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TransformedTransitionKernel applies a bijector to the MCMC's state space.

Inherits From: TransitionKernel

The TransformedTransitionKernel TransitionKernel enables fitting a tfp.bijectors.Bijector which serves to decorrelate the Markov chain Monte Carlo (MCMC) event dimensions thus making the chain mix faster. This is particularly useful when the geometry of the target distribution is unfavorable. In such cases it may take many evaluations of the target_log_prob_fn for the chain to mix between faraway states.

The idea of training an affine function to decorrelate chain event dims was presented in [Parno and Marzouk (2014)][1]. Used in conjunction with the HamiltonianMonteCarlo TransitionKernel, the [Parno and Marzouk (2014)][1] idea is an instance of Riemannian manifold HMC [(Girolami and Calderhead, 2011)][2].

The TransformedTransitionKernel enables arbitrary bijective transformations of arbitrary TransitionKernels, e.g., one could use bijectors tfp.bijectors.Affine, tfp.bijectors.RealNVP, etc. with transition kernels tfp.mcmc.HamiltonianMonteCarlo, tfp.mcmc.RandomWalkMetropolis, etc.

Transforming nested kernels

TransformedTransitionKernel can operate on multiply nested kernels, as in the following example:

      ... # doesn't matter

Upon construction, TransformedTransitionKernel searches the given inner_kernel and the "stack" of nested kernels in any inner_kernel fields thereof until it finds one with a field called target_log_prob_fn, and replaces this with the transformed function. If no inner_kernel has such a target log prob a ValueError is raised.

Mathematical Details

TransformedTransitionKernel enables Markov chains which operate in "unconstrained space." Since we interpret the bijector as mapping "unconstrained space" to "user space", this means that the MCMC transformed target_log_prob is:

target_log_prob(bij.forward(x)) + bij.forward_log_det_jacobian(x)

Recall that tfp.distributions.TransformedDistribution uses the inverse to compute its log_prob. Despite this difference, the use of forward in TransformedTransitionKernel is perfectly consistent with TransformedDistribution following the TFP convention of "sampling" being what defines semantics. The apparent difference is because TransformedDistribution.log_prob is derived from a user provided distribution while in TransformedTransitionKernel samples are derived from target_log_prob_fn. That is, in TransformedDistribution we do:

x ~ NoiseDistribution()
y = bij.forward(x)
log_prob_y = NoiseDistribution().log_prob(bij.inverse(y))

             + bij.inverse_log_det_jacobian(y)

yet in TransformedTransitionKernel we do:

x ~ MCMC()
y = bij.forward(x)
log_prob_y = log_prob(y) + bij.forward_log_det_jacobian(x)

In other words (and in general), tfp.mcmc is derived from a log_prob which what induces a seeming direction convention change. Aside from TFP convention, that Bijectors should adhere to "sample first" semantics is important because it mitigates pervasive necessity of tfp.bijectors.Invert in user code.


RealNVP + HamiltonianMonteCarlo
  • a 1-layer RealNVP is a pretty weak density model, since it can't change the density of the masked dimensions
  • we're not actually training the bijector to do anything useful.
from tensorflow_probability.python.internal.backend import jax as tf
import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.jax
tfd = tfp.distributions
tfb = tfp.bijectors

def make_likelihood(true_variances):
  return tfd.MultivariateNormalDiag(

dims = 10
dtype = np.float32
true_variances = tf.linspace(dtype(1), dtype(3), dims)
likelihood = make_likelihood(true_variances)

realnvp_hmc = tfp.mcmc.TransformedTransitionKernel(
          hidden_layers=[512, 512])))

states, kernel_results = tfp.mcmc.sample_chain(

# Compute sample stats.
sample_mean = tf.reduce_mean(states, axis=0)
sample_var = tf.reduce_mean(
    tf.squared_difference(states, sample_mean),


[1]: Matthew Parno and Youssef Marzouk. Transport map accelerated Markov chain Monte Carlo. arXiv preprint arXiv:1412.5492, 2014.

[2]: Mark Girolami and Ben Calderhead. Riemann manifold langevin and hamiltonian monte carlo methods. In Journal of the Royal Statistical Society, 2011.

inner_kernel TransitionKernel-like object that either has a target_log_prob_fn argument, or wraps around another inner_kernel with said argument.
bijector tfp.distributions.Bijector or list of tfp.distributions.Bijectors. These bijectors use forward to map the inner_kernel state space to the state expected by inner_kernel.target_log_prob_fn.
name Python str name prefixed to Ops created by this function. Default value: None (i.e., "transformed_kernel").



is_calibrated Returns True if Markov chain converges to specified distribution.

TransitionKernels which are "uncalibrated" are often calibrated by composing them with the tfp.mcmc.MetropolisHastings TransitionKernel.


parameters Return dict of __init__ arguments and their values.



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Returns an object with the same type as returned by one_step.

Unlike other TransitionKernels, TransformedTransitionKernel.bootstrap_results has the option of initializing the TransformedTransitionKernelResults from either an initial state, eg, requiring computing bijector.inverse(init_state), or directly from transformed_init_state, i.e., a Tensor or list of Tensors which is interpretted as the bijector.inverse transformed state.

init_state Tensor or Python list of Tensors representing the a state(s) of the Markov chain(s). Must specify init_state or transformed_init_state but not both.
transformed_init_state Tensor or Python list of Tensors representing the a state(s) of the Markov chain(s). Must specify init_state or transformed_init_state but not both.

kernel_results A (possibly nested) tuple, namedtuple or list of Tensors representing internal calculations made within this function.

ValueError if none of the nested inner_kernel results contain the member "target_log_prob".


To use transformed_init_state in context of tfp.mcmc.sample_chain, you need to explicitly pass the previous_kernel_results, e.g.,

transformed_kernel = tfp.mcmc.TransformedTransitionKernel(...)
init_state = ...        # Doesnt matter.
transformed_init_state = ... # Does matter.
results = tfp.mcmc.sample_chain(


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Non-destructively creates a deep copy of the kernel.

**override_parameter_kwargs Python String/value dictionary of initialization arguments to override with new values.

new_kernel TransitionKernel object of same type as self, initialized with the union of self.parameters and override_parameter_kwargs, with any shared keys overridden by the value of override_parameter_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).


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Runs one iteration of the Transformed Kernel.

current_state Tensor or Python list of Tensors representing the current state(s) of the Markov chain(s), after application of bijector.forward. The first r dimensions index independent chains, r = tf.rank(target_log_prob_fn(*current_state)). The inner_kernel.one_step does not actually use current_state, rather it takes as input previous_kernel_results.transformed_state (because TransformedTransitionKernel creates a copy of the input inner_kernel with a modified target_log_prob_fn which internally applies the bijector.forward).
previous_kernel_results collections.namedtuple containing Tensors representing values from previous calls to this function (or from the bootstrap_results function.)
seed Optional, a seed for reproducible sampling.

next_state Tensor or Python list of Tensors representing the state(s) of the Markov chain(s) after taking exactly one step. Has same type and shape as current_state.
kernel_results collections.namedtuple of internal calculations used to advance the chain.