Estimate the value of a sigmoid function over expectation

Question

I would like to estimate the function value of the sigmoid over an expectation, that is: \begin{equation} \sigma(\mathbb{E}_{p(x)}[f(x)]), \end{equation} 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, Apr 28 '22 at 12:31
@whube Let's discuss the case when p(x) is Bernoulli or categorical distribution — jzin, Apr 29 '22 at 03:51

score 1 · Answer 1 · answered May 06 '22 at 10:57

1

No unbiased estimator exists, when $p(x)$ is Categorical distribution:

Unbiased estimator of exponential of measure of a set?

For the binomial distribution, why does no unbiased estimator exist for $1/p$?

Thank @whuber for the given link.

answered May 06 '22 at 10:57

jzin

337

Estimate the value of a sigmoid function over expectation

1 Answers1