BAYESIAN INFERENCE Sampling techniques

BAYESIAN INFERENCE Sampling techniquesAndreas SteingtterMotivation & BackgroundExact inference is intractable, so we have to resort to some form of approximation

Motivation & Backgroundvariational Bayes deterministic approximation not exact in principle

Alternative approximation:Perform inference by numerical sampling, also known as Monte Carlo techniques.

Motivation & Background

Motivation & Background

Classical Monte Carlo approx

approximationMotivation & Background

How to do sampling?Basic Sampling algorithmsRestricted mainly to 1- / 2- dimensional problems

Markov chain Monte CarloVery general and powerful framework

Basic sampling

Random samplingComputers can generate only pseudorandom numbersCorrelation of successive valuesLack of uniformity of distributionPoor dimensional distribution of output sequenceDistance between where certain values occur are distributed differently from those in a random sequence distribution

Random sampling from theUniform DistributionAssumption: good pseudo-random generator for uniformly distributed data is implemented

Alternative: http://www.random.orgtrue random numbers with randomness coming from atmospheric noise

Random sampling from a standard non-uniform distributionRandom sampling from a standard non-uniform distribution

Rejection sampling

Rejection samplingRejection sampling

Adaptive rejection sampling Adaptive rejection sampling

Slope

Offset k Adaptive rejection sampling Importance sampling

Importance sampling

Importance sampling

Importance sampling Markov Chain Monte Carlo (MCMC) sampling Markov Chain Monte Carlo (MCMC) sampling

MCMC - Metropolis algorithm

MCMC - Metropolis algorithm

Metropolis algorithm Examples: Metropolis algorithm Implementation in R:Elliptical distibution

Examples: Metropolis algorithm Implementation in R:Initialization [-2,2], step size = 0.3

n=1500

n=15000Examples: Metropolis algorithm Implementation in R:Initialization [-2,2], step size = 0.5

n=1500

n=15000Examples: Metropolis algorithm Implementation in R:Initialization [-2,2], step size = 1

n=1500

n=15000Validation of MCMChomogeneousz(1)z(2)z(m)z(m+1)

Invariant(stationary)

Validation of MCMChomogeneousdetailed balance

Invariant(stationary)

SufficientreversibleValidation of MCMCergodicityinvariantProperties and validation of MCMC

k - Mixing coefficients

Metropolis-Hastings algorithm

If symmetryMetropolis-Hastings algorithm Gaussian centered on current stateSmall variance -> high acceptance, slow walk, dependent samplesLarge variance -> high rejection rate

Gibbs sampling

repeated by cycling randomly choose variable to be updatedGibbs sampling

Gibbs samplingz(1)z(2)z(3)Gibbs samplingObtain m independent samples:Sample MCMC during a burn-in period to remove dependence on initial valuesThen, sample at set time points (e.g. every Mth sample)The Gibbs sequence converges to a stationary (equilibrium) distribution that is independent of the starting values, By construction this stationary distribution is the target distribution we are trying to simulate.

Gibbs sampling