Question 1

What is the difference between prior and posterior probability?

Accepted Answer

Prior probability P(H) is your belief about the hypothesis before seeing evidence — based on base rates, historical data, or expert judgment. Posterior probability P(H|E) is your updated belief after observing evidence E. Bayesian inference is the formal process of updating priors with evidence to obtain posteriors.

Question 2

What is the base rate fallacy?

Accepted Answer

The base rate fallacy occurs when people ignore the prior probability (base rate) and focus only on specific evidence. Example: if a disease affects 0.1% of people and a test has 99% accuracy, a positive test still corresponds to roughly 91% probability of not having the disease, because the condition is so rare. Bayes' theorem prevents this error by formally incorporating the prior.

Question 3

What is a likelihood ratio?

Accepted Answer

A likelihood ratio (LR) = P(evidence|hypothesis true) / P(evidence|hypothesis false). LR > 1 means evidence supports the hypothesis; LR < 1 means evidence undermines it. For the medical test example: LR = sensitivity / (1 − specificity) = 0.90 / 0.05 = 18. A positive test makes disease 18 times more likely, but the absolute probability depends on the prior.

Question 4

What is a conjugate prior?

Accepted Answer

A conjugate prior is a prior distribution that, when combined with a particular likelihood function, produces a posterior of the same distributional family. For example, a Beta prior with a Binomial likelihood produces a Beta posterior. Conjugate priors simplify Bayesian calculations and allow analytical (closed-form) solutions.

Question 5

How is Bayesian probability different from frequentist probability?

Accepted Answer

Frequentists define probability as long-run relative frequency — only repeatable events have probabilities. Bayesians define probability as a degree of belief — any uncertain proposition can have a probability, even one-time events. In practice: frequentists compute p-values and confidence intervals; Bayesians compute posterior distributions and credible intervals. Both frameworks are valid and used in research.

Question 6

Can I chain multiple Bayesian updates?

Accepted Answer

Yes — this is one of Bayesian inference's most powerful features. The posterior from one update becomes the prior for the next. If you observe three independent pieces of evidence E₁, E₂, E₃, you update: P(H) → P(H|E₁) → P(H|E₁,E₂) → P(H|E₁,E₂,E₃). Each new piece of evidence refines your estimate.

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