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 ) 

Used in the notebooks

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.ScaleMatvecTriL, 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:

tfp.mcmc.TransformedTransitionKernel( inner_kernel=tfp.mcmc.SimpleStepSizeAdaptation( inner_kernel=tfp.mcmc.HamiltonianMonteCarlo( ... # doesn't matter ), num_adaptation_steps=9) bijector=tfb.Identity())) 

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.

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

Methods

bootstrap_results

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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.

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), trace_fn=None, kernel=transformed_kernel) 

copy

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copy( **override_parameter_kwargs ) 

Non-destructively creates a deep copy of the kernel.

experimental_with_shard_axes

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experimental_with_shard_axes( shard_axis_names ) 

Returns a copy of the kernel with the provided shard axis names.

one_step

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one_step( current_state, previous_kernel_results, seed=None ) 

Runs one iteration of the Transformed Kernel.