Learn MAP vs MLE Estimation | Bayesian Estimation Methods

Maximum Likelihood Estimation (MLE) and Maximum A Posteriori (MAP) estimation are two fundamental approaches for estimating model parameters in probabilistic modeling. Both methods aim to find the "best" parameter values given observed data, but they differ in how they treat prior information.

The MLE method seeks the parameter value that maximizes the likelihood function, which is the probability of observing the data given the parameter. Mathematically, if you have data $D$ and parameter $θ$ , the MLE is defined as:

\hat{\theta}_{MLE} = \underset{\theta}{\arg\max} \; p(D \mid \theta)

This means you choose the parameter $θ$ that makes the observed data most probable under the model, without taking any prior beliefs about $θ$ into account.

In contrast, MAP estimation combines the likelihood with a prior distribution over the parameter. It finds the parameter value that maximizes the posterior probability, which is proportional to the product of the likelihood and the prior. The MAP estimate is defined as:

\hat{\theta}_{MAP} = \underset{\theta}{\arg\max} \; p(\theta \mid D) = \underset{\theta}{\arg\max} \; \left[ p(D \mid \theta) \cdot p(\theta) \right]

Here, $p(θ)$ is the prior distribution, reflecting your beliefs about $θ$ before observing the data. The MAP estimator thus incorporates both the observed data and any prior knowledge or assumptions about the parameter.

The key difference is that MAP estimation uses prior information, while MLE does not. This distinction can lead to different parameter estimates, especially when the data is limited or the prior is informative.

Assumptions

MLE: assumes no prior knowledge about the parameter; relies solely on the observed data;
MAP: assumes a prior distribution over the parameter, reflecting beliefs or domain knowledge before seeing the data.

Results

MLE: yields the parameter value that makes the observed data most likely;
MAP: yields the parameter value that is most probable given both the data and the prior.

Use Cases

MLE: preferred when you have a large amount of data or no reliable prior information;
MAP: preferred when prior information is available or data is scarce.

1. Which of the following best describes the conceptual difference between Maximum Likelihood Estimation (MLE) and Maximum A Posteriori (MAP) estimation?

2. Under what condition do MAP and MLE yield the same parameter estimate?

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Section 2. Chapter 3

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\hat{\theta}_{MLE} = \underset{\theta}{\arg\max} \; p(D \mid \theta)