Bayesian statistics is defined as a branch of statistics in which hypotheses are tested or model parameters are estimated using methods based on which theorem?

Bayesian statistics is defined as a branch of statistics in which hypotheses are tested or model parameters are estimated using methods based on which theorem?

• A. Fisher's theorem
• B. Bayes' theorem
• C. Central limit theorem
• D. Law of large numbers

Answer: (B)

Bayesian inference rests on Bayes' theorem to update beliefs about hypotheses or model parameters as data are observed. This theorem links what we believed before seeing the data (the prior) with what the data tell us (the likelihood) to produce a revised belief (the posterior). In practice, you start with a prior distribution for the parameter or hypothesis, multiply by the likelihood of the observed data, and normalize to get the posterior distribution. Inference then comes from the posterior—summary measures like the posterior mean, credible intervals, or probabilities assigned to hypotheses. For example, if you’re unsure about a coin’s bias, you can start with a prior distribution for p (the probability of heads). After flipping the coin, you update to a posterior distribution that reflects both your prior and the observed outcomes. This posterior directly supports probability statements about p and decisions based on those probabilities. The other options don’t provide the updating rule that Bayesian methods rely on. They are associated with different foundational ideas (such as how sample means behave or convergence properties in frequentist contexts) and do not form the core mechanism for updating beliefs that Bayesian statistics use.

Bayesian inference rests on Bayes' theorem to update beliefs about hypotheses or model parameters as data are observed. This theorem links what we believed before seeing the data (the prior) with what the data tell us (the likelihood) to produce a revised belief (the posterior). In practice, you start with a prior distribution for the parameter or hypothesis, multiply by the likelihood of the observed data, and normalize to get the posterior distribution. Inference then comes from the posterior—summary measures like the posterior mean, credible intervals, or probabilities assigned to hypotheses.

For example, if you’re unsure about a coin’s bias, you can start with a prior distribution for p (the probability of heads). After flipping the coin, you update to a posterior distribution that reflects both your prior and the observed outcomes. This posterior directly supports probability statements about p and decisions based on those probabilities.

The other options don’t provide the updating rule that Bayesian methods rely on. They are associated with different foundational ideas (such as how sample means behave or convergence properties in frequentist contexts) and do not form the core mechanism for updating beliefs that Bayesian statistics use.