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BAYESIAN INFERENCE Sampling techniques Andreas Steingtter

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Motivation & Background Exact inference is intractable, so we have to resort to some form of approximation

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Motivation & Background variational Bayes deterministic approximation not exact in principle Alternative approximation: Perform inference by numerical sampling, also known as Monte Carlo techniques.

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Motivation & Background

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Classical Monte Carlo approx approximation

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Motivation & Background

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How to do sampling? 1.Basic Sampling algorithms Restricted mainly to 1- / 2- dimensional problems 2.Markov chain Monte Carlo Very general and powerful framework

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Basic sampling

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Random sampling Computers can generate only pseudorandom numbers Correlation of successive values Lack of uniformity of distribution Poor dimensional distribution of output sequence Distance between where certain values occur are distributed differently from those in a random sequence distribution

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Random sampling from the Uniform Distribution Assumption: good pseudo-random generator for uniformly distributed data is implemented Alternative: http://www.random.org http://www.random.org true random numbers with randomness coming from atmospheric noise

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Random sampling from a standard non-uniform distribution

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Rejection sampling

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Adaptive rejection sampling

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Slope Offset k

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Adaptive rejection sampling

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Importance sampling

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Markov Chain Monte Carlo (MCMC) sampling

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MCMC - Metropolis algorithm

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Metropolis algorithm

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Examples: Metropolis algorithm Implementation in R : Elliptical distibution

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Examples: Metropolis algorithm Implementation in R : Initialization [-2,2], step size = 0.3 n=1500n=15000

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Examples: Metropolis algorithm Implementation in R : Initialization [-2,2], step size = 0.5 n=1500 n=15000

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Examples: Metropolis algorithm Implementation in R : Initialization [-2,2], step size = 1 n=1500 n=15000

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Validation of MCMC homogeneous z (1) z (2) z (m) z (m+1) Invariant (stationary)

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Validation of MCMC homogeneous detailed balance Invariant (stationary) Sufficient reversible

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Validation of MCMC ergodicity invariant

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Properties and validation of MCMC k - Mixing coefficients

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Metropolis-Hastings algorithm If symmetry

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Metropolis-Hastings algorithm Gaussian centered on current state Small variance -> high acceptance, slow walk, dependent samples Large variance -> high rejection rate

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Gibbs sampling repeated by cycling randomly choose variable to be updated

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Gibbs sampling

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z (1) z (2) z (3)

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Gibbs sampling Obtain m independent samples: 1.Sample MCMC during a burn-in period to remove dependence on initial values 2.Then, sample at set time points (e.g. every M th 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.

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Gibbs sampling