Relativistic MCMC

class pints.RelativisticMCMC(x0, sigma0=None)[source]

Implements Relativistic Monte Carlo as described in [1].

Uses a physical analogy of a particle moving across a landscape under Hamiltonian dynamics to aid efficient exploration of parameter space. Introduces an auxilary variable – the momentum (p_i) of a particle moving in dimension i of negative log posterior space – which supplements the position (q_i) of the particle in parameter space. The particle’s motion is dictated by solutions to Hamilton’s equations,

\[\begin{split}dq_i/dt &= \partial H/\partial p_i\\ dp_i/dt &= - \partial H/\partial q_i.\end{split}\]

The Hamiltonian is given by,

\[\begin{split}H(q,p) &= U(q) + KE(p)\\ &= -\text{log}(p(q|X)p(q)) + mc^2 (\Sigma_{i=1}^{d} p_i^2 / (m^2 c^2) + 1)^{0.5}\end{split}\]

where d is the dimensionality of model, m is the scalar ‘mass’ given to each particle (chosen to be 1 as default) and c is the speed of light (chosen to be 10 by default).

To numerically integrate Hamilton’s equations, it is essential to use a sympletic discretisation routine, of which the most typical approach is the leapfrog method,

\[\begin{split}p_i(t + \epsilon/2) &= p_i(t) - (\epsilon/2) d U(q_i(t))/dq_i\\ q_i(t + \epsilon) &= q_i(t) + \epsilon M^{-1}(p_i(t + \epsilon/2)) p_i(t + \epsilon/2)\\ p_i(t + \epsilon) &= p_i(t + \epsilon/2) - (\epsilon/2) d U(q_i(t + \epsilon))/dq_i\end{split}\]

where relativistic mass (a scalar) is,

\[M(p) = m (\Sigma_{i=1}^{d} p_i^2 / (m^2 c^2) + 1)^{0.5}\]

In particular, the algorithm we implement follows eqs. in section 2.1 of [1].

Extends SingleChainMCMC.

References

ask()[source]

See SingleChainMCMC.ask().

divergent_iterations()[source]

Returns the iteration number of any divergent iterations

epsilon()[source]

Returns epsilon used in leapfrog algorithm

hamiltonian_threshold()[source]

Returns threshold difference in Hamiltonian value from one iteration to next which determines whether an iteration is divergent.

in_initial_phase()

For methods that need an initial phase (see needs_initial_phase()), this method returns True if the method is currently configured to be in its initial phase. For other methods a NotImplementedError is returned.

leapfrog_step_size()[source]

Returns the step size for the leapfrog algorithm.

leapfrog_steps()[source]

Returns the number of leapfrog steps to carry out for each iteration.

mass()[source]

Returns mass which is the rest mass of particle.

n_hyper_parameters()[source]

See TunableMethod.n_hyper_parameters().

name()[source]

See pints.MCMCSampler.name().

needs_initial_phase()

Returns True if this method needs an initial phase, for example an adaptation-free period for adaptive covariance methods, or a warm-up phase for DREAM.

needs_sensitivities()[source]

See pints.MCMCSampler.needs_sensitivities().

replace(current, current_log_pdf, proposed=None)

Replaces the internal current position, current LogPDF, and proposed point (if any) by the user-specified values.

This method can only be used once the initial position and LogPDF have been set (so after at least 1 round of ask-and-tell).

This is an optional method, and some samplers may not support it.

scaled_epsilon()[source]

Returns scaled epsilon used in leapfrog algorithm

set_epsilon(epsilon)[source]

Sets epsilon for the leapfrog algorithm

set_hamiltonian_threshold(hamiltonian_threshold)[source]

Sets threshold difference in Hamiltonian value from one iteration to next which determines whether an iteration is divergent.

set_hyper_parameters(x)[source]

The hyper-parameter vector is [leapfrog_steps, leapfrog_step_size, mass, c].

See TunableMethod.set_hyper_parameters().

set_initial_phase(in_initial_phase)

For methods that need an initial phase (see needs_initial_phase()), this method toggles the initial phase algorithm. For other methods a NotImplementedError is returned.

set_leapfrog_step_size(step_size)[source]

Sets the step size for the leapfrog algorithm.

set_leapfrog_steps(steps)[source]

Sets the number of leapfrog steps to carry out for each iteration.

set_mass(mass)[source]

Sets scalar mass.

set_speed_of_light(c)[source]

Sets speed of light.

speed_of_light()[source]

Returns speed of light.

tell(reply)[source]

See pints.SingleChainMCMC.tell().