## Background

Suppose one has data \boldsymbol Y = \{ y_1, y_2, \ldots y_N \}, y_i \in \{0, 1\} and some covariates \boldsymbol X in the form of a N \times P matrix. Suppose further that the model to explain \boldsymbol Y using \boldsymbol X is a logistic regression. For instance, we could be interested in explaining the probability of death from a certain disease given some explanatory variable(s).

The questions here pertain to the elicitation of (marginal) prior distributions for the coefficients, \boldsymbol \beta by manipulating odds ratios (\operatorname{OR}), risk ratios (\operatorname{RR}) and the baseline prevalence, p_0.

The **main idea** here is that one has a data set of moderate size that one wishes to analyse while incorporating extensive knowledge about the baseline prevalence/risk and an informative guess about the maximum risk ratio of a given covariate.

## Calculations

For simplicity, let’s concentrate on elicitation for a single coefficient.

Assuming we know p_0, we can use the identity \beta = \log(\operatorname{OR}) to write the risk ratio, as \operatorname{RR} = \frac{\exp(\beta)}{(1-p_0) +p_0 \exp(\beta)}. Suppose we have a maximum postulated risk ratio \operatorname{RR}_m, where \operatorname{RR}_m < 1/p_0 . If we place a prior \pi_B(\beta) on the coefficient, we can truncate the prior at \beta_m = \log\left( \frac{p_0\operatorname{RR}_m - \operatorname{RR}_m}{p_0\operatorname{RR}_m - 1}\right), creating a new prior \pi_B^\ast(\beta) = \pi_B(\beta)/F_B(\beta_m), with F_B(x) = \int_0^x \pi_B(x)\, dx.

This prior distribution incorporates knowledge about p_0 and translates the informative guess about risk ratios to the coefficients. Notice that other probabilistic constraints can also be incorporated this way, for instance \operatorname{Pr}(\operatorname{RR} < 2) = 1/2 \to \operatorname{Pr}\left(\beta < \log\left( \frac{2p_0 - 2}{2p_0 - 1}\right)\right) = 1/2.

## Questions

- Has anyone seen this method of constructing informative priors for the coefficients in logistic regression?
- Is this a good idea?

I understand a major criticism of this approach is that we seldom know the baseline risk. But I posit there are, indeed, situations where we have a pretty good idea and can use that information to our advantage when creating prior distributions that incorporate important physical constraints.