Package 'LangevinFlow'

Metropolis-Adjusted Langevin Algorithm

Description

Asymptotically exact sampler for $\pi(x) \propto \exp(-\beta U(x))$. At each step a Langevin proposal is generated as in ula and then accepted or rejected by the Metropolis-Hastings rule so that $\pi$ is exactly invariant. Optimal acceptance rates are typically near 0.574 in high dimension (Roberts & Rosenthal, 1998).

Usage

mala(init_x, U, grad_u, step_size, n_iter, beta = 1, burn_in = 0L)

Arguments

init_x	Numeric vector. Starting state of the chain.
U	Function. Takes a numeric vector of length length(init_x) and returns the scalar potential $U(x) = -\log \pi(x)$ (additive constants do not matter).
grad_u	Function. Returns the gradient of U; same shape convention as for ula.
step_size	Positive numeric scalar. Discretization step $\gamma$.
n_iter	Positive integer. Number of iterations.
beta	Positive numeric scalar. Inverse temperature; defaults to 1.
burn_in	Non-negative integer, strictly less than n_iter. Defaults to 0.

Value

An object of class "langevin_chain" (see ula for structure). The acceptance_rate component is populated and accepted is a logical vector indicating per-iteration outcomes (post-burn-in).

References

Roberts, G. O., & Rosenthal, J. S. (1998). Optimal scaling of discrete approximations to Langevin diffusions. Journal of the Royal Statistical Society, Series B, 60(1), 255-268.

See Also

ula

Examples

# Standard 2D Gaussian: U(x) = 0.5 * ||x||^2.
set.seed(1)
U      <- function(x) 0.5 * sum(x^2)
grad_u <- function(x) x
fit <- mala(init_x = c(3, -3), U = U, grad_u = grad_u,
            step_size = 0.3, n_iter = 2000, burn_in = 500)
summary(fit)

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Plot a Langevin Chain

Description

Trace plot for each dimension, plus a marginal histogram. For high-dimensional chains, only the first max_dim coordinates are shown.

Usage

## S3 method for class 'langevin_chain'
plot(x, max_dim = 4L, ...)

Arguments

x	An object of class "langevin_chain".
max_dim	Maximum number of coordinates to display. Defaults to 4.
...	Passed to plot.

Value

Called for side effect. Returns x invisibly.

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Summarize a Langevin Chain

Description

Computes per-coordinate posterior summaries (mean, standard deviation, and selected quantiles) from a fitted Langevin chain.

Usage

## S3 method for class 'langevin_chain'
summary(object, probs = c(0.025, 0.5, 0.975), ...)

Arguments

object	An object of class "langevin_chain".
probs	Numeric vector of probabilities for quantile computation.
...	Currently unused.

Value

Invisibly, a data frame of summaries. Printed by default.

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Unadjusted Langevin Algorithm

Description

Draws a Markov chain whose stationary distribution approximates $\pi(x) \propto \exp(-\beta U(x))$ using the Euler-Maruyama discretization of the overdamped Langevin diffusion. The discretization introduces a bias that vanishes as step_size tends to zero; for an asymptotically exact sampler, use mala.

Usage

ula(init_x, grad_u, step_size, n_iter, beta = 1, burn_in = 0L)

Arguments

init_x	Numeric vector. Starting state of the chain. Its length defines the dimension.
grad_u	Function. Takes a numeric vector of length length(init_x) and returns the gradient of $U(x) = -\log \pi(x)$, as a numeric vector of the same length. See the sign convention in ?LangevinFlow.
step_size	Positive numeric scalar. Discretization step $\gamma$.
n_iter	Positive integer. Number of iterations to run.
beta	Positive numeric scalar. Inverse temperature; defaults to 1.
burn_in	Non-negative integer. Number of initial samples to discard. Must be strictly less than n_iter. Defaults to 0.

Value

An object of class "langevin_chain", which is a list with components:

samples

Numeric matrix of post-burn-in samples; rows index iterations and columns index dimensions.

algorithm

Character string "ULA".

step_size, beta, n_iter, burn_in, dimension

Echoed inputs.

acceptance_rate

NA for ULA (no accept/reject step).

elapsed_secs

Wall-clock runtime of the sampler in seconds.

References

Roberts, G. O., & Tweedie, R. L. (1996). Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli, 2(4), 341-363.

See Also

mala

Examples

# Standard 2D Gaussian target: U(x) = 0.5 * ||x||^2, grad_U(x) = x.
set.seed(1)
grad_u <- function(x) x
fit <- ula(init_x = c(3, -3), grad_u = grad_u,
           step_size = 0.05, n_iter = 2000, burn_in = 500)
summary(fit)

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