cmp::mcmc Namespace Reference

Classes
class	EvolutionaryMarkovChain
	Implements an Evolutionary Markov Chain Monte Carlo sampler. More...
class	HamiltonianMarkovChain
	Implements the Hamiltonian Monte Carlo (HMC) algorithm with the No-U-Turn Sampler (NUTS) and Dual Averaging. More...
class	MarkovChain
	Represents a single Markov Chain utilizing Delayed Rejection Adaptive Metropolis (DRAM) for sampling. More...

Functions
double	log1m (double a)
template<typename ProposalType >
double	multiChainDiagnosis (const std::vector< MarkovChain< ProposalType > > &chains)
	Computes the Gelman-Rubin convergence diagnostic metric (R-hat).

Function Documentation

log1m()

double cmp::mcmc::log1m ( double  a )

inline

multiChainDiagnosis()

template<typename ProposalType >

double cmp::mcmc::multiChainDiagnosis

(

const std::vector< MarkovChain< ProposalType > > &

chains

)

Computes the Gelman-Rubin convergence diagnostic metric (R-hat).

Mathematical Formulation For \(M\) chains of length \(N\), it computes within-chain variance \(W\) and between-chain variance \(B\):

\[ W = \frac{1}{M} \sum_{m=1}^M s_m^2, \quad B = \frac{N}{M-1} \sum_{m=1}^M (\bar{\theta}_{m} - \bar{\theta}_{\cdot})^2 \]

The posterior marginal variance is estimated as:

\[ \widehat{\text{Var}}(\theta \mid y) = \frac{N-1}{N} W + \frac{1}{N} B \]

The potential scale reduction factor (PSRF or \(\hat{R}\)) is:

\[ \hat{R} = \sqrt{\frac{\widehat{\text{Var}}(\theta \mid y)}{W}} \]

An \(\hat{R} \to 1.0\) indicates successful convergence.

Implementation Algorithm Computes coordinate-wise mean and variance vectors across all active chains, calculates the within-chain and between-chain variances, and returns the maximum coordinate value of \(\hat{R}\).