offset in logistic regression to take into account a priori individual probability estimates I am performing a logistic regression (LR) model on a set of data on which an initial probability estimate for event occurrence, sai $\hat{p_i}^{base}$ has been provided. I have supplementary regressors and I want to update my model starting from the initial estimate. I was thinking to insert $logit\left(\hat{p_i}^{base}\right)$ as offset of the (LR) model, thus $logit\left(\hat{p_i}^{new}\right)=1*logit\left(\hat{p_i}^{base}\right)+x_i^T*\beta$. I am asking whether the offset form (logit) is the right functional form to allow for an initial "guessing" of the probability and if this is the most appropriate approach to update a logistic regression model, given initial updates. Thanks in advance for the attention.
 A: If you wish to conduct Bayesian logistic regression with prior probabilities and updating, you must do Bayesian logistic regression with prior probabilities and updating. 
A note about offsets: their general rationale is to re-express an outcome which is a ratio or difference of two values, one varying and one "fixed", by subtracting or dividing the "fixed" value from the LHS to incorporate it on the RHS with a fixed coefficient. This usually requires GLMs with collapsible links, like linear or Poisson regression. A difference in logit probabilities doesn't have any tangible meaning, we usually think of risk ratios or risk differences as meaningful comparative measures of two probabilities.
But more importantly, the reason why I believe it wouldn't work is that there is unaccounted correlation between your new (candidate) predictors and the old risk-prediction. For instance, suppose I predict cancer by age and obtain an cancer odds ratio of, say, 1.20 for a 5-year difference in age and linearity holds appropriately. I then obtain risk predictions for cancer. In another cohort, I measure smoking status in individuals. Suppose further there is a generational effect of smoking so that older individuals tend to smoke more, and thus partly explain the age-association with cancer. In a second model jointly adjusting for age and smoking status, the odds ratio for age is now reduced. In fact, because logistic regression is non-collapsible, coefficients will change upon further inclusion of uncorrelated predictors.
Hence, by including new covariates you cannot be guaranteed that the prior logit will have a coefficient of 1 against the posterior logit.
