Maximum Likelihood Estimation (MLE) is a statistical method used for parameter estimation to find the distribution that best describes observed data. For example, if observed data closely follows the shape of a Gaussian bell curve, one might ask: What is the best-fitting Gaussian distribution for this data? The MLE method enables us to find the mean and variance of the Gaussian distribution that most accurately represents the data. In other words, what is the center (mean) and width(variance) of this distribution.
In this article, we will explore this concept through the following visual steps:
- Generate synthetic Gaussian histogram data.
- Plot the Maximum Likelihood function for the mean.
- Identify the Maximum Likelihood Estimate (MLE) for the mean from the plot.
- Plot the Maximum Likelihood function for the variance.
- Identify the MLE for the variance from the plot.
- Finally, construct the Gaussian distribution using the two estimated parameters: mean and variance.
We’ll plot this function for a range of μ values, keeping σ² fixed (using some initial guess ).
The vertical red dashed line represents the Maximum Likelihood Estimate (MLE) for the mean, which is approximately μ=4.91