# Estimate the value of a sigmoid function over expectation

I would like to estimate the function value of the sigmoid over an expectation, that is: $$$$\sigma(\mathbb{E}_{p(x)}[f(x)]),$$$$ where $$\sigma(x) = \frac{1}{1 + e^{-x}}$$, and $$p(x)$$ is the one we only access its samples but cannot evaluate its density.

To estimate $$\sigma(\mathbb{E}_{p(x)}[f(x)])$$, we could use $$\sigma(\frac{1}{n}\sum_{i=1}^n f(x_i))$$, but it is biased. My question is, how could we define an unbiased estimator.

• You need to postulate a specific family of distributions governing the random samples $x_i.$ What would that be?
– whuber
Commented Apr 28, 2022 at 12:31
• @whube Let's discuss the case when p(x) is Bernoulli or categorical distribution
– jzin
Commented Apr 29, 2022 at 3:51
• The solution to that is well known and instructive.
– whuber
Commented Apr 29, 2022 at 11:39

No unbiased estimator exists, when $$p(x)$$ is Categorical distribution:
For the binomial distribution, why does no unbiased estimator exist for $1/p$?