# Using beta prior when agnostic about correct functional form of covariate

I have a logistic regression setting, where I’m interested in predictive power most of all . I’m unsure about what functional form of one of my covariates will guarantee maximum predictive power out-of-sample. I know that the covariate at hand matters a-priori, so I don’t want to use a horseshoe because I run the risk of selecting the variable out entirely.

Inspired by the BYM2 identification strategy (as seen for example in Riebler, A., Sørbye, S. H., Simpson, D., & Rue, H. (2016). An intuitive Bayesian spatial model for disease mapping that accounts for scaling ) I thought an appropriate model might be a mixture of the covariate effects under different functional forms:

y_i \sim \mbox{Bernoulli} (\pi_i )
\mbox{logit}(\pi_i) = \alpha_i + \mbox{mix}_i
\mbox{mix}_i = \sqrt{\rho} \beta_1 x_i + \sqrt{1-\rho} * \beta_2 f(x_i)
\rho \sim \mbox{Beta}(0.5,0.5)
\beta_{1,2} \sim N(0,1)

where f is any function that is not linear. I might also want to extend this to a Dirichlet prior if there are more than 1 candidate f s.

Is this approach valid ? Do you have any other suggestion as to how to go about a similar exercise ?

What do you mean exactly by valid? If there is a justification for that choice, from a modeling point of view it should be valid, to me the question than becomes: are the parameters identifiable? (it’s sometimes difficult to identify mixtures confidently).
It seems like this is still a logistic regression, just you are replacing the linear term with the coefficients with a mixture of x and a function of x, so there is nothing weird going on with the model itself, I think, it would depend on what the functional form f(x_i) actually takes.

In sum, if the model formulation makes sense in general, it’s probably best to run the inference, see if there are any identifiability issues, and then evaluate its performance.

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