Clipping Bias Estimate

Problem

Compute the expected value of a Gaussian gradient after clipping to [-clip, clip].

For g ~ N(mu, sigma ** 2), return mu * (Phi(z1) - Phi(z2)) + sigma * (phi(z2) - phi(z1)) + clip * (1 - Phi(z1) - Phi(z2)) with z1 = (clip - mu) / sigma, z2 = (-clip - mu) / sigma, and Phi, phi the standard normal CDF and PDF. For sigma <= 0, clip mu directly.

Hint

Split the expectation into the truncated body and the two clipped tails.

Test cases

case 1

in: [0,1,2]

expected: 0

case 2

in: [1,1,2]

expected: 0.9170666837293614

case 3

in: [3,1,2]

expected: 1.916684582873969

case 4

in: [0.5,2,1]

expected: 0.1896444392971461

Starter codePython

def clipping_bias_estimate(mu, sigma, clip):
    # Your code here
    pass

4 test cases · run your code against them in the editor