Key Points

Short introduction to Bayesian statistics

Likelihood determines the probability of data conditional on the model parameters.
Prior encodes beliefs about the model parameters without considering data.
Posterior quantifies the probability of parameter values conditional on the data.
The posterior is a compromise between the data and prior. The less data available, the greater the impact of the prior.
The grid approximation is a method for inferring the (approximate) posterior distribution.
Posterior information can be summarized with point estimates and posterior intervals.
The marginal posterior is accessed by integrating over nuisance parameters.
Usually, Bayesian models are fitted using methods that generate samples from the posterior.

Stan is a tool for efficient posterior distribution sample generation.
A Stan program is specified in a separate .stan file consisting of code blocks, and can be compiled and sampled from R using cmdstanr.
Subsequent analysis of the posterior samples is performed outside Stan.

Markov chain Monte Carlo methods can be used to generate samples from a posterior distribution.
Values of the chain are generated from a proposal distribution.
Proposals towards higher areas of the target distribution are accepted with higher probability.
MCMC convergence should always be monitored.

Hierarchical models are appropriate for scenarios where the study population naturally divides into subgroups.
Hierarchical models borrow statistical strength across the population groups.
Population distributions hold information about the variation of the model parameters over the whole population.

Bayesian model comparison can be performed (for example) with posterior predictive checks, information criteria, and cross-validation.

Survival analysis models time until an event while handling censoring and aims to describe and compare time-to-event patterns across covariates or groups.
Time-to-event outcomes can be modeled parametrically with a Weibull model that specifies survival and hazard functions or semiparametrically with a Cox proportional hazards model that leaves the baseline hazard unspecified while estimating relative risk.
The hazard summarizes instantaneous event risk, hazard ratios compare risk between groups or covariate levels, and Kaplan–Meier curves provide a nonparametric survival estimate often shown with a log-rank test.

There are several R packages that provide more user-friendly ways of using Stan.
brms package can be used to fit a vast array of different Bayesian models.
bayesplot package is a library for various plotting tools.
Approximate leave-one-out cross-validation can be performed with the loo package.