Runs one step of Metropolis-adjusted Langevin algorithm.

Inherits From: `TransitionKernel`

Metropolis-adjusted Langevin algorithm (MALA) is a Markov chain Monte Carlo (MCMC) algorithm that takes a step of a discretised Langevin diffusion as a proposal. This class implements one step of MALA using Euler-Maruyama method for a given `current_state` and diagonal preconditioning `volatility` matrix. Mathematical details and derivations can be found in [Roberts and Rosenthal (1998)][1] and [Xifara et al. (2013)][2].

See `UncalibratedLangevin` class description below for details on the proposal generating step of the algorithm.

The `one_step` function can update multiple chains in parallel. It assumes that all leftmost dimensions of `current_state` index independent chain states (and are therefore updated independently). The output of `target_log_prob_fn(*current_state)` should reduce log-probabilities across all event dimensions. Slices along the rightmost dimensions may have different target distributions; for example, `current_state[0, :]` could have a different target distribution from `current_state[1, :]`. These semantics are governed by `target_log_prob_fn(*current_state)`. (The number of independent chains is `tf.size(target_log_prob_fn(*current_state))`.)

#### Examples:

##### Simple chain with warm-up.

In this example we sample from a standard univariate normal distribution using MALA with `step_size` equal to 0.75.

``````import tensorflow.compat.v2 as tf
import tensorflow_probability as tfp
import numpy as np
import matplotlib.pyplot as plt

tfd = tfp.distributions
dtype = np.float32

# Target distribution is Standard Univariate Normal
target = tfd.Normal(loc=dtype(0), scale=dtype(1))

def target_log_prob(x):
return target.log_prob(x)

# Define MALA sampler with `step_size` equal to 0.75
samples = tfp.mcmc.sample_chain(
num_results=1000,
current_state=dtype(1),
target_log_prob_fn=target_log_prob,
step_size=0.75),
num_burnin_steps=500,
trace_fn=None,
seed=42)

sample_mean = tf.reduce_mean(samples, axis=0)
sample_std = tf.sqrt(
tf.reduce_mean(
tf.math.squared_difference(samples, sample_mean),
axis=0))

print('sample mean', sample_mean)
print('sample standard deviation', sample_std)

plt.title('Traceplot')
plt.plot(samples.numpy(), 'b')
plt.xlabel('Iteration')
plt.ylabel('Position')
plt.show()
``````
##### Sample from a 3-D Multivariate Normal distribution.

In this example we also consider a non-constant volatility function.

``````import tensorflow.compat.v2 as tf
import tensorflow_probability as tfp
import numpy as np

dtype = np.float32
true_mean = dtype([0, 0, 0])
true_cov = dtype([[1, 0.25, 0.25], [0.25, 1, 0.25], [0.25, 0.25, 1]])
num_results = 500
num_chains = 500

# Target distribution is defined through the Cholesky decomposition
chol = tf.linalg.cholesky(true_cov)
target = tfd.MultivariateNormalTriL(loc=true_mean, scale_tril=chol)

# Here we define the volatility function to be non-constant
def volatility_fn(x):
# Stack the input tensors together
return 1. / (0.5 + 0.1 * tf.math.abs(x))

# Initial state of the chain
init_state = np.ones([num_chains, 3], dtype=dtype)

# Run MALA with normal proposal for `num_results` iterations for
# `num_chains` independent chains:
states = tfp.mcmc.sample_chain(
num_results=num_results,
current_state=init_state,
target_log_prob_fn=target.log_prob,
step_size=.1,
volatility_fn=volatility_fn),
num_burnin_steps=200,
num_steps_between_results=1,
trace_fn=None,
seed=42)

sample_mean = tf.reduce_mean(states, axis=[0, 1])
x = (states - sample_mean)[..., tf.newaxis]
sample_cov = tf.reduce_mean(
tf.matmul(x, tf.transpose(x, [0, 1, 3, 2])), [0, 1])

print('sample mean', sample_mean.numpy())
print('sample covariance matrix', sample_cov.numpy())
``````

#### References

[1]: Gareth Roberts and Jeffrey Rosenthal. Optimal Scaling of Discrete Approximations to Langevin Diffusions. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60: 255-268, 1998. https://doi.org/10.1111/1467-9868.00123

[2]: T. Xifara et al. Langevin diffusions and the Metropolis-adjusted Langevin algorithm. arXiv preprint arXiv:1309.2983, 2013. https://arxiv.org/abs/1309.2983

`target_log_prob_fn` Python callable which takes an argument like `current_state` (or `*current_state` if it's a list) and returns its (possibly unnormalized) log-density under the target distribution.
`step_size` `Tensor` or Python `list` of `Tensor`s representing the step size for the leapfrog integrator. Must broadcast with the shape of `current_state`. Larger step sizes lead to faster progress, but too-large step sizes make rejection exponentially more likely. When possible, it's often helpful to match per-variable step sizes to the standard deviations of the target distribution in each variable.
`volatility_fn` Python callable which takes an argument like `current_state` (or `*current_state` if it's a list) and returns volatility value at `current_state`. Should return a `Tensor` or Python `list` of `Tensor`s that must broadcast with the shape of `current_state` Defaults to the identity function.
`parallel_iterations` the number of coordinates for which the gradients of the volatility matrix `volatility_fn` can be computed in parallel. Default value: `None` (i.e., use system default).
`experimental_shard_axis_names` A structure of string names indicating how members of the state are sharded.
`name` Python `str` name prefixed to Ops created by this function. Default value: `None` (i.e., 'mala_kernel').

`ValueError` if there isn't one `step_size` or a list with same length as `current_state`.
`TypeError` if `volatility_fn` is not callable.

`experimental_shard_axis_names` The shard axis names for members of the state.
`is_calibrated` Returns `True` if Markov chain converges to specified distribution.

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

`name`

`parallel_iterations`

`parameters` Return `dict` of `__init__` arguments and their values.
`step_size`

`target_log_prob_fn`

`volatility_fn`

## Methods

### `bootstrap_results`

View source

Creates initial `previous_kernel_results` using a supplied `state`.

### `copy`

View source

Non-destructively creates a deep copy of the kernel.

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

Returns
`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)`.

### `experimental_with_shard_axes`

View source

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

Args
`shard_axis_names` a structure of strings indicating the shard axis names for each component of this kernel's state.

Returns
A copy of the current kernel with the shard axis information.

### `one_step`

View source

Runs one iteration of MALA.

Args
`current_state` `Tensor` or Python `list` of `Tensor`s representing the current state(s) of the Markov chain(s). The first `r` dimensions index independent chains, `r = tf.rank(target_log_prob_fn(*current_state))`.
`previous_kernel_results` `collections.namedtuple` containing `Tensor`s representing values from previous calls to this function (or from the `bootstrap_results` function.)
`seed` PRNG seed; see `tfp.random.sanitize_seed` for details.

Returns
`next_state` Tensor or Python list of `Tensor`s 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.

Raises
`ValueError` if there isn't one `step_size` or a list with same length as `current_state` or `diffusion_drift`.

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