Agenda

<aside> 📌 Reminders

• Quiz today! </aside>

Maximum Likelihood Estimate (MLE)

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Maximum Likelihood Estimate (MLE): Used to estimate the parameters of a probability distribution by maximizing a likelihood function. It seeks to find the parameter values that make the observed data most probable.

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Likelihood function: The probability of observing the given data under a specific parameter. For a set of data points $x_1, x_2, ..., x_n$, and some parameter $\theta$, the likelihood function is given by: $L(\theta) = P(x_1, x_2, ..., x_n | \theta)$

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Steps to Compute MLE:

1. Define the Likelihood Function: Write down the likelihood function based on the distribution of the data and the parameter(s) to be estimated.
2. Take the Log of the Likelihood: This simplifies the calculation since multiplying probabilities becomes summing log-probabilities.
3. Maximize the Log-Likelihood: Find the value of $\theta$ that maximizes the log-likelihood. This can often be done by taking the derivative of the log-likelihood with respect to $\theta$ (Score function) and setting it to zero (solving the equation).
4. Solve for the Parameter: The solution gives the maximum likelihood estimate for the parameter.

A useful theorem:

Practice with MLE