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Gelman and Rubin (1992)'s potential scale reduction for chain convergence.

Given 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)][1]; [Brooks and Gelman (1998)][2].

Some guidelines:

  • 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 C does improves 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)][2].
  • 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)][2].


  • chains_states: Tensor or Python list of Tensors representing the state(s) of a Markov Chain at each result step. The ith state is assumed to have shape [Ni, Ci1, Ci2,...,CiD] + A. Dimension 0 indexes the Ni > 1 result steps of the Markov Chain. Dimensions 1 through D index the Ci1 x ... x CiD independent chains to be tested for convergence to the same target. The remaining dimensions, A, can have any shape (even empty).
  • independent_chain_ndims: Integer type Tensor with value >= 1 giving the number of giving the number of dimensions, from dim = 1 to dim = D, holding independent chain results to be tested for convergence.
  • name: String name to prepend to created tf. Default: potential_scale_reduction.


Tensor or Python list of Tensors representing the R-hat statistic for the state(s). Same dtype as state, and shape equal to state.shape[1 + independent_chain_ndims:].


  • ValueError: If independent_chain_ndims < 1.


Diagnosing convergence by monitoring 10 chains that each attempt to sample from a 2-variate normal.

import tensorflow as tf
import tensorflow_probability as tfp
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(
==> (1000, 10, 2)

rhat = tfp.mcmc.diagnostic.potential_scale_reduction(
    chains_states, independent_chain_ndims=1)

# The second dimension needed a longer burn-in.
==> [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 E, 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.


[1]: Stephen P. Brooks and Andrew Gelman. General Methods for Monitoring Convergence of Iterative Simulations. Journal of Computational and Graphical Statistics, 7(4), 1998.

[2]: Andrew Gelman and Donald B. Rubin. Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7(4):457-472, 1992.