This short course introduces students to Bayesian statistical methods in biomedical settings, and provides skills for designing, assessing and interpreting Bayesian analyses using the R and JAGS statistical software. The emphasis throughout will be on practical, applied modelling: code to carry out analyses will be provided.
The course runs on the Moodle online learning platform, and involves 7 sessions:
- Quantifying uncertainty with probability
- Bayesian inference
- Bayesian regression models
- Critiquing and comparing Bayesian models
- Hierarchical models
- Modelling with missing and censored data
- Integrating multiple sources of data
The course will be delivered via the Moodle online learning platform. The course consists of 7 half-days worth of content, and will take place over 7 half-days across 3 weeks starting on Friday 17th November 2023.
It will consist of some on-demand content and some timetabled (live) sessions.
- The talks in the course are pre-recorded and available to watch on-demand whenever is convenient. The talks for each session last up to 90 minutes in total.
- The computer practical sessions (around 1.5 hours) can be completed whenever is convenient. Full solutions are available to participants.
Live interactive sessions:
- There is an associated live online drop-in session for each half-day to come and ask questions (1pm-2.30pm UK time each day). Specifically these will be held on:
- Session 1: Friday 17th November 2023
- Session 2: Monday 20th November 2023
- Session 3: Wednesday 22nd November 2023
- Session 4: Friday 24th November 2023
- Session 5: Monday 27th November 2023
- Session 6: Wednesday 29th November 2023
- Session 7: Friday 1st December 2023
- These are not compulsory, but are supplementary to the core content, and are an opportunity to engage with the course tutors.
Questions can also be asked during the course on a dedicated Slack channel.
The target audience of the course is statisticians and people doing statistical analysis in any subject area.
No experience of Bayesian methods is assumed, but we do we assume familiarity with key statistical concepts:
- Basic probability concepts: discrete and continuous random variables; probability density functions; expectation; variance; familiarity with standard probability distributions (e.g. normal, binomial, uniform).
- A good understanding of classical (ie non-Bayesian) statistical modelling: likelihood (as in maximum likelihood estimation) and sampling distributions; linear regression; generalised linear models including logistic regression; assessment of model fit using residuals.
No experience with specialist Bayesian software will be assumed, but you should be comfortable using the statistical software R:
- While all code is provided, we think that you will find it easier to follow if you have a good familiarity with R, and we may not be able to fully support users with no R experience.
Statisticians working in any application area who are familiar with Bayesian modelling and the BUGS or JAGS software.
Course learning outcomes
- Understand the principles of Bayesian statistics: learning from data and judgements through probability distributions on parameters in models.
- Design a range of Bayesian models for health science problems, including appropriate selection of prior distributions.
- Implement the models in standard software, and assess the performance of computational algorithms used for this.
- Summarise and accurately interpret the output of Bayesian analyses.
- Understand the assumptions being made in Bayesian models, and effectively appraise and compare them both qualitatively and quantitatively using standard methods.
Bayesian Statistics online short course = Session List:
- Half-day 1: Quantifying uncertainty with probability
- The basic concept behind Bayesian methods: describing uncertainty in knowledge using probability statements on parameters of models. Constructing prior distributions on model parameters. Contrast with the frequentist view in which probabilities can only describe observables.
- Making predictions and decisions using probabilistic judgements.
- Introduction to probabilistic programming using JAGS.
Practical exercises 1: constructing priors, making probability statements about parameters and predictions, making decisions under uncertainty, in simple beta or normal examples.
- Half-day 2: Bayesian inference
- Bayes’ theorem. Bayesian inference for a proportion and a normally-distributed quantity.
- Markov chain Monte Carlo (MCMC) – Gibbs sampling. Convergence diagnosis.
Practical 2: conjugate inference, specifying models in JAGS, convergence assessment, Gibbs sampling.
- Half-day 3: Bayesian regression models
- Linear regression from a Bayesian perspective. Defining priors on regression coefficients and error variance: vague, weakly informative or informative.
- Logistic regression, count data regression, non-linear regression, and prior specification. Understanding importance of model assumptions, and appropriately interpreting summaries of posterior distributions.
Practical 3: specifying, fitting and summarising models, and informally comparing models with different assumptions.
- Half-day 4: Critiquing and comparing Bayesian models
- Prediction: out-of-sample prediction; prediction as imputing missing data. Model checking: checking the likelihood (goodness of fit / checking for outliers). In-sample prediction for model checking; Bayesian p-values; out-of-sample prediction for model checking (cross-validation); Bayesian residuals for model assessment.
- Model checking: deviance summaries for model assessment. Checking the prior: sensitivity analyses to the prior; prior-data conflict. Model comparison: penalised deviances.
Practical 4: exploring prediction to understand its role in both in- and out-of-sample prediction for prediction itself and model assessment; practice manipulating posterior samples to derive residuals and posterior-predictive p-values for model checking; exploring deviance residuals, prior sensitivity and prior-data conflict; practice comparing models using penalised deviances.
- Half-day 5: Hierarchical models
- Normal hierarchical models
- Hierarchical regression models
- Basic meta analysis. Priors in hierarchical models
Practical 5: JAGS specification of hierarchical models. Practical involving specifying, fitting and summarising models, and comparing models with different assumptions.
- Half-day 6: Modelling with missing and censored data
- Bayesian methods for handling missing data in regression models, Outcomes missing at random. Covariates missing at random: fully Bayesian modelling and Bayesian multiple imputation
- Bayesian methods for handling data missing not at random.
- Bayesian methods for censored data and survival analysis.
Practical 6: JAGS specification of these models. Practical involving specifying, fitting and summarising models, and comparing models with different assumptions.
- Half-day 7: Integrating multiple sources of data
- Key concepts of evidence synthesis, generalising meta-analysis, illustrated through simple examples; formulating likelihood from multiple data sources.
- Model criticism for evidence synthesis: conflict resolution via bias modelling; robustifying inference using over-dispersion; cross-validatory mixed-predictive checks; systematic bias adjustment.
Practical 7: Practise building a simple evidence synthesis; understand that synthesising evidence can result in lack-of-fit to the data, if some of the evidence is conflicting; practise using posterior-predictive p-values to detect conflict and the DIC to compare models. Practise resolving conflict by introducing bias parameters; practise specifying robust models which account for over-dispersion. Practise detecting outliers using cross-validatory mixed-predictive checks; and practise systematic bias adjustment to improve inference.