Metropolis-adjusted Langevin Algorithm¶
Table of contents
Algorithm Description¶
The Metropolis-adjusted Langevin algorithm (MALA) extends the Random Walk Metropolis-Hasting algorithm by generating proposal draws via Langevin diffusions.
Let \(\theta^{(i)}\) denote a \(d\)-dimensional vector of stored values at stage \(i\) of the algorithm. MALA proceeds in two steps.
(Proposal Step) Let
\[\mu(\theta^{(i)}) := \theta^{(i)} + \frac{\epsilon^2}{2} \times \mathbf{M} \left[ \nabla_\theta \ln K(\theta^{(i)} | X) \right]\]where \(K\) denotes the posterior kernel function; \(\nabla_\theta\) denotes the gradient operator; \(\mathbf{M}\) is a pre-conditioning matrix, set via
mala_settings.precond_mat
; and \(\epsilon\) is a scaling value, set viamala_settings.step_size
.Generate a proposal draw, \(\theta^{(*)}\). using:
\[\theta^{(*)} = \mu(\theta^{(i)}) + c \times \mathbf{M}^{1/2} W\]where \(W \sim N(0,I_d)\).
(Accept/Reject Step) Denote the proposal density by \(q(\theta^{(*)} | \theta^{(i)}) := \phi(\theta^{(*)}; \mu(\theta^{(i)}), \epsilon^2 \mathbf{M})\) and let
\[\alpha = \min \left\{ 1, [ K(\theta^{(*)} | X) q(\theta^{(i)} | \theta^{(*)})] / [ K(\theta^{(i)} | X) q(\theta^{(*)} | \theta^{(i)})] \right\}\]Then
\[\theta^{(i+1)} = \begin{cases} \theta^{(*)} & \text{ with probability } \alpha \\ \theta^{(i)} & \text{ else } \end{cases}\]
The algorithm stops when the number of draws reaches n_burnin_draws
+ n_keep_draws
, and returns the final n_keep_draws
number of draws.
Function Declarations¶
-
bool mala(const ColVec_t &initial_vals, std::function<fp_t(const ColVec_t &vals_inp, ColVec_t *grad_out, void *target_data)> target_log_kernel, Mat_t &draws_out, void *target_data)¶
The Metropolis-adjusted Langevin Algorithm (MALA)
- Parameters
initial_vals – a column vector of initial values.
target_log_kernel – the log posterior kernel function of the target distribution, taking three arguments:
vals_inp
a vector of inputs; andgrad_out
a vector to store the gradient; andtarget_data
additional data passed to the user-provided function.
draws_out – a matrix of posterior draws, where each row represents one draw.
target_data – additional data passed to the user-provided function.
- Returns
a boolean value indicating successful completion of the algorithm.
-
bool mala(const ColVec_t &initial_vals, std::function<fp_t(const ColVec_t &vals_inp, ColVec_t *grad_out, void *target_data)> target_log_kernel, Mat_t &draws_out, void *target_data, algo_settings_t &settings)¶
The Metropolis-adjusted Langevin Algorithm (MALA)
- Parameters
initial_vals – a column vector of initial values.
target_log_kernel – the log posterior kernel function of the target distribution, taking three arguments:
vals_inp
a vector of inputs; andgrad_out
a vector to store the gradient; andtarget_data
additional data passed to the user-provided function.
draws_out – a matrix of posterior draws, where each row represents one draw.
target_data – additional data passed to the user-provided function.
settings – parameters controlling the MCMC routine.
- Returns
a boolean value indicating successful completion of the algorithm.
Control Parameters¶
The basic control parameters are:
size_t mala_settings.n_burnin_draws
: number of burn-in draws.size_t mala_settings.n_keep_draws
: number of draws to keep (post sample burn-in period).bool vals_bound
: whether the search space of the algorithm is bounded. Iftrue
, thenColVec_t lower_bounds
: defines the lower bounds of the search space.ColVec_t upper_bounds
: defines the upper bounds of the search space.
Additional settings:
int mala_settings.omp_n_threads
: the number of OpenMP threads to use.Default value:
-1
(use all available threads divided by 2).
fp_t mala_settings.step_size
: scaling parameter for the proposal step.Default value:
1.0
.
Mat_t mala_settings.precond_mat
: preconditioning matrix for the proposal step.Default value: diagonal matrix.
Examples¶
Gaussian Distribution¶
Code to run this example is given below.
Armadillo (Click to show/hide)
#define MCMC_ENABLE_ARMA_WRAPPERS
#include "mcmc.hpp"
struct norm_data_t {
arma::vec x;
};
double ll_dens(const arma::vec& vals_inp, arma::vec* grad_out, void* ll_data)
{
const double pi = arma::datum::pi;
const double mu = vals_inp(0);
const double sigma = vals_inp(1);
norm_data_t* dta = reinterpret_cast<norm_data_t*>(ll_data);
const arma::vec x = dta->x;
const int n_vals = x.n_rows;
//
const double ret = - n_vals * (0.5 * std::log(2*pi) + std::log(sigma)) - arma::accu( arma::pow(x - mu,2) / (2*sigma*sigma) );
//
if (grad_out) {
grad_out->set_size(2,1);
//
const double m_1 = arma::accu(x - mu);
const double m_2 = arma::accu( arma::pow(x - mu,2) );
(*grad_out)(0,0) = m_1 / (sigma*sigma);
(*grad_out)(1,0) = (m_2 / (sigma*sigma*sigma)) - ((double) n_vals) / sigma;
}
//
return ret;
}
double log_target_dens(const arma::vec& vals_inp, arma::vec* grad_out, void* ll_data)
{
return ll_dens(vals_inp,grad_out,ll_data);
}
int main()
{
const int n_data = 1000;
const double mu = 2.0;
const double sigma = 2.0;
norm_data_t dta;
arma::vec x_dta = mu + sigma * arma::randn(n_data,1);
dta.x = x_dta;
arma::vec initial_val(2);
initial_val(0) = mu + 1; // mu
initial_val(1) = sigma + 1; // sigma
//
mcmc::algo_settings_t settings;
settings.mala_settings.step_size = 0.08;
settings.mala_settings.n_burnin_draws = 2000;
settings.mala_settings.n_keep_draws = 2000;
//
arma::mat draws_out;
mcmc::mala(initial_val, log_target_dens, draws_out, &dta, settings);
//
std::cout << "mala mean:\n" << arma::mean(draws_out) << std::endl;
std::cout << "acceptance rate: " << static_cast<double>(settings.mala_settings.n_accept_draws) / settings.mala_settings.n_keep_draws << std::endl;
//
return 0;
}
Eigen (Click to show/hide)
#define MCMC_ENABLE_EIGEN_WRAPPERS
#include "mcmc.hpp"
inline
Eigen::VectorXd
eigen_randn_colvec(size_t nr)
{
static std::mt19937 gen{ std::random_device{}() };
static std::normal_distribution<> dist;
return Eigen::VectorXd{ nr }.unaryExpr([&](double x) { (void)(x); return dist(gen); });
}
struct norm_data_t {
Eigen::VectorXd x;
};
double ll_dens(const Eigen::VectorXd& vals_inp, Eigen::VectorXd* grad_out, void* ll_data)
{
const double pi = 3.14159265358979;
const double mu = vals_inp(0);
const double sigma = vals_inp(1);
norm_data_t* dta = reinterpret_cast<norm_data_t*>(ll_data);
const Eigen::VectorXd x = dta->x;
const int n_vals = x.size();
//
const double ret = - n_vals * (0.5 * std::log(2*pi) + std::log(sigma)) - (x.array() - mu).pow(2).sum() / (2*sigma*sigma);
//
if (grad_out) {
grad_out->resize(2,1);
//
const double m_1 = (x.array() - mu).sum();
const double m_2 = (x.array() - mu).pow(2).sum();
(*grad_out)(0,0) = m_1 / (sigma*sigma);
(*grad_out)(1,0) = (m_2 / (sigma*sigma*sigma)) - ((double) n_vals) / sigma;
}
//
return ret;
}
double log_target_dens(const Eigen::VectorXd& vals_inp, Eigen::VectorXd* grad_out, void* ll_data)
{
return ll_dens(vals_inp,grad_out,ll_data);
}
int main()
{
const int n_data = 1000;
const double mu = 2.0;
const double sigma = 2.0;
norm_data_t dta;
Eigen::VectorXd x_dta = mu + sigma * eigen_randn_colvec(n_data).array();
dta.x = x_dta;
Eigen::VectorXd initial_val(2);
initial_val(0) = mu + 1; // mu
initial_val(1) = sigma + 1; // sigma
mcmc::algo_settings_t settings;
settings.mala_settings.step_size = 0.08;
settings.mala_settings.n_burnin_draws = 2000;
settings.mala_settings.n_keep_draws = 2000;
//
Eigen::MatrixXd draws_out;
mcmc::mala(initial_val, log_target_dens, draws_out, &dta, settings);
//
std::cout << "mala mean:\n" << draws_out.colwise().mean() << std::endl;
std::cout << "acceptance rate: " << static_cast<double>(settings.mala_settings.n_accept_draws) / settings.mala_settings.n_keep_draws << std::endl;
//
return 0;
}