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Gelman and Rubin (1992)'s potential scale reduction for chain convergence.
tfp.substrates.numpy.mcmc.potential_scale_reduction( chains_states, independent_chain_ndims=1, split_chains=False, validate_args=False, name=None )
N > 1 states from each of
C > 1 independent chains, the potential
scale reduction factor, commonly referred to as R-hat, measures convergence of
the chains (to the same target) by testing for equality of means.
Specifically, R-hat measures the degree to which variance (of the means)
between chains exceeds what one would expect if the chains were identically
distributed. See [Gelman and Rubin (1992)]; [Brooks and Gelman (1998)].
- The initial state of the chains should be drawn from a distribution overdispersed with respect to the target.
- If all chains converge to the target, then as
N --> infinity, R-hat --> 1. Before that, R-hat > 1 (except in pathological cases, e.g. if the chain paths were identical).
- The above holds for any number of chains
C > 1. Increasing
Cdoes improve effectiveness of the diagnostic.
- Sometimes, R-hat < 1.2 is used to indicate approximate convergence, but of course this is problem-dependent. See [Brooks and Gelman (1998)].
- R-hat only measures non-convergence of the mean. If higher moments, or other statistics are desired, a different diagnostic should be used. See [Brooks and Gelman (1998)].
Diagnosing convergence by monitoring 10 chains that each attempt to sample from a 2-variate normal.
from tensorflow_probability.python.internal.backend import numpy as tf import tensorflow_probability as tfp; tfp = tfp.substrates.numpy tfd = tfp.distributions target = tfd.MultivariateNormalDiag(scale_diag=[1., 2.]) # Get 10 (2x) overdispersed initial states. initial_state = target.sample(10) * 2. ==> (10, 2) # Get 1000 samples from the 10 independent chains. chains_states = tfp.mcmc.sample_chain( num_burnin_steps=200, num_results=1000, current_state=initial_state, trace_fn=None, kernel=tfp.mcmc.HamiltonianMonteCarlo( target_log_prob_fn=target.log_prob, step_size=0.05, num_leapfrog_steps=20)) chains_states.shape ==> (1000, 10, 2) rhat = tfp.mcmc.diagnostic.potential_scale_reduction( chains_states, independent_chain_ndims=1) # The second dimension needed a longer burn-in. rhat.eval() ==> [1.05, 1.3]
To see why R-hat is reasonable, let
X be a random variable drawn uniformly
from the combined states (combined over all chains). Then, in the limit
N, C --> infinity, with
Var denoting expectation and variance,
R-hat = ( E[Var[X | chain]] + Var[E[X | chain]] ) / E[Var[X | chain]].
Using the law of total variance, the numerator is the variance of the combined states, and the denominator is the total variance minus the variance of the the individual chain means. If the chains are all drawing from the same distribution, they will have the same mean, and thus the ratio should be one.
: Stephen P. Brooks and Andrew Gelman. General Methods for Monitoring Convergence of Iterative Simulations. Journal of Computational and Graphical Statistics, 7(4), 1998.
: Andrew Gelman and Donald B. Rubin. Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7(4):457-472, 1992.
: Aki Vehtari, Andrew Gelman, Daniel Simpson, Bob Carpenter, Paul-Christian Bürkner. Rank-normalization, folding, and localization: An improved R-hat for assessing convergence of MCMC, 2021. Bayesian analysis, 16(2):667-718.