Maximum likelihood and Bayesian estimation

Understanding Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method used in statistics to estimate the parameters of a statistical model.
This technique seeks to find the parameter values that maximize the likelihood of making the observations given the model.

Think of MLE as playing a guessing game, where you try to figure out the settings that make the most sense based on what you observe.

To understand how MLE works, imagine you have a bag with different colored marbles, and you’re trying to figure out the proportion of each color based on a random handful you picked.

You would want to determine the percentages that would most likely result in the handful you got, and this is exactly what MLE does.

Why Use Maximum Likelihood Estimation?

MLE is widely used for several reasons.
Firstly, it provides a consistent method for parameter estimation, meaning that with enough data, MLE findings tend to be close to the true parameter values.

Secondly, MLE is quite flexible and can be applied to a wide variety of models, whether they reference normal distributions, binomial, or other types.

Additionally, MLE often has desirable mathematical properties such as efficiency (minimizing variance) and sufficiency (using all the data provides good estimates).

All these factors make it a preferred method for statisticians and data scientists when working with statistical models.

How Maximum Likelihood Works with An Example

To illustrate MLE, let’s consider flipping a coin.
Suppose we want to know if the coin is fair, meaning it has an equal probability of landing on heads or tails.

We represent heads with parameter `p` and tails with `1-p`.
Now we flip the coin several times and record the results.

The goal of MLE is to find the value of `p` that best explains our results.
This is done by calculating the likelihood, which is the probability of observing the data, given a specific parameter value.

If, after several flips, you recorded mostly heads, the MLE would lean towards a value of `p` closer to 1, suggesting it’s less likely that the coin is fair.

Exploring Bayesian Estimation

Bayesian estimation is another approach to parameter estimation, but it’s quite different from MLE.
It incorporates both the data and prior beliefs or knowledge about the parameters into the estimation process.

In Bayesian estimation, we update our beliefs with new data using probability distributions, which is based on Bayes’ Theorem.

The result is a posterior distribution representing the updated belief after considering the new evidence.

Understanding Bayes’ Theorem

Bayes’ Theorem is at the core of Bayesian estimation.
It states:

\[ P(\text{Parameter}|\text{Data}) = \frac{P(\text{Data}|\text{Parameter}) \times P(\text{Parameter})}{P(\text{Data})} \]

In simpler terms, it tells us how to update the probability of a hypothesis, given new evidence.

The left side of the equation, the posterior `P(Parameter|Data)`, is the updated belief about the parameter after seeing the data.

The right side involves multiplying the likelihood `P(Data|Parameter)` with the prior `P(Parameter)`, which represents initial beliefs before observing the data.
Lastly, `P(Data)` is the marginal likelihood, ensuring probabilities sum up to one.

Advantages of Bayesian Estimation

Bayesian estimation has several appealing attributes.
It allows for the incorporation of prior knowledge, which is beneficial in situations where you already have some insights based on past experience or research.

Furthermore, it provides a complete probability distribution of the parameter, offering more information than a single point estimate like MLE does.

Decision-makers find value as these probabilities offer insight into the uncertainty and range of plausible values for the parameters.

Additionally, Bayesian methods can be applied iteratively, updating with new data over time, which is practical for dynamic systems.

Example of Bayesian Estimation in Action

Let’s revisit the coin flip scenario.
Suppose you believe the coin might be slightly biased towards heads, with a prior estimate `p = 0.6`.

By flipping the coin several times and observing the outcomes, Bayesian estimation updates this belief into a posterior distribution.

This new probability distribution considers both the initial assumption and the collected data, resulting in a range of likely values for the parameter `p`.

Comparing Maximum Likelihood and Bayesian Estimation

Both MLE and Bayesian estimation are valuable methods, but they have distinct approaches and are used in different contexts.

MLE focuses purely on the data at hand, aiming for parameter estimates that maximize the likelihood of that data.

Meanwhile, Bayesian estimation combines data with prior beliefs to update what we know about the parameters.

In practice, MLE often requires less computational resources and can be simpler to implement, making it suitable for models with large datasets or limited prior information.

Bayesian estimation, however, provides a richer understanding when prior knowledge is available, or when one needs to account for uncertainty.

Different scenarios and data science applications might call for one method over the other.
Yet, both methods highlight the importance of using statistics to derive meaningful conclusions from data.

Whether you lean toward maximum likelihood or Bayesian estimation, understanding these methods will enhance your ability to solve statistical problems and improve data-driven decision-making.