quick script to show likelihood stuff

mle.py

import numpy as np
import random
import plotly.graph_objects as go
from plotly.subplots import make_subplots


class GaussianDistribution:

    def __init__(self, mean, sigma):
        self.u = mean
        self.o = sigma

    def sample(self):
        xp = np.random.normal(self.u, self.o)
        return GaussianDistribution(xp, self.o)

    def likelihood(self, x):
        s = 1 / (self.o * np.sqrt(2 * np.pi))
        e = np.exp(-(1/2) * ((x - self.u) / self.o) ** 2)
        return s * e

    def negative_log_likelihood(self, x):
        # we could just do this
        # return -np.log(self.likelihood(x))

        # but let's write out the whole thing instead
        log_s = np.log(1 / (self.o * np.sqrt(2 * np.pi)))
        log_e = -(1/2) * ((x - self.u) / self.o) ** 2
        return -(log_s + log_e)


def likelihood(measurements, x):
    l = 1.0
    for m in measurements:
        l *= m.likelihood(x)
    return l


def negative_log_likelihood(measurements, x):
    nll = 0
    for m in measurements:
        nll += m.negative_log_likelihood(x)
    return nll


def main():
    true_mean = 16.573  # cm
    sigma = 1  # cm
    true_distribution = GaussianDistribution(true_mean, sigma)

    # let's simulate N measurements (we choose N)
    num_measurements = 10
    measurements = []
    for i in range(num_measurements):
        measurements.append(true_distribution.sample())

    # let's plot the likelihood and log-likelihood for a number of x values
    xs = np.linspace(0, 20, num=100)
    fig = make_subplots(specs=[[{"secondary_y": True}]])

    # Add Likelihood Plot
    ls = [likelihood(measurements, x) for x in xs]
    fig.add_trace(go.Scatter(x=xs, y=ls, name='likelihood'))

    # Add Negative Log Likelihood plot
    nlls = [negative_log_likelihood(measurements, x) for x in xs]
    fig.add_trace(go.Scatter(x=xs, y=nlls, name='neg-log likelihood'), secondary_y=True)

    # Let's visualize the predicted MLE between both plots
    mle = max((l, x) for x, l in zip(xs, ls))
    fig.add_vline(x=mle[1], annotation_text="MLE")

    fig.show()


if __name__ == "__main__":
    main()