Bayesian Analysis #biostatistics #shorts

Core Idea We start with a prior distribution that encodes what we believe (or assume) about a parameter $\theta$ before seeing data. After observing data $x$, we update to a posterior distribution via Bayes’ theorem: $$p(\theta \mid x) = \frac{p(x \mid \theta)\, p(\theta)}{p(x)}$$ $p(\theta)$: prior $p(x \mid \theta) $: likelihood $p(x) $: marginal likelihood (evidence) $p(\theta \mid x) $: posterior The posterior becomes the new prior if more data arrive (sequential updating). Predictions for new observations are obtained by integrating over the posterior (posterior predictive distribution). #BayesianAnalysis #statistics #datascience #machinelearning #statisticalmodeling #BayesianInference #ProbabilisticModeling #dataanalysis #researchmethods #quantitativeresearch