Bayesian Statistics Blog2025-01-15

Introduction to Bayesian Statistics

Learn the fundamentals of Bayesian inference and how it differs from frequentist approaches.

Bayesian statistics provides a powerful framework for reasoning under uncertainty. Unlike frequentist statistics, which treats parameters as fixed but unknown values, Bayesian statistics treats parameters as random variables with probability distributions.

Key Concepts

Prior Probability

The prior probability represents our belief about a parameter before seeing the data. This can be based on:

  • Previous studies or expert knowledge
  • Theoretical considerations
  • Weak priors when we have little prior information

Likelihood

The likelihood function describes how probable the observed data is for different parameter values. This is the same concept used in frequentist statistics.

Posterior Probability

The posterior probability combines the prior and likelihood using Bayes' theorem:

P(θ|data) = P(data|θ) × P(θ) / P(data)

Where:

  • P(θ|data) is the posterior probability
  • P(data|θ) is the likelihood
  • P(θ) is the prior probability
  • P(data) is the marginal likelihood

Why Use Bayesian Methods?

Bayesian statistics offers several advantages:

  1. Incorporates prior information - Use existing knowledge to improve estimates
  2. Intuitive interpretation - Direct probability statements about parameters
  3. Uncertainty quantification - Full posterior distributions, not just point estimates
  4. Sequential learning - Today's posterior becomes tomorrow's prior

Example: Estimating a Proportion

Suppose we want to estimate the success rate of a new treatment. We run a trial with 10 patients and observe 7 successes.

Frequentist Approach

The maximum likelihood estimate would be 7/10 = 0.7, with a 95% confidence interval.

Bayesian Approach

We start with a prior (e.g., Beta(1,1) = uniform), update it with the data to get a posterior Beta(8,4), and can make direct probability statements like "There's a 95% probability the true success rate is between X and Y."

Getting Started

To begin your Bayesian journey, we recommend:

  1. Learning probability theory fundamentals
  2. Understanding Bayes' theorem deeply
  3. Practicing with simple conjugate models
  4. Exploring computational methods (MCMC, variational inference)

Stay tuned for more in-depth tutorials on Bayesian methods!