SOCI 620: Quantitative methods 2

Agenda

Monte-carlo methods

  1. Administrative
  2. MAP/ML
  3. MCMC
  4. Hamiltonian MC
  5. What can go wrong
  6. Hands on:
    Convergence in brms and lme4

Slides are licensed under CC BY-NC-SA 4.0

Administrative

Worksheet 4

  • Forthcoming!
  • Hopefully later today

Maximizing probability

still from the 1960s Batman. Batman and Robin appear to be scaling a wall holding a thin rope. A man sticks his head out the window.

Exploring the posterior

Recall: simple binomial model

5 trials; 4 ‘successes’

This is the same as an intercept-only logistic distribution with a logit-transformed uniform for the prior on

Bayes' rule

Exploring the posterior

Recall: simple binomial model

5 trials; 4 ‘successes’

This is the same as an intercept-only logistic distribution with a logit-transformed uniform for the prior on

Bayes' rule

Hill climbing (MAP/ML)

Hill climbing (MAP/ML)

Hill climbing (MAP/ML)

Hill climbing (MAP/ML)

Hill climbing (MAP/ML)

Hill climbing (MAP/ML)

Markov
chain
Monte
Carlo

A man (Jackie Chan) in early 20th century Chinese costume lists to the side as if drunk while holding his hand out in an awkward but aggressive manner. His face is red and he has a silly grin.

Markov chain Monte Carlo

Markov chain Monte Carlo

Markov chain Monte Carlo

Markov chain Monte Carlo

Markov chain Monte Carlo

Markov chain Monte Carlo

Markov chain Monte Carlo

Markov chain Monte Carlo

Density based on 50,000 posterior observations

Hamil­tonian Monte Carlo

Video clip of a skateboarder (a young Tony Hawk) in late 1980s clothes scating casually in an empty, rounded swimming pool.

Hamiltonian Monte Carlo

Simulate a
physical system

“Energy” at any point in the parameter space is proportional to the negative log posterior probability:


Get random draws
by ‘perturbing’ a particle in that field

Place a particle in that system, give it a push in a random direction, and use Hamiltonian dynamics to simulate its motion.


Wherever the particle ends up after a fixed amount of time is the next candidate draw from the posterior.

Hamiltonian Monte Carlo

Hamiltonian Monte Carlo

HMC vs. MCMC

Takes advantage
of gradient

Gradient (slope) information helps HMC adjust to the shape of the posterior.

Reduces autocorrelation

HMC tends to explore the plausible areas of the parameter space much more quickly than ‘standard’ MCMC like Metropolis–Hastings. It is not likely to spend too much time in one small area.

“No-U-Turn sampler”
(NUTS)

A version of HMC that automatically optimizes some of the meta-parameters of the algorithm.

What can possibly
go wrong?

A man (Harrison Ford playing Indiana Jones) in a leather coat and dirty fedora prepares to replace a golden idol on an altar with a bag of sand.

What can go wrong?

Autocorrelation

  • Because each posterior sample depends on the previous sample, HMC usually displays some autocorrelation
  • A sample of 1,000 autocorrelated samples will have less information than a sample of 1,000 independent samples
  • Relevant quantity:
    effective sample size (ESS)

What can go wrong?

Non-convergence

  • Sampling may have trouble “converging” (properly representing the posterior)
  • Many possible causes
    Bad model specification
    Insufficient iterations
    Badly tuned sampler
  • Diagnose with R̂ (“Rhat”) on multiple chains to check agreement between multiple chains

What can go wrong?

Divergent transistions

  • HMC can ‘overshoot’ if the posterior density has sharp edges
  • Common in multilevel models
  • Stan (and brms) will warn you about these
  • Address with re-specification of model or finer-grained steps

Image credit

Figures by Peter McMahan (source code)

still from the 1960s Batman. Batman and Robin appear to be scaling a wall holding a thin rope. A man sticks his head out the window.

Still from Batman (1966)

A man (Jackie Chan) in early 20th century Chinese costume lists to the side as if drunk while holding his hand out in an awkward but aggressive manner. His face is red and he has a silly grin.

Still from The Legend of the Drunken Master (1994)

Video clip of a skateboarder (a young Tony Hawk) in late 1980s clothes scating casually in an empty, rounded swimming pool.

Clip from Gleaming the Cube (1989)

A man (Harrison Ford playing Indiana Jones) in a leather coat and dirty fedora prepares to replace a golden idol on an altar with a bag of sand.

Clip from Raiders of the Lost Ark (1981)