Difference in logits to difference in probabilities

Question

a) Is there any way to prove that you cannot compute a probability difference when you know a logit difference?

b)In general, is there some acceptable way to convert distances in logodds space to distances in probability space? I have a bound for a difference of logits ($logit(p_1)-logit(p_2))$ when my problem calls for a bound in $p_1-p_2$, and I find no straightforward way to bound this difference.

Answer 1

Here are two pairs of probabilities with $\operatorname{logit} p_1 - \operatorname{logit} p_2$ equal but $p_1 - p_2$ not equal:

• $p_1 = \tfrac{3}{4}$, $p_2 = \tfrac{1}{2}$
  • $p_1 - p_2 = \tfrac{1}{4}$
  • $\operatorname{logit} p_1 - \operatorname{logit} p_2 = \ln 3$
• $p_1 = \tfrac{9}{10}$, $p_2 = \tfrac{3}{4}$
  • $p_1 - p_2 = \tfrac{3}{20}$
  • $\operatorname{logit} p_1 - \operatorname{logit} p_2 = \ln 3$