01
Prior
What uncertainty exists before the new data?
A prior distribution represents model uncertainty about an unknown quantity before incorporating the current evidence.
Prior · likelihood · posterior · prediction
Bayesian Statistics
Bayesian statistics represents uncertainty with probability distributions and updates that uncertainty when data arrive. The result is conditional on a prior model, a likelihood for the data, and the observed evidence, making assumptions part of the calculation rather than invisible scenery.
Bayesian update cycle
Evidence does not replace uncertainty. It reshapes a distribution over possibilities.
The familiar one-line Bayes formula is the discrete shadow of a broader workflow: specify uncertainty, specify how data would arise, update, then propagate the posterior into prediction or decision-making.
02
Likelihood
How compatible is the observed data with each parameter value?
The likelihood comes from a data model: it describes how the observed evidence would behave under different possible parameter values.
03
Posterior
How should uncertainty change after observing the data?
Bayes’ rule combines prior and likelihood, then normalizes the result into a posterior probability distribution.
04
Predict / update
What does the posterior imply for future observations or decisions?
Posterior predictive distributions propagate parameter uncertainty into predictions and can become the prior for a later update.
Evidence-update lab
Imagine 1,000 cases. Set the prior rate and how strongly a positive observation favors the hypothesis.
posterior = true positives / all positives
Prior P(H)10%
How common is H before this evidence?
P(+ | H)90%
True-positive rate / sensitivity.
P(+ | not H)5%
False-positive rate.
These sliders define a simple two-hypothesis model. The posterior is conditional on this model and these rates being appropriate for the situation.
H is true100 cases
true +: 90false −: 10.0
H is false900 cases
false +: 45true −: 855
Condition on a positive result
True +
90False +
45P(H | +)
66.7%P(H|+) = 90 / (90 + 45)
Credible statements
A posterior distribution supports direct probability statements about model parameters.
For example, a 95% posterior credible interval can be constructed so that 95% of the posterior probability for a parameter lies inside the interval. That interpretation differs from the repeated-sampling coverage definition of a frequentist 95% confidence interval.
Compare frequentist intervalsBayes’ rule
p(θ | y) ∝ p(y | θ) p(θ)
Posterior ∝ likelihood × prior. The missing proportionality constant is the marginal probability of the observed data, which normalizes the posterior so it integrates or sums to 1.
Model discipline
A posterior can be mathematically exact and still depend on a poor model.
01The prior is part of the model
Priors can encode previous information, structural constraints, regularization, or deliberately weak information. Their influence should be examined rather than hidden.
02The likelihood carries the sampling model
Bayesian updating does not bypass model assumptions. A misspecified likelihood can produce a precise posterior about the wrong model.
03Posterior probability is conditional
Statements such as P(θ > 0 | data) are probabilities conditional on the prior, likelihood, observed data, and other model choices.
04Check sensitivity and fit
Posterior predictive checks, alternative priors, alternative likelihoods, and model comparison help reveal whether conclusions depend strongly on fragile assumptions.