python - Maximum Likelihood Estimate pseudocode

Question

I need to code a Maximum Likelihood Estimator to estimate the mean and variance of some toy data. I have a vector with 100 samples, created with numpy.random.randn(100). The data should have zero mean and unit variance Gaussian distribution.

I checked Wikipedia and some extra sources, but I am a little bit confused since I don't have a statistics background.

Is there any pseudo code for a maximum likelihood estimator? I get the intuition of MLE but I cannot figure out where to start coding.

Wiki says taking argmax of log-likelihood. What I understand is: I need to calculate log-likelihood by using different parameters and then I'll take the parameters which gave the maximum probability. What I don't get is: where will I find the parameters in the first place? If I randomly try different mean & variance to get a high probability, when should I stop trying?

Answer 1

I just came across this, and I know its old, but I'm hoping that someone else benefits from this. Although the previous comments gave pretty good descriptions of what ML optimization is, no one gave pseudo-code to implement it. Python has a minimizer in Scipy that will do this. Here's pseudo code for a linear regression.

# import the packages
import numpy as np
from scipy.optimize import minimize
import scipy.stats as stats
import time

# Set up your x values
x = np.linspace(0, 100, num=100)

# Set up your observed y values with a known slope (2.4), intercept (5), and sd (4)
yObs = 5 + 2.4*x + np.random.normal(0, 4, 100)

# Define the likelihood function where params is a list of initial parameter estimates
def regressLL(params):
    # Resave the initial parameter guesses
    b0 = params[0]
    b1 = params[1]
    sd = params[2]

    # Calculate the predicted values from the initial parameter guesses
    yPred = b0 + b1*x

    # Calculate the negative log-likelihood as the negative sum of the log of a normal
    # PDF where the observed values are normally distributed around the mean (yPred)
    # with a standard deviation of sd
    logLik = -np.sum( stats.norm.logpdf(yObs, loc=yPred, scale=sd) )

    # Tell the function to return the NLL (this is what will be minimized)
    return(logLik)

# Make a list of initial parameter guesses (b0, b1, sd)    
initParams = [1, 1, 1]

# Run the minimizer
results = minimize(regressLL, initParams, method='nelder-mead')

# Print the results. They should be really close to your actual values
print results.x

This works great for me. Granted, this is just the basics. It doesn't profile or give CIs on the parameter estimates, but its a start. You can also use ML techniques to find estimates for, say, ODEs and other models, as I describe here.

I know this question was old, hopefully you've figured it out since then, but hopefully someone else will benefit.