Does it make sense to generate prediction intervals for the estimates of a logistic regression? Say I have a binary outcome of 0 or 1 and suppose I were to use logistic regression model to estimate the probability a new sample will have an outcome of 1.
I have read answers (for example here: Computing prediction intervals for logistic regression) that indicate it's nonsensical to compute prediction intervals, as the outcome can only be 0 or 1.
However, is there nevertheless a sensible way to compute a prediction interval in log-odds space, before transforming it using the logistic function into a probability interval?
My goal is to be able to communicate a level of uncertainty in each new prediction I make (e.g,. "This new product is estimated at a 0.40 probability of having the desired outcome, but the prediction interval around this estimate is [0.3, 0.5]").
 A: In the log-odds space, every outcome is either $-\infty$ or $\infty$, so a prediction interval still makes no sense. You can achieve your goal by giving a confidence interval for the probability.
A: That's a little bit confusing. If you are doing logistic regression, you predict binary class memberships, e.g., 0 and 1. The class label is basically the outcome of a thresholding function that is 
$sigmoid(z) \ge 0.5 \rightarrow 1 $, and 0 otherwise.
where $z = w^Tx$ (w=weights, x=sample), and $sigmoid$ is the inverse-logit function $\frac{1}{1 + e^{-z}}$
or equivalently to save this one step of computation you can directly compute:
$z \ge 0 \rightarrow 1 $, and 0 otherwise.
The conditional probability $p = sigmoid(z)$ is basically your "confidence". I think what you want is the confidence of this probability? You could do this by calculating the standard error of your prediction on the linear scale $w^Tx$, and then calculate the upper and lower bounds of your, e.g., 95%, confidence interval via $[ pred. value +/- 1.96\times std. err ]$. After you obtained the upper and lower bounds, you can use the sigmoid function to transform those onto the logit scale.
