| Title: | Metropolis Adjusted Langevin Trajectories |
|---|---|
| Description: | Implements the sampling algorithm: Metropolis Adjusted Langevin Trajectories (Riou-Durand and Vogrinc 2022). Details available at: <arXiv:2202.13230>. The MALT algorithm is a robust extension of Hamiltonian Monte Carlo, for which robustness of tuning is enabled by a positive choice of damping parameter (a.k.a friction). |
| Authors: | Lionel Riou-Durand [aut, cre] |
| Maintainer: | Lionel Riou-Durand <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 0.9 |
| Built: | 2025-02-15 05:10:09 UTC |
| Source: | https://github.com/lrioudurand/malt |
computes effective sample sizes of the chain
ess_summary(output)ess_summary(output)
output |
an output of the malt function |
a list of effective sample sizes when estimating the first and second moments
d=50 sigma=((d:1)/d)^(1/2) init=rnorm(d)*sigma U=function(x){sum(0.5*x^2/sigma^2)} grad=function(x){x/sigma^2} n=10^4 g=1.5 h=0.20 L=8 output=malt(init,U,grad,n,g,h,L) ess_summary(output)d=50 sigma=((d:1)/d)^(1/2) init=rnorm(d)*sigma U=function(x){sum(0.5*x^2/sigma^2)} grad=function(x){x/sigma^2} n=10^4 g=1.5 h=0.20 L=8 output=malt(init,U,grad,n,g,h,L) ess_summary(output)
Implements the sampling algorithm: Metropolis Adjusted Langevin Trajectories (MALT) as described in Riou-Durand and Vogrinc (2022).
malt(init, U, grad, n, g, h, L, warm = FALSE)malt(init, U, grad, n, g, h, L, warm = FALSE)
init |
Real vector. Initial values for the sampling algorithm. |
U |
A potential function to return the log-density of the distribution to be sampled from, up to an additive constant. It should input a real vector of the same length as |
grad |
A function to return the gradient of the potential. It should input and output a real vector of the same length as |
n |
The number of samples to be generated. Positive integer. |
g |
The friction, a.k.a damping parameter. Non-negative real number. The choice g=0 boils down to Hamiltonian Monte Carlo. |
h |
The time step. Positive real number. |
L |
The number of steps per trajectory. Positive integer. The choice L=1 boils down to the Metropolis Adjusted Langevin Algorithm. |
warm |
Should the chain be warmed up? Logical. If TRUE, the samples are generated after a warm-up phase of n successive trajectories. The first half of the warm-up phase is composed by unadjusted trajectories. |
Generates approximate samples from a distribution with density
A Markov chain is generated by drawing successive Langevin trajectories. Each trajectory starts from a fresh Gaussian velocity and is faced with an accept-reject test known as Metropolis adjustment. The Hamiltonian Monte Carlo (HMC) algorithm is recovered as a particular case when the damping parameter is set to zero. A positive choice of damping can ensure robustness of tuning, see Riou-Durand and Vogrinc (2022) for further details.
Returns a list with the following objects:
samples |
a matrix whose rows are the samples generated. |
draw |
a vector corresponding to the last draw of the chain. |
accept |
the acceptance rate of the chain. An acceptance rate close to zero/one indicates that the time step chosen is respectively too large/small. Optimally, the time step should be tuned to obtain an acceptance rate slightly above 65%. |
param |
the input parameters of the malt algorithm. |
Riou-Durand and Vogrinc (2022). Available at: https://arxiv.org/abs/2202.13230.
trajectory, which draws a Langevin trajectory and computes the numerical error along the path.
d=50 sigma=((d:1)/d)^(1/2) init=rnorm(d)*sigma U=function(x){sum(0.5*x^2/sigma^2)} grad=function(x){x/sigma^2} n=10^4 g=1.5 h=0.20 L=8 malt(init,U,grad,n,g,h,L)d=50 sigma=((d:1)/d)^(1/2) init=rnorm(d)*sigma U=function(x){sum(0.5*x^2/sigma^2)} grad=function(x){x/sigma^2} n=10^4 g=1.5 h=0.20 L=8 malt(init,U,grad,n,g,h,L)
print.malt
## S3 method for class 'malt' print(x, digits = max(3L, getOption("digits") - 3L), ...)## S3 method for class 'malt' print(x, digits = max(3L, getOption("digits") - 3L), ...)
x |
an output of the malt function |
digits |
the number of significative digits |
... |
other parameters |
Draws a Langevin trajectory starting from a Gaussian velocity and computes its numerical error. with target density
Given a potential function and its gradient evaluation, draws a trajectory and computes its numerical error. The trajectory drawn corresponds to the proposal in the sampling algorithm: Metropolis Adjusted Langevin Trajectories (Riou-Durand and Vogrinc 2022). Details available at: https://arxiv.org/abs/2202.13230.
trajectory(init, U, grad, g, h, L)trajectory(init, U, grad, g, h, L)
init |
Real vector. Initial values for the Langevin trajectory. |
U |
A potential function to return the log-density of the distribution to be sampled from, up to an additive constant. It should input a real vector of the same length as |
grad |
A function to return the gradient of the potential. It should input and output a real vector of the same length as |
g |
Non-negative real number. The friction, a.k.a damping parameter. The choice g=0 boils down to Hamiltonian Monte Carlo. |
h |
Positive real number. The time step. |
L |
Positive integer. The number of steps per trajectory. The choice L=1 boils down to the Metropolis Adjusted Langevin Algorithm. |
Returns a list with the following objects:
path |
a matrix whose rows are the successive steps of the trajectory. |
draw |
a vector containing the output of the trajectory. |
num_error |
a vector containing the cumulative numerical errors along the path. The numerical error is measured by the energy difference incurred by the leapfrog integrator. |
d=50 sigma=((d:1)/d)^(1/2) init=rnorm(d)*sigma U=function(x){sum(0.5*x^2/sigma^2)} grad=function(x){x/sigma^2} g=1.5 h=0.20 L=8 trajectory(init,U,grad,g,h,L)d=50 sigma=((d:1)/d)^(1/2) init=rnorm(d)*sigma U=function(x){sum(0.5*x^2/sigma^2)} grad=function(x){x/sigma^2} g=1.5 h=0.20 L=8 trajectory(init,U,grad,g,h,L)