Bayesians and Frequentists

As a statistician ,data analyst or scientist, how do you define your probability. That's the ultimate difference between the Bayesians and Frequentists.

Frequentist

Frequentists define probability with the concept of frequencies. To a Frequentist, saying "there's a 30% chance of rain tomorrow" means that in 100 days with identical atmospheric conditions, it would rain on approximately 30 days of them. Probability uses the idea property of repetition on random processes. This way only works correctly for coin flips and dice rolls but becomes strained when dealing with one time events. Frequentist methods rely entirely on the data collected, without using any prior information. Common examples include t-tests, ANOVA, and confidence intervals.

Bayesian

The Bayesian approach treats probability as a degree of belief or confidence in an event. It considers parameters as random variables that can change as new data becomes available. Bayesian methods start with a prior belief about a parameter, then update this belief using observed data to produce a posterior probability. This process, based on Bayes’ theorem, allows continuous learning as new information is introduced.

Real World Example

Imagine a doctor testing the effectiveness of a new drug.
A Frequentist would collect data from a large number of patients and test whether the difference in recovery rates between the drug and placebo is statistically significant (example, p-value < 0.05).
A Bayesian would start with prior knowledge about similar drugs, then update the probability that the drug is effective as more patient data becomes available.

Summary

In summary, the Frequentist approach focuses on objective, long-term patterns that emerge from repeated experiments, while the Bayesian approach centers on updating beliefs as new evidence becomes available. Each method has its own strengths . Frequentist techniques are straightforward and widely used, whereas Bayesian methods are more flexible and allow for continuous learning. Ultimately, the best approach depends on the nature of the data, how much prior information is available, and the specific goals of the analysis.