One trick is to simulate from your priors and see whether you think they make a priori sense.

Let's assume we have data $x$ in the range -5 to 5. Below I try three sets of priors for $\alpha, \beta$ in the usual logistic model, increasing in informativeness:

• $\alpha \sim N(0, 100), \beta \sim N(0, 10)$
• $\alpha \sim N(0, 10), \beta \sim N(0, 1)$
• $\alpha \sim N(0, 1), \beta \sim N(1, 1)$

The plot below demonstrates the implied models.

The mostly-flat priors are nonsense. Many of the models predict all 0 or all 1 across the range in question, and where they don't do that they are much too steep in my opoinion. The medium informative priors are perhaps what I would go with in this case, having no strong opinions about direction or steepness of this curve. In the last case I consider what it would look like if I expected $\beta$ to be positive.

This type of thing is difficult to do with many predictor, then it would be better to simulate data from the implied models and compare with your own data: That is, simulate $y$ from $$y_i ~ \mathrm{Bernoulli}( \mathrm{logistic}( \alpha + \beta x_i))$$ and compare with your own $y$s, perhaps grouped by something you know about beforehand. If you observe 5 zeroes out of 10 then a set of priors that predicts only 10 ones might be too strange.

set.seed(20200228)

par(mfrow=c(1,3))

logistic <- function(z) 1/(1 + exp(-z))

# really wide priors
a <- rnorm(50, 0, 100)
b <- rnorm(50, 0, 10)


plot(NULL, xlim=c(-5, 5), ylim=0:1, xlab = "x", ylab="p(y | x)",
     main = "Almost flat priors")
for (i in 1:50) {
  curve(logistic(a[i] + b[i]*x), add=T, col="grey", lwd=2)
}

# more informative
a <- rnorm(50, 0, 10)
b <- rnorm(50, 0, 1)
plot(NULL, xlim=c(-5, 5), ylim=0:1, xlab = "x", ylab="p(y | x)",
     main = "Weakly informative priors")
for (i in 1:50) {
  curve(logistic(a[i] + b[i]*x), add=T, col="grey", lwd=2)
}


a <- rnorm(50, 0, 1)
b <- rnorm(50, 1, 1)  # note: expecting a positive b now

plot(NULL, xlim=c(-5, 5), ylim=0:1, xlab = "x", ylab="p(y | x)",
     main = "Informative priors, \n prior assumption b is positive")
for (i in 1:50) {
  curve(logistic(a[i] + b[i]*x), add=T, col="grey", lwd=2)
}

^{Created on 2020-02-28 by the reprex package (v0.2.1)}