tfp.mcmc.TransformedTransitionKernel

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

Inherits From: TransitionKernel

tfp.mcmc.TransformedTransitionKernel(
    inner_kernel, bijector, name=None
)

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.

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.

Examples

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.
import tensorflow as tf
import tensorflow_probability as tfp
tfd = tfp.distributions
tfb = tfp.bijectors

def make_likelihood(true_variances):
  return tfd.MultivariateNormalDiag(
      scale_diag=tf.sqrt(true_variances))

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

realnvp_hmc = tfp.mcmc.TransformedTransitionKernel(
    inner_kernel=tfp.mcmc.HamiltonianMonteCarlo(
      target_log_prob_fn=likelihood.log_prob,
      step_size=0.5,
      num_leapfrog_steps=2),
    bijector=tfb.RealNVP(
      num_masked=2,
      shift_and_log_scale_fn=tfb.real_nvp_default_template(
          hidden_layers=[512, 512])))

states, kernel_results = tfp.mcmc.sample_chain(
    num_results=1000,
    current_state=tf.zeros(dims),
    kernel=realnvp_hmc,
    num_burnin_steps=500)

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

References

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

[2]: Mark Girolami and Ben Calderhead. Riemann manifold langevin and hamiltonian monte carlo methods. In Journal of the Royal Statistical Society, 2011. https://doi.org/10.1111/j.1467-9868.2010.00765.x

Args:

  • inner_kernel: TransitionKernel-like object which has a target_log_prob_fn 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").

Attributes:

  • bijector
  • inner_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.

  • name

  • parameters: Return dict of __init__ arguments and their values.

Methods

bootstrap_results

View source

bootstrap_results(
    init_state=None, transformed_init_state=None
)

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.

Args:

  • 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.

Returns:

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

Raises:

  • ValueError: if inner_kernel results doesn't contain the member "target_log_prob".

Examples

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(
    num_results=...,
    current_state=init_state,
    previous_kernel_results=transformed_kernel.bootstrap_results(
        transformed_init_state=transformed_init_state),
    kernel=transformed_kernel)

one_step

View source

one_step(
    current_state, previous_kernel_results
)

Runs one iteration of the Transformed Kernel.

Args:

  • 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.)

Returns:

  • 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.