Bayesian Statistics and Credibility Theory

Visual recap of conditional probability using a grid of blue and pink squares with plus signs, illustrating how conditional and Bayes probabilities relate.

Apply bayesian statistics to a simple emoji texting example to compute the probability that a text with an emoji came from Candice using p(b|a) = p(a|b)p(b)/p(a), yielding 46.51 percent.

Combine prior information with observed data to form the posterior distribution. The posterior is proportional to the prior times the likelihood, narrowing estimates between prior and observed evidence.

Learn the notation for prior and posterior distributions in Bayesian statistics, linking theta, the parameter, and the sample x via the proportional relationship to prior and likelihood.

Explore a Bayesian prior–posterior example using exponential battery lifetimes to determine whether five batteries are good, based on a 50/50 prior and 30-hour test that yields 91.68%.

Conjugate priors yield a posterior in the same family after updating with the likelihood; illustrated by geometric likelihood and beta prior producing a beta posterior with updated alpha and beta.

Explore loss functions: Bayesian versus classical mean square error, minimizing expected posterior loss. Use squared error for the mean, absolute error for the median, and zero for the mode.

Explore credibility theory and its connection to bayesian statistics in setting premiums by weighting the sample mean and population mean with the credibility factor Z.

Explore Bayesian credibility by combining prior collateral data with new sample data to form a posterior using a credibility factor, with gamma-Poisson and normal models.

Explore empirical Bayes credibility theory, comparing Model 1 with equal weights and Model 2 weighted by business volume, and focus on functions of the parameter rather than distributions.

Walk through an exam-style empirical based credibility theory question, computing the credibility factor from four years across three risks. Apply variance of mean function formulas from the Orange Book.