Hamiltonian Monte Carlo¶
Table of contents
Algorithm Description¶
The Hamiltonian Monte Carlo (HMC) algorithm is a Markov Chain Monte Carlo method based on principles of Hamiltonian Dynamics.
Let \(\theta^{(i)}\) denote a \(d\)-dimensional vector of stored values at stage \(i\) of the algorithm. The HMC algorithm proceeds in three steps.
(Initialization) Sample \(p^{(i)} \sim N(0,\mathbf{M})\), and set: \(\theta^{(*)} = \theta^{(i)}\) and \(p^{(*)} = p^{(i)}\).
(Leapfrog Steps) for \(k \in \{ 1, \ldots,\)
n_leap_steps
\(\}\) do:
Momentum Update Half-Step.
\[p^{(*)} = p^{(*)} + \epsilon \times \nabla_\theta \ln K(\theta^{(*)} | X) / 2\]where \(K\) denotes the posterior kernel function and \(\epsilon\) is a scaling value set via
hmc_settings.step_size
.
Position Update Step.
\[\theta^{(*)} = \theta^{(*)} + \epsilon \times \mathbf{M}^{-1} p^{(*)}\]where \(\mathbf{M}\) is a pre-conditioning matrix set via
hmc_settings.precond_mat
.
Momentum Update Half-Step.
\[p^{(*)} = p^{(*)} + \epsilon \times \nabla_\theta \ln K(\theta^{(*)} | X) / 2\]
(Accept/Reject Step) Denote the Hamiltonian by
\[H(\theta, p) := \frac{1}{2} \log \left\{ (2 \pi)^d | \mathbf{M} | \right\} + \frac{1}{2} p^\top \mathbf{M}^{-1} p - \ln K(\theta | X)\]and define
\[\alpha = \min \left\{ 1, \exp( H(\theta^{(i)}, p^{(i)}) - H(\theta^{(*)}, p^{(*)}) ) \right\}\]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 hmc(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 Hamiltonian Monte Carlo (HMC) MCMC Algorithm.
- 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 hmc(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 Hamiltonian Monte Carlo (HMC) MCMC Algorithm.
- 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 hmc_settings.n_burnin_draws
: number of burn-in draws.size_t hmc_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 hmc_settings.omp_n_threads
: the number of OpenMP threads to use.Default value:
-1
(use all available threads divided by 2).
size_t hmc_settings.n_leap_steps
: the number of leapfrog steps.Default value:
1
.
fp_t hmc_settings.step_size
: scaling parameter for the leapfrog step.Default value:
1.0
.
Mat_t hmc_settings.precond_mat
: preconditioning matrix for the leapfrog step.Default value: a 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.hmc_settings.step_size = 0.08;
settings.hmc_settings.n_burnin_draws = 2000;
settings.hmc_settings.n_keep_draws = 2000;
arma::mat draws_out;
mcmc::hmc(initial_val, log_target_dens, draws_out, &dta, settings);
//
std::cout << "hmc mean:\n" << arma::mean(draws_out) << std::endl;
std::cout << "acceptance rate: " << static_cast<double>(settings.hmc_settings.n_accept_draws) / settings.hmc_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.hmc_settings.step_size = 0.08;
settings.hmc_settings.n_burnin_draws = 2000;
settings.hmc_settings.n_keep_draws = 2000;
//
Eigen::MatrixXd draws_out;
mcmc::hmc(initial_val, log_target_dens, draws_out, &dta, settings);
//
std::cout << "hmc mean:\n" << draws_out.colwise().mean() << std::endl;
std::cout << "acceptance rate: " << static_cast<double>(settings.hmc_settings.n_accept_draws) / settings.hmc_settings.n_keep_draws << std::endl;
//
return 0;
}