gen-define-bayesian-versus-frequentist-inference-and-give-one-sh-8054

Define Bayesian versus frequentist inference and give one short worked example.

# Bayesian vs Frequentist Inference ## Definitions **Frequentist inference**: Treats parameters as fixed (unknown) constants. Probability refers only to long-run frequencies of repeatable events. Data is random; parameters are not. **Bayesian inference**: Treats parameters as random variables with distributions. Probability represents degree of belief. Updates prior beliefs about parameters using data via Bayes' theorem. ## Worked Example: Estimating Coin Bias **Setup**: You flip a coin 10 times and get 7 heads. What is the probability of heads (θ)? ### Frequentist Approach - **Point estimate**: θ̂ = 7/10 = 0.7 (maximum likelihood) - **95% confidence interval**: Using normal approximation: - SE = √[0.7(0.3)/10] = 0.145 - CI = 0.7 ± 1.96(0.145) = [0.42, 0.98] - **Interpretation**: If we repeated this experiment many times, 95% of such intervals would contain the true θ (which is fixed but unknown) ### Bayesian Approach - **Prior**: Assume θ ~ Beta(2, 2) (slightly favor fair coins) - **Likelihood**: 7 heads in 10 flips - **Posterior**: θ | data ~ Beta(2+7, 2+3) = Beta(9, 5) - Mean = 9/14 = 0.64 - 95% credible interval = [0.38, 0.86] - **Interpretation**: Given the data, there's a 95% probability that θ lies in [0.38, 0.86] **Key difference**: Bayesian interval makes direct probability statements about the parameter; frequentist interval does not.