Gaussian Log-Likelihood

Medium

~20 min

code completion

A Gaussian (Normal) distribution with mean $μ$ and standard deviation $σ$ has probability density:

p (x) = \frac{1}{σ 2 π} exp (- \frac{( x - μ ) ^{2}}{2 σ ^{2}})

In practice we work with the log-likelihood to avoid numerical underflow:

ln p (x) = - \frac{( x - μ ) ^{2}}{2 σ ^{2}} - ln (σ) - \frac{1}{2} ln (2 π)

For a dataset of independent observations $x_{1}, \dots, x_{n}$ , the total log-likelihood is the sum.

Your task:

Implement gaussian_log_likelihood(x, mu, sigma) that returns the total (summed) log-likelihood over all observations in x.

Example Tests

Single point at the mean: maximum log-likelihood for sigma=1

Input: {"x":[0],"mu":0,"sigma":1}

Expected: -0.91894

Two points equidistant from mean

Input: {"x":[-1,1],"mu":0,"sigma":1}

Expected: -2.83788

Larger sigma reduces log-likelihood at mean

Input: {"x":[0],"mu":0,"sigma":2}

Expected: -1.61209

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