hopefully sample in Stan.

I think there should be another exp in the denominator of G.

Maybe a Beta regression would be appropriate? I think there is also a zero-one-inflated version in brms.