""" Various Monte Carlo equilibrium sampling algorithms, which simulate only one Markov chain. Here is how to sample from a PDF using the L{HMCSampler} class. In the following snippet we draw 5000 samples from a 1D normal distribution and plot them: >>> import numpy >>> from csb.io.plots import Chart >>> from csb.statistics.pdf import Normal >>> from csb.statistics.samplers import State >>> from csb.statistics.samplers.mc.singlechain import HMCSampler >>> initial_state = State(numpy.array([1.])) >>> grad = lambda q, t: q >>> timestep = 1.5 >>> nsteps = 30 >>> nsamples = 5000 >>> sampler = HMCSampler(Normal(), initial_state, grad, timestep, nsteps) >>> states = [] >>> for i in range(nsamples): sampler.sample() states.append(sampler.state) >>> print('acceptance rate:', sampler.acceptance_rate) 0.8 >>> states = [state.position[0]for state in states] >>> chart = Chart() >>> chart.plot.hist([numpy.random.normal(size=5000), states], bins=20, normed=True) >>> chart.plot.legend(['numpy.random.normal', 'HMC']) >>> chart.show() First, several things which are being needed are imported. As every sampler in this module implements a Markov Chain, an initial state has to be chosen. In the following lines, several parameters are set: - the gradient of the negative log-probability of the PDF under consideration - the integration timestep - the number of integration steps to be performed in each iteration, that is, the HMC trajectory length - the number of samples to be drawn The empty list states is initialized. It will serve to store the samples drawn. In the loop, C{sampler.sample()} is repeatedly called. After each call of C{sampler.sample()}, the current state of the Markov Chain is stored in sampler.state and this state is appended to the sample storage list. Then the acceptance rate is printed, the numeric values are being extracted from the L{State} objects in states, a histogram is created and finally plotted. """ import numpy import csb.numeric import csb.core from abc import ABCMeta, abstractmethod from csb.statistics.samplers import State from csb.statistics.samplers.mc import AbstractMC, MCCollection, augment_state from csb.statistics.samplers.mc.propagators import MDPropagator from csb.statistics.samplers.mc.neqsteppropagator import NonequilibriumStepPropagator from csb.numeric.integrators import FastLeapFrog from csb.numeric import InvertibleMatrix class AbstractSingleChainMC(AbstractMC): """ Abstract class for Monte Carlo sampling algorithms simulating only one ensemble. @param pdf: probability density function to sample from @type pdf: subclass of L{csb.statistics.pdf.AbstractDensity} @param state: Initial state @type state: L{State} @param temperature: Pseudo-temperature of the Boltzmann ensemble M{p(x) = 1/N * exp(-1/T * E(x))} with the pseudo-energy defined as M{E(x) = -log(p(x))} where M{p(x)} is the PDF under consideration @type temperature: float """ __metaclass__ = ABCMeta def __init__(self, pdf, state, temperature=1.): super(AbstractSingleChainMC, self).__init__(state) self._pdf = pdf self._temperature = temperature self._nmoves = 0 self._accepted = 0 self._last_move_accepted = None def _checkstate(self, state): if not isinstance(state, State): raise TypeError(state) def sample(self): """ Draw a sample. @rtype: L{State} """ proposal_communicator = self._propose() pacc = self._calc_pacc(proposal_communicator) accepted = None if numpy.random.random() < pacc: accepted = True else: accepted = False if accepted == True: self._accept_proposal(proposal_communicator.proposal_state) self._update_statistics(accepted) self._last_move_accepted = accepted return self.state @abstractmethod def _propose(self): """ Calculate a new proposal state and gather additional information needed to calculate the acceptance probability. @rtype: L{SimpleProposalCommunicator} """ pass @abstractmethod def _calc_pacc(self, proposal_communicator): """ Calculate probability with which to accept the proposal. @param proposal_communicator: Contains information about the proposal and additional information needed to calculate the acceptance probability @type proposal_communicator: L{SimpleProposalCommunicator} """ pass def _accept_proposal(self, proposal_state): """ Accept the proposal state by setting it as the current state of the sampler object @param proposal_state: The proposal state @type proposal_state: L{State} """ self.state = proposal_state def _update_statistics(self, accepted): """ Update the sampling statistics. @param accepted: Whether or not the proposal state has been accepted @type accepted: boolean """ self._nmoves += 1 self._accepted += int(accepted) @property def energy(self): """ Negative log-likelihood of the current state. @rtype: float """ return -self._pdf.log_prob(self.state.position) @property def acceptance_rate(self): """ Acceptance rate. """ if self._nmoves > 0: return float(self._accepted) / float(self._nmoves) else: return 0.0 @property def last_move_accepted(self): """ Information whether the last MC move was accepted or not. """ return self._last_move_accepted @property def temperature(self): return self._temperature class HMCSampler(AbstractSingleChainMC): """ Hamilton Monte Carlo (HMC, also called Hybrid Monte Carlo by the inventors, Duane, Kennedy, Pendleton, Duncan 1987). @param pdf: Probability density function to be sampled from @type pdf: L{csb.statistics.pdf.AbstractDensity} @param state: Inital state @type state: L{State} @param gradient: Gradient of the negative log-probability @type gradient: L{AbstractGradient} @param timestep: Timestep used for integration @type timestep: float @param nsteps: Number of integration steps to be performed in each iteration @type nsteps: int @param mass_matrix: Mass matrix @type mass_matrix: n-dimensional L{InvertibleMatrix} with n being the dimension of the configuration space, that is, the dimension of the position / momentum vectors @param integrator: Subclass of L{AbstractIntegrator} to be used for integrating Hamiltionian equations of motion @type integrator: L{AbstractIntegrator} @param temperature: Pseudo-temperature of the Boltzmann ensemble M{p(x) = 1/N * exp(-1/T * E(x))} with the pseudo-energy defined as M{E(x) = -log(p(x))} where M{p(x)} is the PDF under consideration @type temperature: float """ def __init__(self, pdf, state, gradient, timestep, nsteps, mass_matrix=None, integrator=FastLeapFrog, temperature=1.): super(HMCSampler, self).__init__(pdf, state, temperature) self._timestep = None self.timestep = timestep self._nsteps = None self.nsteps = nsteps self._mass_matrix = None self._momentum_covariance_matrix = None self._integrator = integrator self._gradient = gradient self._setup_matrices(mass_matrix) self._propagator = self._propagator_factory() def _setup_matrices(self, mass_matrix): self._d = len(self.state.position) self._mass_matrix = mass_matrix if self.mass_matrix is None: self.mass_matrix = InvertibleMatrix(numpy.eye(self._d), numpy.eye(self._d)) self._momentum_covariance_matrix = self._temperature * self.mass_matrix def _propagator_factory(self): """ Factory which produces a L{MDPropagator} object initialized with the MD parameters given in __init__(). @return: L{MDPropagator} instance @rtype: L{MDPropagator} """ return MDPropagator(self._gradient, self._timestep, mass_matrix=self._mass_matrix, integrator=self._integrator) def _propose(self): current_state = self.state.clone() current_state = augment_state(current_state, self.temperature, self.mass_matrix) proposal_state = self._propagator.generate(current_state, self._nsteps).final return SimpleProposalCommunicator(current_state, proposal_state) def _hamiltonian(self, state): """ Evaluates the Hamiltonian consisting of the negative log-probability and a quadratic kinetic term. @param state: State on which the Hamiltonian should be evaluated @type state: L{State} @return: Value of the Hamiltonian (total energy) @rtype: float """ V = lambda q: -self._pdf.log_prob(q) T = lambda p: 0.5 * numpy.dot(p.T, numpy.dot(self.mass_matrix.inverse, p)) return T(state.momentum) + V(state.position) def _calc_pacc(self, proposal_communicator): cs = proposal_communicator.current_state ps = proposal_communicator.proposal_state pacc = csb.numeric.exp(-(self._hamiltonian(ps) - self._hamiltonian(cs)) / self.temperature) if self.state.momentum is None: proposal_communicator.proposal_state.momentum = None else: proposal_communicator.proposal_state.momentum = self.state.momentum return pacc @property def timestep(self): return self._timestep @timestep.setter def timestep(self, value): self._timestep = float(value) if "_propagator" in dir(self): self._propagator.timestep = self._timestep @property def nsteps(self): return self._nsteps @nsteps.setter def nsteps(self, value): self._nsteps = int(value) @property def mass_matrix(self): return self._mass_matrix @mass_matrix.setter def mass_matrix(self, value): self._mass_matrix = value if "_propagator" in dir(self): self._propagator.mass_matrix = self._mass_matrix class RWMCSampler(AbstractSingleChainMC): """ Random Walk Metropolis Monte Carlo implementation (Metropolis, Rosenbluth, Teller, Teller 1953; Hastings, 1970). @param pdf: Probability density function to be sampled from @type pdf: L{csb.statistics.pdf.AbstractDensity} @param state: Inital state @type state: L{State} @param stepsize: Serves to set the step size in proposal_density, e.g. for automatic acceptance rate adaption @type stepsize: float @param proposal_density: The proposal density as a function f(x, s) of the current state x and the stepsize s. By default, the proposal density is uniform, centered around x, and has width s. @type proposal_density: callable @param temperature: Pseudo-temperature of the Boltzmann ensemble M{p(x) = 1/N * exp(-1/T * E(x))} with the pseudo-energy defined as M{E(x) = -log(p(x))} where M{p(x)} is the PDF under consideration @type temperature: float """ def __init__(self, pdf, state, stepsize=1., proposal_density=None, temperature=1.): super(RWMCSampler, self).__init__(pdf, state, temperature) self._stepsize = None self.stepsize = stepsize if proposal_density == None: self._proposal_density = lambda x, s: x.position + \ s * numpy.random.uniform(size=x.position.shape, low=-1., high=1.) else: self._proposal_density = proposal_density def _propose(self): current_state = self.state.clone() proposal_state = self.state.clone() proposal_state.position = self._proposal_density(current_state, self.stepsize) return SimpleProposalCommunicator(current_state, proposal_state) def _calc_pacc(self, proposal_communicator): current_state = proposal_communicator.current_state proposal_state = proposal_communicator.proposal_state E = lambda x: -self._pdf.log_prob(x) pacc = csb.numeric.exp((-(E(proposal_state.position) - E(current_state.position))) / self.temperature) return pacc @property def stepsize(self): return self._stepsize @stepsize.setter def stepsize(self, value): self._stepsize = float(value) class AbstractNCMCSampler(AbstractSingleChainMC): """ Implementation of the NCMC sampling algorithm (Nilmeier et al., "Nonequilibrium candidate Monte Carlo is an efficient tool for equilibrium simulation", PNAS 2011) for sampling from one ensemble only. Subclasses have to specify the acceptance probability, which depends on the kind of perturbations and propagations in the protocol. @param state: Inital state @type state: L{State} @param protocol: Nonequilibrium protocol with alternating perturbations and propagations @type protocol: L{Protocol} @param reverse_protocol: The reversed version of the protocol, that is, the order of perturbations and propagations in each step is reversed @type reverse_protocol: L{Protocol} """ __metaclass__ = ABCMeta def __init__(self, state, protocol, reverse_protocol): self._protocol = None self.protocol = protocol self._reverse_protocol = None self.reverse_protocol = reverse_protocol pdf = self.protocol.steps[0].perturbation.sys_before.hamiltonian temperature = self.protocol.steps[0].perturbation.sys_before.hamiltonian.temperature super(AbstractNCMCSampler, self).__init__(pdf, state, temperature) def _pick_protocol(self): """ Picks either the protocol or the reversed protocol with equal probability. @return: Either the protocol or the reversed protocol @rtype: L{Protocol} """ if numpy.random.random() < 0.5: return self.protocol else: return self.reverse_protocol def _propose(self): protocol = self._pick_protocol() gen = NonequilibriumStepPropagator(protocol) traj = gen.generate(self.state) return NCMCProposalCommunicator(traj) def _accept_proposal(self, proposal_state): if self.state.momentum is not None: proposal_state.momentum *= -1.0 else: proposal_state.momentum = None super(AbstractNCMCSampler, self)._accept_proposal(proposal_state) @property def protocol(self): return self._protocol @protocol.setter def protocol(self, value): self._protocol = value @property def reverse_protocol(self): return self._reverse_protocol @reverse_protocol.setter def reverse_protocol(self, value): self._reverse_protocol = value class SimpleProposalCommunicator(object): """ This holds all the information needed to calculate the acceptance probability in both the L{RWMCSampler} and L{HMCSampler} classes, that is, only the proposal state. For more advanced algorithms, one may derive classes capable of holding the neccessary additional information from this class. @param current_state: Current state @type current_state: L{State} @param proposal_state: Proposal state @type proposal_state: L{State} """ __metaclass__ = ABCMeta def __init__(self, current_state, proposal_state): self._current_state = current_state self._proposal_state = proposal_state @property def current_state(self): return self._current_state @property def proposal_state(self): return self._proposal_state class NCMCProposalCommunicator(SimpleProposalCommunicator): """ Holds all information (that is, the trajectory with heat, work, Hamiltonian difference and jacbian) needed to calculate the acceptance probability in the AbstractNCMCSampler class. @param traj: Non-equilibrium trajectory stemming from a stepwise protocol @type traj: NCMCTrajectory """ def __init__(self, traj): self._traj = None self.traj = traj super(NCMCProposalCommunicator, self).__init__(traj.initial, traj.final)