Weakly informative priors for logistic regression

On a previous post, I have mentioned what is called the separation problem [1]. It can happen for example in a logistic regression, when a predictor (or combination of predictors) can perfectly predicts (separate) the data, leading to infinite Maximum Likelihood Estimate (MLE) due to a flat likelihood.

I also mentioned that one (possibly) naive solution to the problem could be to blindly exclude the predictors responsible for the problem. Other more elegant solutions include a penalized likelihood approach [1] and the use of weakly informative priors [2]. In this post, I would like to discuss the latter.

Model setup

Our model of interest here is a simple logistic regression

\displaystyle y_t \sim Bin(n, p_t), \quad p_t = logit^{-1}(\eta_t)

\displaystyle \eta_t = \beta_0 + \sum_{i=1}^{k}\beta_i

and since we are talking about Bayesian statistics the only thing left to complete our model specification is to assign prior distributions to {\beta_i}‘s. If you are not used to the above notation take a look here to see logistic regression from a more (non-Bayesian) Machine Learning oriented viewpoint.

Weakly informative priors

The idea proposed by Andrew Gelman and co-authors in [2] is to use minimal generic prior knowledge, enough to regularize the extreme inference that are obtained from maximum likelihood estimation. More specifically, they realized that we rarely encounter situations where a typical change in an input {x} corresponds to the probability of the outcome {y_t} changing from 0.01 to 0.99. Hence, we are willing to assign a prior distribution to the coefficient associated with {x} that gives low probability to changes of 10 on logistic scale.

After some experimentation they settled with a Cauchy prior with scale parameter equal to {2.5} (Figure above) for the coefficients {\beta_i}, {i=1,...,k}. When combined with pre-processed inputs with standard deviation equal to 0.5, this implies that the absolute difference in logit probability should be less then 5, when moving from one standard deviation below the mean, to one standard deviation above the mean, in any input variable. A Cauchy prior with scale parameter equal to {10} was proposed for the intercept {\beta_0}. The difference is because if we use a Cauchy with scale {2.5} for {\beta_0} it would mean that {p_t} would probably be between {1\%} and {99\%} for units that are average for all inputs and as a default prior this might be too strong assumption. With scale equal to 10, {p_t} is probably within {10^{-9}} and {1-10^{-9}} in such a case.

There is also a nice (and important) discussion about the pre-processing of input variables in [2] that I will keep for a future post.

Conclusion

I am in favor of the idea behind weakly informative priors. If we have some sensible information about the problem at hand we should find a way to encode it in our models. And Bayesian statistics provides an ideal framework for such a task. In the particular case of the separation problem in logistic regression, it was able to avoid the infinite estimates obtained with MLE and give sensible solutions to a variety of problems just by adding sensible generic information relevant to logistic regression.

References:

[1] Zorn, C. (2005). A solution to separation in binary response models. Political Analysis, 13(2), 157-170.
[2] Gelman, A., Jakulin, A., Pittau, M.G. and Su, Y.S. (2008). A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics, 1360-1383.

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