BAYESIAN AND FREQUENTISTS

Bayesian and frequentist are two different approaches to statistical inference, differing primarily in how they define and use probability to interpret uncertainty. The frequentist approach considers probability as the long-run frequency of an event and views population parameters as fixed but unknown. In contrast, the Bayesian approach treats probability as a degree of belief and considers parameters to be random variables that can be updated with new evidence using prior beliefs and observed data.

Frequentist approach

Probability:

• Views probability as the long-run frequency of an event if an experiment were repeated many times. Parameters:
• Treats parameters of a model as fixed, but unknown, values. Key output:
• Focuses on estimating parameters based on the observed data, often using methods like maximum likelihood estimation. It provides a single best estimate for the parameter. Uncertainty:
• Quantifies uncertainty through confidence intervals, which describe the range that would contain the true parameter in a high percentage of repeated experiments. Example:
• When testing a coin, the frequentist approach would ask, "What is the probability of getting this result, given a fair coin?" The probability is a property of the data, not the hypothesis itself.

Bayesian approach

Probability:

• Views probability as a degree of belief or certainty about an unknown event or parameter. Parameters:
• Treats parameters as random variables with their own probability distributions. Key output:
• Updates the probability distribution of a parameter based on new evidence, combining prior beliefs with observed data through Bayes' theorem. Uncertainty:
• Quantifies uncertainty through a posterior distribution, which is a probability distribution of the parameter after considering the data. Example:
• When testing a coin, the Bayesian approach would ask, "What is the probability that the coin is biased, given the results of my experiment?" It starts with a prior belief about the coin and updates it with each flip.