Lesson 1 - The Basics

Basics of Bayesian Inference

• Bayes theorem for events
  • Exercise - which terms should go into nominator/denumerator?
• Bayes theorem for models

\[\begin{gather*} \pi_\text{joint}(y, \theta) = \pi_\text{obs}(y | \theta) \pi_\text{prior}(\theta)\\ \pi_\text{marg}\left(y \right) = \int_\Theta \mathrm{d} \theta \: \pi_{\text{obs}}(y | \theta) \pi_\text{prior}(\theta)\\ \pi_\text{post}(\theta | y) = \frac{\pi_\text{obs}(y | \theta) \pi_\text{prior}(\theta)}{\pi_\text{marg}\left(y \right)}. \end{gather*}\]

• Bayesian statistics \(\neq\) Bayesian epistemology
  • Problems with catchall hypothesis
  • Problems in computation of Bayes factors
  • Problems with selective inference
  • Compare to modern falsificationist ideas, notably Mayo (e.g. Mayo 2018, SIST — I can lend you a copy, or Mayo & Spanos 2011)
    • Liftoff
• Modelling notation, e.g.

\[ y \sim N(\mu,\sigma) \\ \mu \sim N(0, 1) \\ \sigma \sim HalfN(0, 2) \]

• Modelled vs. unmodelled data

• Great intro to the underlying philosophical ideas: Chapters 1 - 2 of McElreath’s “Statistical Rethinking” - available freely online at https://xcelab.net/rmpubs/sr2/statisticalrethinking2_chapters1and2.pdf

  • McElreath also has great free course on Bayesian statistics at https://github.com/rmcelreath/stat_rethinking_2024

• Extracting information from probability distributions via expectation values (see Mike Betancourt’s writing for a very detailed coverage)

• MCMC algorithms allow us to sample posterior

  • What is a chain
  • Convergence, Rhat, ESS

• HMC - show example, requires gradient

  • Conceptually two components — random initial momentum for each trajectory makes the chain explore, preference for high density regions makes the chain spend more time there.
  • Wax poetic how HMC is great
  • HMC will work great, if gradient is useful
  • HMC has good diagnostic - the “divergence”

• What is needed to be proficient at probabilistic programming?

  • Mathematical probability/statistics
  • Numerics
  • A little software engineering

Basics of Stan

• A decent tutorial covering similar ground as we did (and some more) is at https://betanalpha.github.io/assets/case_studies/stan_intro.html Note that the example R code there uses the rstan package, while we will use the newer cmdstanr package.
• Stan documentation:
  • language reference - how the language and algorithms work
  • function reference - individual functions available in the language
  • user’s guide - worked out implementations of many model classes
• Stan has very helpful community at https://discourse.mc-stan.org
• Fundamentally a Stan program computes logarithm of the density (and its gradient, Hessian via autodiff)
  • Why logarithms?
• Stan is statically typed (and generally quite restrictive). Discuss pros and cons.
• Stan architecture:
  • Math library + autodiff
  • The language
  • The interfaces
  • compiler to C++, compile C++
• Go through the “hello world” model for the lesson.
• Main program blocks
  • data (both modelled and non-modelled)
  • parameters
  • model
• target += statements
• _lpdf

The tasks

• Start a project to contain your coursework
• In general: we expect you to use online documentation etc. a lot, just like in “real life”
• Task description - separate page

Summary after tasks

• Constraints
• Divergences as helpful diagnostics
• Discuss project proposals