2.-Statistics/bayesian-vs-frequentist.md

+## 2.1.18 Bayesian vs Frequentist Methods
+
+**Bayesian** and **Frequentist** approaches represent two different schools of thought in statistical inference. They differ primarily in how they interpret probability and incorporate prior information into their models. Both approaches have their own strengths, depending on the problem at hand.
+
+### Frequentist Methods
+
+The **frequentist approach** defines probability as the long-run frequency of an event occurring in repeated experiments. In this framework, model parameters are considered fixed but unknown, and inference is made purely based on the data observed.
+
+#### Key Characteristics:
+- **Fixed Parameters**: Parameters are fixed and unknown. The aim is to estimate them using the observed data.
+- **No Priors**: Frequentist methods do not incorporate prior beliefs or knowledge about the parameters.
+- **P-Values and Confidence Intervals**: Statistical inference is based on p-values and confidence intervals, derived from the sampling distribution of the estimator.
+
+#### Example: Maximum Likelihood Estimation (MLE)
+
+In the frequentist approach, parameters are typically estimated using **maximum likelihood estimation (MLE)**, which finds the parameter values that maximize the likelihood of the observed data:
+
+$$
+\hat{\theta}_{\text{MLE}} = \arg \max_\theta L(\theta | X)
+$$
+
+Where:
+- $L(\theta | X)$ is the likelihood of the data $X$ given the parameter $\theta$.
+
+### Bayesian Methods
+
+The **Bayesian approach** defines probability as a degree of belief in an event, based on prior knowledge and the observed data. Bayesian inference updates prior beliefs about the parameters in light of new data using **Bayes' theorem**:
+
+$$
+P(\theta | X) = \frac{P(X | \theta) P(\theta)}{P(X)}
+$$
+
+Where:
+- $P(\theta | X)$ is the posterior probability of the parameter $\theta$ given the data $X$,
+- $P(X | \theta)$ is the likelihood of the data given the parameter,
+- $P(\theta)$ is the prior probability of the parameter, and
+- $P(X)$ is the marginal likelihood.
+
+#### Key Characteristics:
+- **Parameters as Random Variables**: In the Bayesian approach, parameters are treated as random variables with a probability distribution.
+- **Use of Priors**: Bayesian methods incorporate prior knowledge or beliefs about the parameters, which are updated as new data becomes available.
+- **Posterior Distributions**: The result of Bayesian inference is a posterior distribution, which combines the prior with the likelihood of the observed data.
+
+### Example: Bayesian Estimation
+
+In Bayesian estimation, we update our prior beliefs about the parameter $\theta$ using the data $X$:
+
+$$
+P(\theta | X) = \frac{L(X | \theta) \cdot P(\theta)}{P(X)}
+$$
+
+Where:
+- $L(X | \theta)$ is the likelihood of the data given the parameter $\theta$,
+- $P(\theta)$ is the prior belief about $\theta$, and
+- $P(\theta | X)$ is the posterior distribution of $\theta$ after observing the data.
+
+### Differences Between Bayesian and Frequentist Approaches
+
+$$
+\begin{array}{|l|c|c|}
+\hline
+\textbf{Aspect} & \textbf{Frequentist Approach} & \textbf{Bayesian Approach} \\
+\hline
+\text{Interpretation of Probability} & \text{Long-run frequency of events} & \text{Degree of belief or certainty} \\
+\hline
+\text{Parameters} & \text{Fixed and unknown} & \text{Random variables with distributions} \\
+\hline
+\text{Use of Prior Information} & \text{No use of prior information} & \text{Incorporates prior beliefs} \\
+\hline
+\text{Inference} & \text{P-values, confidence intervals} & \text{Posterior distributions, credible intervals} \\
+\hline
+\text{Objective} & \text{Find point estimates (e.g., MLE)} & \text{Find posterior distribution of parameters} \\
+\hline
+\end{array}
+$$
+
+### Common Use Cases
+
+- **Frequentist Methods**:
+  - Hypothesis testing with large datasets, where prior information is either unavailable or not necessary.
+  - When interpreting results using p-values and confidence intervals.
+  - Used in many traditional statistical models like linear regression, ANOVA, and t-tests.
+
+- **Bayesian Methods**:
+  - When you have prior knowledge or strong assumptions about the parameter(s) of interest.
+  - In situations where data is limited, and incorporating prior information can improve model performance.
+  - Used in hierarchical models, decision-making, and when probabilistic interpretation is desired.
+
+### Common Issues
+
+- **Frequentist Issues**:
+  - **Over-reliance on p-values**: Frequentist inference heavily depends on p-values, which can be misleading, especially with small or unbalanced datasets.
+  - **No Prior Information**: The frequentist approach doesn’t leverage any prior knowledge, even if it’s available.
+
+- **Bayesian Issues**:
+  - **Subjectivity of Priors**: The choice of priors can significantly influence the results. Poor or uninformed priors can lead to misleading conclusions.
+  - **Computational Intensity**: Bayesian methods, especially for complex models, can be computationally expensive due to the need for techniques like MCMC (Markov Chain Monte Carlo) to approximate the posterior distribution.
+
+### Best Practices
+
+- **Use Frequentist Methods**:
+  - When you have large datasets with minimal prior knowledge, or when traditional hypothesis testing is required.
+  - For well-established models where assumptions about the data are clear, and frequentist methods can provide efficient estimates.
+
+- **Use Bayesian Methods**:
+  - When you have prior information that can inform the model or when working with small datasets.
+  - For complex models, hierarchical structures, or when you need probabilistic interpretations of parameters.
+