# tfp.experimental.substrates.numpy.mcmc.ReplicaExchangeMC

## Class `ReplicaExchangeMC`

Runs one step of the Replica Exchange Monte Carlo.

Inherits From: `TransitionKernel`

Replica Exchange Monte Carlo is a Markov chain Monte Carlo (MCMC) algorithm that is also known as Parallel Tempering. This algorithm takes multiple samples (from tempered distributions) in parallel, then swaps these samples according to the Metropolis-Hastings criterion. See also the review paper .

The `K` replicas are parameterized in terms of `inverse_temperature`'s, `(beta, beta, ..., beta[K-1])`. If the target distribution has probability density `p(x)`, the `kth` replica has density `p(x)**beta_k`.

Typically `beta = 1.0`, and `1.0 > beta > beta > ... > 0.0`. Trying geometrically decaying `beta` is good starting point.

• `beta == 1` ==> First replicas samples from the target density, `p`.
• `beta[k] < 1`, for `k = 1, ..., K-1` ==> Other replicas sample from "flattened" versions of `p` (peak is less high, valley less low). These distributions are somewhat closer to a uniform on the support of `p`.

By default, samples from adjacent replicas `i`, `i + 1` are used as proposals for each other in a Metropolis step. This allows the lower `beta` samples, which explore less dense areas of `p`, to eventually swap state with the `beta == 1` chain, allowing it to explore these new regions.

Samples from replica 0 are returned, and the others are discarded.

#### Examples

##### Sampling from the Standard Normal Distribution.
``````import numpy as np
from tensorflow_probability.python.internal.backend import numpy as tf
import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy
tfd = tfp.distributions

dtype = tf.float32

target = tfd.Normal(loc=dtype(0), scale=dtype(1))

# Geometric decay is a good rule of thumb.
inverse_temperatures = 0.5**tf.range(4, dtype=dtype)

# If everything was Normal, step_size should be ~ sqrt(temperature).
step_size = 0.5 / tf.sqrt(inverse_temperatures)

def make_kernel_fn(target_log_prob_fn, seed):
return tfp.mcmc.HamiltonianMonteCarlo(
target_log_prob_fn=target_log_prob_fn,
seed=seed, step_size=step_size, num_leapfrog_steps=3)

remc = tfp.mcmc.ReplicaExchangeMC(
target_log_prob_fn=target.log_prob,
inverse_temperatures=inverse_temperatures,
make_kernel_fn=make_kernel_fn)

def trace_swaps(unused_state, results):

tfp.mcmc.sample_chain(
num_results=1000,
current_state=1.0,
kernel=remc,
num_burnin_steps=500,
trace_fn=trace_swaps)
)

# conditional_swap_prob[k] = P[ExchangeAccepted | ExchangeProposed],
# for the swap between replicas k and k+1.
conditional_swap_prob = (
/
``````
##### Sampling from a 2-D Mixture Normal Distribution.
``````import numpy as np
from tensorflow_probability.python.internal.backend import numpy as tf
import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy
import matplotlib.pyplot as plt
tfd = tfp.distributions

dtype = tf.float32

target = tfd.MixtureSameFamily(
mixture_distribution=tfd.Categorical(probs=[0.5, 0.5]),
components_distribution=tfd.MultivariateNormalDiag(
loc=[[-1., -1], [1., 1.]],
scale_identity_multiplier=[0.1, 0.1]))

inverse_temperatures = 0.5**tf.range(4, dtype=dtype)

# step_size must broadcast with all batch and event dimensions of target.
# Here, this means it must broadcast with:
#  [len(inverse_temperatures)] + target.event_shape
step_size = 0.5 / tf.reshape(tf.sqrt(inverse_temperatures), shape=(4, 1))

def make_kernel_fn(target_log_prob_fn, seed):
return tfp.mcmc.HamiltonianMonteCarlo(
target_log_prob_fn=target_log_prob_fn,
seed=seed, step_size=step_size, num_leapfrog_steps=3)

remc = tfp.mcmc.ReplicaExchangeMC(
target_log_prob_fn=target.log_prob,
inverse_temperatures=inverse_temperatures,
make_kernel_fn=make_kernel_fn)

samples = tfp.mcmc.sample_chain(
num_results=1000,
# Start near the [1, 1] mode.  Standard HMC would get stuck there.
current_state=tf.ones(2, dtype=dtype),
kernel=remc,
trace_fn=None,
num_burnin_steps=500)

plt.figure(figsize=(8, 8))
plt.xlim(-2, 2)
plt.ylim(-2, 2)
plt.plot(samples_[:, 0], samples_[:, 1], '.')
plt.show()
``````

: David J. Earl, Michael W. Deem Parallel Tempering: Theory, Applications, and New Perspectives https://arxiv.org/abs/physics/0508111

## `__init__`

View source

``````__init__(
target_log_prob_fn,
inverse_temperatures,
make_kernel_fn,
swap_proposal_fn=default_swap_proposal_fn(1.0),
seed=None,
validate_args=False,
name=None
)
``````

Instantiates this object.

#### Args:

• `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.
• `inverse_temperatures`: `Tensor` of inverse temperatures to temper each replica. The leftmost dimension is the `num_replica` and the second dimension through the rightmost can provide different temperature to different batch members, doing a left-justified broadcast.
• `make_kernel_fn`: Python callable which takes target_log_prob_fn and seed args and returns a TransitionKernel instance.
• `swap_proposal_fn`: Python callable which take a number of replicas, and returns `swaps`, a shape `[num_replica] + batch_shape` `Tensor`, where axis 0 indexes a permutation of `{0,..., num_replica-1}`, designating replicas to swap.
• `seed`: Python integer to seed the random number generator. Default value: `None` (i.e., no seed).
• `validate_args`: Python `bool`, default `False`. When `True` distribution parameters are checked for validity despite possibly degrading runtime performance. When `False` invalid inputs may silently render incorrect outputs.
• `name`: Python `str` name prefixed to Ops created by this function. Default value: `None` (i.e., "remc_kernel").

#### Raises:

• `ValueError`: `inverse_temperatures` doesn't have statically known 1D shape.

## Properties

### `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`.

### `parameters`

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

## Methods

### `bootstrap_results`

View source

``````bootstrap_results(init_state)
``````

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

#### Args:

• `init_state`: `Tensor` or Python `list` of `Tensor`s representing the initial state(s) of the Markov chain(s).

#### Returns:

• `kernel_results`: A (possibly nested) `tuple`, `namedtuple` or `list` of `Tensor`s representing internal calculations made within this function. This inculdes replica states.

### `num_replica`

View source

``````num_replica()
``````

Integer (`Tensor`) number of replicas being tracked.

### `one_step`

View source

``````one_step(
current_state,
previous_kernel_results
)
``````

Takes one step of the TransitionKernel.

#### Args:

• `current_state`: `Tensor` or Python `list` of `Tensor`s representing the current state(s) of the Markov chain(s).
• `previous_kernel_results`: A (possibly nested) `tuple`, `namedtuple` or `list` of `Tensor`s representing internal calculations made within the previous call to this function (or as returned by `bootstrap_results`).

#### Returns:

• `next_state`: `Tensor` or Python `list` of `Tensor`s representing the next state(s) of the Markov chain(s).
• `kernel_results`: A (possibly nested) `tuple`, `namedtuple` or `list` of `Tensor`s representing internal calculations made within this function. This inculdes replica states.