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Bayesian Statistics

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Great choice! Bayesian Statistics is a powerful and elegant framework for updating beliefs in the presence of evidence. Here's a structured overview of Bayesian Statistics – perfect for coursework, self-study, or applied work. Let me know if you'd like examples, slides, or code in Python/R!

📘 Bayesian Statistics Overview

Bayesian statistics is a branch of statistics based on Bayes’ Theorem, which describes how to update the probabilities of hypotheses when given new evidence.

🧠 Core Concepts

🔄 Bayes’ Theorem:

P(θ∣D)=P(D∣θ)⋅P(θ)P(D)P(\theta | D) = \frac{P(D | \theta) \cdot P(\theta)}{P(D)}

Where:

  • P(θ∣D)P(\theta | D) = Posterior (updated belief after seeing data)
  • P(D∣θ)P(D | \theta) = Likelihood (probability of data given the parameter)
  • P(θ)P(\theta) = Prior (initial belief before seeing data)
  • P(D)P(D) = Evidence or Marginal Likelihood

🔑 Key Terms

Term Description
Prior Your belief about the parameter before observing data
Likelihood How likely the observed data is, given the parameter
Posterior Updated belief after observing the data
Bayesian Inference Updating beliefs using Bayes’ theorem
Credible Interval Bayesian version of a confidence interval
Conjugate Prior A prior that, when combined with the likelihood, gives a posterior of the same family

🧪 Bayesian vs. Frequentist

Feature Bayesian Frequentist
Interpretation Probability = degree of belief Probability = long-run frequency
Uses Prior? ✅ Yes ❌ No
Output Posterior distribution Point estimate, CI, p-values
Flexibility Very flexible (especially with small data) Rigid but objective

🔢 Common Distributions in Bayesian Stats

  • Beta distribution: Often used as a prior for binomial problems
  • Normal distribution: Common for continuous variables
  • Gamma distribution: Prior for Poisson rate or precision of Normal
  • Dirichlet distribution: Multivariate generalization of Beta

📐 Bayesian Methods

  1. Bayesian Parameter Estimation
    • Estimate parameter distributions instead of point values.
    • E.g., posterior distribution of mean μ\mu
  2. Bayesian Hypothesis Testing
    • Use Bayes Factors instead of p-values.
    • BF10=P(D∣H1)P(D∣H0)BF_{10} = \frac{P(D | H_1)}{P(D | H_0)}
  3. Markov Chain Monte Carlo (MCMC)
    • Used to sample from complex posterior distributions.
    • Algorithms: Metropolis-Hastings, Gibbs Sampling
  4. Hierarchical Bayesian Models
    • Incorporates multiple levels of parameters.
    • Useful in psychology, medicine, and sports analytics.

🛠️ Tools for Bayesian Analysis

🐍 Python

  • PyMC / PyMC3 / PyMC4
  • NumPyro
  • TensorFlow Probability
  • scikit-learn (some Bayesian tools)

🐘 R

  • rstan
  • brms (high-level wrapper for Stan)
  • BayesFactor
  • coda, MCMCpack

🧠 Other

  • Stan (probabilistic programming language)
  • BUGS/JAGS (older Bayesian tools)

📊 Applications of Bayesian Statistics

  • Medical diagnostics (updating disease probabilities)
  • Machine learning (Bayesian Neural Networks)
  • A/B testing (with more nuance than p-values)
  • Forecasting (e.g., Bayesian time series models)
  • Sports analytics, epidemiology, finance

📚 Suggested Resources

  • 📘 Bayesian Data Analysis by Gelman et al.
  • 📘 Doing Bayesian Data Analysis by John Kruschke (great for beginners)
  • 📘 Statistical Rethinking by Richard McElreath (with R/Stan)
  • 🎓 Online: Coursera (Bayesian Methods for Data Science), edX, YouTube (StatQuest, Kruschke’s lectures)

Would you like:

  • A practical example (e.g., coin toss with Bayesian inference)?
  • Code walkthrough (Python or R)?
  • Visuals or handouts?

Let me know how you'd like to explore this further!