PhD Courses in Denmark

A student who has met the objectives of the course will be able to conduct a custom Bayesian statstical analysis of realistic biological data using the latest available theoretical and computational methods.

A student who has met the objectives of the course will be able to:

• Explain common Bayesian statistics terms such as probability space, prior distribution, measurement model, joint distribution, posterior distribution, Markov Chain Monte Carlo, regression, linear model, generalised linear model
• Determine the space in which some data or parameters live (unconstrained Euclidean space, positive-constrained, simplex, etc.) and apply appropriate transformations
• Describe how gradient-based Markov Chain Monte Carlo works
• Fit a Bayesian statistical model with an embedded dynamical system based on a fed-batch fermentation process.
• Diagnose non-convergence, excessive autocorrelation and solver failures from gradient based Markov Chain Monte Carlo outputs
• Evaluate how well a statistical model fits some data
• Choose the right software for a Bayesian statistical modelling problem: high-level formula-based interface, probabilistic programming language or plain JAX.
• Carry out an independent project involving Bayesian statistics, then write up the results in a report
• Contribute to improving computational Bayesian software

The course takes place over three weeks. The aim of the first week is to cover most of the substantive material. During the second week the students will deepen their knowledge by practicing higher-level tasks like diagnostics, evaluation, software choice and project planning. The third week is set aside for carrying out a supervised independent project. Week 1: Bayesian statistics? Day 1 Theme: how and why to take this course Topics: Introduction, course resources, Bayesian statistics in context Activities: Bayesian statistics tarot reading Role play as fermentation stakeholders Day 2 Theme: Probability as a model for measurement noise and information Topics: Probability spaces, constrained measurements, quantitative vs qualitative belief Activities: Discuss: What is… A space? A probability space? Noise? Belief? Exercise: how to model different kinds of measurement? Including optical density, isotopologue compositions and quantitative metabolomics Computer setup Exercise: calibrate an OD meter Day 3 Theme: Gradient-based Markov chain Monte Carlo Topics: Dynamical systems, Numerical solving Activities: Break a symplectic integrator Fix a broken Bayesian computation Day 4 Themes: automatic differentiation and model evaluation Topics: automatic differentiation, push-forward evaluations, loss functions Activities: Automatic differentiation tarot Stakeholder committee critiques a model Day 5 Theme: embedded dynamical systems Topics: MCMC and automatic differentiation with embedded numerical solving Activities: Fit a simple model of a fermentation Project ideation Week 2: Bayesian statistics in practice Day 6 Theme: Software for Bayesian statistics Topics: types of software, rules of thumb for choosing between alternatives Activities: Research a Bayesian software library Improve the fermentation model and update the course repository Day 7 Theme: Difficult models Topics: difficult measurements (missing data, saturation, heteroskedasticity), difficult priors (colinearity, dependence, tail information, non-parametric models) Activity: fit a difficult model and add it to the course repository Day 8 Theme: The state of the art Topics: recent algorithms, models and applications Activities Some interesting topics in Bayesian statistics research Find and read an interesting recent Bayesian statistics paper Decide in committees how to improve the main model based on findings Day 9 Theme: Making small improvements Activities: Project planning Make a concrete plan for improving Bayesian statistics. For example, find a live issue in a Bayesian software library and try to fix it, or draft a proposal for making conferences more inclusive. Day 10 Theme: Gaps in the field Activities: Discuss and present: the biggest problem with computational Bayesian statistics? Present project plans, get and give peer feedback Week 3: Project work