Hybrid estimation for logistic model

Question

Say I have a logistic model, $$ Y_i \sim {\rm Bernoulli}({\rm logistic}(\beta_0 + \beta_1 X_i)), $$ where $logistic(x) = e^x/(1+e^x)$ as usual.

I have

1. $n$ observations $Y_i \in \{0,1\}$ ($i = 1, \ldots, n$), and also
2. $m$ independent guestimates $\hat\pi_i$ ($i = n+1,\ldots,m+n$) of the probabilities $\pi_i = {\rm logistic}(\beta_0 + \beta_1 X_i)$. Let's be generous and say $\hat\pi_i \sim N(\pi_i, \sigma^2)$, where $\sigma^2$ is small (and known, for the sake of argument).

I want to use all this information to get better estimates $\hat\beta_0$ and $\hat\beta_1$ for the model parameters.

I can think of a couple of approaches but this is surely a standard problem. If yes, what is it called? Any useful references?

Thanks!

Edit: so the sample data might look like:

X = -1: observations Y= 0,0,1

X = -2: observations Y= 0,1, $\hat\pi = 0.3$

X = 3: observations $\hat\pi = 0.5$, $\hat\pi =0.53$

Answer 1

You have a prior for the predicted probabilities. Including that information would be a form of Bayesian statistics. In most discussion of Bayesian statistics the prior is on the coefficients instead of the predictions, but I am sure you can either transform your priors to be priors on the coefficients or include them directly. I vaguely remember a discussion of priors on predictions rather than coefficients in Andrew Gelman, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Donald Rubin (2013) Bayesian Data Analysis. CRC press. However, I don't have the book here, so I cannot look it up.