Fit a distribution by maximum likelihoodΒΆ

In this example we are going to estimate the parameters of a parametric by generic numerical optimization of the likelihood.

The likelihood of a sample X according to a parametric density function p_{\underline{\theta}} is:

likelihood(\underline{x}_1, \dots, \underline{x}_n,\underline{\theta}) = \prod_{i=1}^n p_{\underline{\theta}}(\underline{x}_i)

from __future__ import print_function
import openturns as ot
import math as m

Create data from a gaussian pdf with mu=4, sigma=1.5

sample = ot.Normal(4.0, 1.5).getSample(200)

Create the search interval of (mu, sigma): the constraint is sigma>0

lowerBound = [-1.0, 1.0e-4]
upperBound = [-1.0, -1.0]
finiteLowerBound = [False, True]
finiteUpperBound = [False, False]
bounds = ot.Interval(lowerBound, upperBound, finiteLowerBound, finiteUpperBound)

Create the starting point of the research For mu : the first point For sigma : a value evaluated from the two first points

mu0 = sample[0][0]
sigma0 = m.sqrt((sample[1][0] - sample[0][0]) * (sample[1][0] - sample[0][0]))
startingPoint = [mu0, sigma0]


Create the estimator from a parametric pdf

pdf = ot.Normal()
factory = ot.MaximumLikelihoodFactory(pdf)

Set the starting point via the solver

solver = factory.getOptimizationAlgorithm()

Estimate the parametric model

distribution =


'Normal(mu = 3.94775, sigma = 1.49821)'

Retrieve the estimated parameters



Total running time of the script: ( 0 minutes 0.004 seconds)

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