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Bayesian Estimation vs. Maximum Likelihood Estimation (MLE)

Bayesian Estimation and Maximum Likelihood Estimation (MLE) are two popular methods used to estimate parameters of a probability distribution or model.

Maximum Likelihood Estimation (MLE) aims to find the parameter values that maximize the likelihood of observing the given data. It treats parameters as fixed and unknown, relying only on the data. MLE is simple and widely used but does not incorporate any prior information about the parameters.

Bayesian Estimation, on the other hand, treats parameters as random variables and combines prior knowledge (prior probability) with the observed data (likelihood) to compute a posterior distribution using Bayes' Theorem. This method gives a complete distribution for the parameter, not just a point estimate.

Key Differences:

• MLE uses only observed data, while Bayesian Estimation uses both prior beliefs and data.

• MLE provides point estimates; Bayesian gives probability distributions.

• Bayesian estimation is more robust in case of small data or uncertainty.

Example:
If estimating the probability of heads in a coin toss:

• MLE might say the chance is 0.6 based on 6 heads out of 10.

• Bayesian Estimation might say the chance is likely around 0.6 but shows uncertainty using a distribution.

Conclusion:
Bayesian estimation is more informative, but MLE is computationally simpler and widely used.