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<!DOCTYPE html>
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<meta name="description" content="MCMCLib: a C++ MCMC library.">
<meta name="author" content="Keith O'Hara">
<meta name="keywords" content="mcmc, MCMC, Metropolis Hastings, RWMH, Differential Evolution, Metropolis-adjusted Langevin algorithm, MALA, Hamiltonian Monte Carlo, HMC, C++, C++11, Cpp, NYU, New York University, Econometrics, Research" />
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<title>MCMC: RWMH</title>
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<a data-toggle="collapse" href="#collapse1"><h4><strong style="font-size: 120%;">MCMC: Random Walk Metropolis-Hastings</strong></h4></a>
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<a href="#definition">Definition</a> <br>
<a href="#details">Details</a> <br>
<a href="#examples">Examples</a>
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<p>Posterior simulation via a random walk.</p>
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<h3 style="text-align: left;" id="definition"><strong style="font-size: 100%;">Definition and Syntax</strong></h3>
<pre class="brush: cpp;">
bool rwmh(const arma::vec& initial_vals, arma::mat& draws_out, std::function<double (const arma::vec& vals_inp, void* target_data)> target_log_kernel, void* target_data);
bool rwmh(const arma::vec& initial_vals, arma::mat& draws_out, std::function<double (const arma::vec& vals_inp, void* target_data)> target_log_kernel, void* target_data, algo_settings_t& settings);
</pre>
<p><strong>Function arguments:</strong></p>
<ul>
<li><code>initial_vals</code> a column vector of initial values.</li>
<li><code>draws_out</code> a two-dimensional array containing the posterior draws.</li>
<li><code>target_log_kernel</code> the target log-posterior kernel function, taking two arguments:
<ul>
<li><code>vals_inp</code> a vector of input values; and</li>
<!-- <li><code>grad_out</code> an empty vector, as DE does not require the gradient to be known/exist; and</li> -->
<li><code>target_data</code> additional parameters passed to the function.</li>
</ul>
<li><code>target_data</code> additional parameters passed to the posterior kernel.</li>
<li><code>settings</code> parameters controlling the MCMC routine; see below.</li>
</ul>
<p><strong>MCMC control parameters:</strong></p>
<ul>
<li><code>bool vals_bound</code> whether the search space is bounded. If true, then</li>
<ul>
<li><code>arma::vec lower_bounds</code> this defines the lower bounds.</li>
<li><code>arma::vec upper_bounds</code> this defines the upper bounds.</li>
</ul>
<li><code>int rwmh_n_draws</code> number of posterior draws to keep.</li>
<li><code>int rwmh_n_burnin</code> number of burnin draws.</li>
<li><code>double rwmh_par_scale</code> scaling parameter for the proposal covariance matrix <code>rwmh_cov_mat</code>.</li>
<li><code>arma::mat rwmh_cov_mat</code> covariance matrix of the random walk proposals.</li>
</ul>
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="details"><strong style="font-size: 100%;">Details</strong></h3>
<p>Let $\theta^{(i)}$ denote a $d$-dimensional vector of stored values at stage $i$ of the algorithm. The basic RWMH algorithm is comprised of two steps.</p>
<ul>
<li><strong>Proposal Step.</strong> Generate a proposal draw:
$$\theta^{(*)} = \theta^{(i)} + c \Sigma^{1/2} W$$
where $c$ is determined by <code>rwmh_par_scale</code>, $\Sigma$ is determined by <code>rwmh_cov_mat</code>, and $W \sim N(0,I_d)$.</li>
<li><strong>Accept/Reject Step.</strong> Let
$$\alpha = \min \left\{ 1, K(\theta^{(*)} | X) / K(\theta^{(i)} | X) \right\}$$
where $K$ is the posterior kernel. Then
$$\theta^{(i+1)} = \begin{cases} \theta^{(*)} & \text{ if } Z < \alpha \\ \theta^{(i)} & \text{ else } \end{cases}$$
where $Z \sim \text{Unif}(0,1)$.</li>
</ul>
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="examples"><strong style="font-size: 100%;">Examples</strong></h3>
<p>Normal likelihood with normal prior.</p>
<br>
<pre class="brush: cpp;">
#include "mcmc.hpp"
struct norm_data {
double sigma;
arma::vec x;
double mu_0;
double sigma_0;
};
double ll_dens(const arma::vec& vals_inp, void* ll_data)
{
const double mu = vals_inp(0);
const double pi = arma::datum::pi;
norm_data* dta = reinterpret_cast<norm_data*>(ll_data);
const double sigma = dta->sigma;
const arma::vec x = dta->x;
const int n_vals = x.n_rows;
//
const double ret = - ((double) n_vals) * (0.5*std::log(2*pi) + std::log(sigma)) - arma::accu( arma::pow(x - mu,2) / (2*sigma*sigma) );
//
return ret;
}
double log_pr_dens(const arma::vec& vals_inp, void* ll_data)
{
norm_data* dta = reinterpret_cast< norm_data* >(ll_data);
const double mu_0 = dta->mu_0;
const double sigma_0 = dta->sigma_0;
const double pi = arma::datum::pi;
const double x = vals_inp(0);
const double ret = - 0.5*std::log(2*pi) - std::log(sigma_0) - std::pow(x - mu_0,2) / (2*sigma_0*sigma_0);
return ret;
}
double log_target_dens(const arma::vec& vals_inp, void* ll_data)
{
return ll_dens(vals_inp,ll_data) + log_pr_dens(vals_inp,ll_data);
}
int main()
{
const int n_data = 100;
const double mu = 2.0;
norm_data dta;
dta.sigma = 1.0;
dta.mu_0 = 1.0;
dta.sigma_0 = 2.0;
arma::vec x_dta = mu + arma::randn(n_data,1);
dta.x = x_dta;
arma::vec initial_val(1);
initial_val(0) = 1.0;
arma::mat draws_out;
mcmc::rwmh(initial_val,draws_out,log_target_dens,&dta);
return 0;
}
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