#!/usr/bin/env python
# -*- coding: utf-8 -*-
from __future__ import (division, print_function, absolute_import,
unicode_literals)
__all__ = ["PTSampler"]
import numpy as np
import numpy.random as nr
import multiprocessing as multi
from . import autocorr
from .sampler import Sampler
def default_beta_ladder(ndim, ntemps=None, Tmax=None):
"""Returns a ladder of :math:`\beta \equiv 1/T` with temperatures
geometrically spaced with spacing chosen so that a Gaussian
posterior would have a 0.25 temperature swap acceptance rate.
:param ndim:
The number of dimensions in the parameter space.
:param ntemps: (optional)
If set, the number of temperatures to use. If ``None``, the
``Tmax`` argument must be given, and the number of
temperatures is chosen so that the highest temperature is
greater than ``Tmax``.
:param Tmax: (optional)
If ``ntemps`` is not given, this argument controls the number
of temperatures. Temperatures are chosen according to the
spacing criteria until the maximum temperature exceeds
``Tmax``
"""
tstep = np.array([25.2741, 7., 4.47502, 3.5236, 3.0232,
2.71225, 2.49879, 2.34226, 2.22198, 2.12628,
2.04807, 1.98276, 1.92728, 1.87946, 1.83774,
1.80096, 1.76826, 1.73895, 1.7125, 1.68849,
1.66657, 1.64647, 1.62795, 1.61083, 1.59494,
1.58014, 1.56632, 1.55338, 1.54123, 1.5298,
1.51901, 1.50881, 1.49916, 1.49, 1.4813,
1.47302, 1.46512, 1.45759, 1.45039, 1.4435,
1.4369, 1.43056, 1.42448, 1.41864, 1.41302,
1.40761, 1.40239, 1.39736, 1.3925, 1.38781,
1.38327, 1.37888, 1.37463, 1.37051, 1.36652,
1.36265, 1.35889, 1.35524, 1.3517, 1.34825,
1.3449, 1.34164, 1.33847, 1.33538, 1.33236,
1.32943, 1.32656, 1.32377, 1.32104, 1.31838,
1.31578, 1.31325, 1.31076, 1.30834, 1.30596,
1.30364, 1.30137, 1.29915, 1.29697, 1.29484,
1.29275, 1.29071, 1.2887, 1.28673, 1.2848,
1.28291, 1.28106, 1.27923, 1.27745, 1.27569,
1.27397, 1.27227, 1.27061, 1.26898, 1.26737,
1.26579, 1.26424, 1.26271, 1.26121,
1.25973])
dmax = tstep.shape[0]
if ndim > dmax:
# An approximation to the temperature step at large
# dimension
tstep = 1.0 + 2.0*np.sqrt(np.log(4.0))/np.sqrt(ndim)
else:
tstep = tstep[ndim-1]
if ntemps is None and Tmax is None:
raise ValueError('must specify one of ``ntemps`` and ``Tmax``')
elif ntemps is None:
ntemps = int(np.log(Tmax)/np.log(tstep)+2)
return np.exp(np.linspace(0, -(ntemps-1)*np.log(tstep), ntemps))
class PTLikePrior(object):
"""
Wrapper class for logl and logp.
"""
def __init__(self, logl, logp, loglargs=[], logpargs=[], loglkwargs={},
logpkwargs={}):
self.logl = logl
self.logp = logp
self.loglargs = loglargs
self.logpargs = logpargs
self.loglkwargs = loglkwargs
self.logpkwargs = logpkwargs
def __call__(self, x):
lp = self.logp(x, *self.logpargs, **self.logpkwargs)
if lp == float('-inf'):
return lp, lp
return self.logl(x, *self.loglargs, **self.loglkwargs), lp
[docs]class PTSampler(Sampler):
"""
A parallel-tempered ensemble sampler, using :class:`EnsembleSampler`
for sampling within each parallel chain.
:param ntemps:
The number of temperatures. Can be ``None``, in which case
the ``Tmax`` argument sets the maximum temperature.
:param nwalkers:
The number of ensemble walkers at each temperature.
:param dim:
The dimension of parameter space.
:param logl:
The log-likelihood function.
:param logp:
The log-prior function.
:param threads: (optional)
The number of parallel threads to use in sampling.
:param pool: (optional)
Alternative to ``threads``. Any object that implements a
``map`` method compatible with the built-in ``map`` will do
here. For example, :class:`multi.Pool` will do.
:param betas: (optional)
Array giving the inverse temperatures, :math:`\\beta=1/T`,
used in the ladder. The default is chosen so that a Gaussian
posterior in the given number of dimensions will have a 0.25
tswap acceptance rate.
:param a: (optional)
Proposal scale factor.
:param Tmax: (optional)
Maximum temperature for the ladder. If ``ntemps`` is
``None``, this argument is used to set the temperature ladder.
:param loglargs: (optional)
Positional arguments for the log-likelihood function.
:param logpargs: (optional)
Positional arguments for the log-prior function.
:param loglkwargs: (optional)
Keyword arguments for the log-likelihood function.
:param logpkwargs: (optional)
Keyword arguments for the log-prior function.
"""
def __init__(self, ntemps, nwalkers, dim, logl, logp, threads=1,
pool=None, betas=None, a=2.0, Tmax=None, loglargs=[], logpargs=[],
loglkwargs={}, logpkwargs={}):
self.logl = logl
self.logp = logp
self.a = a
self.loglargs = loglargs
self.logpargs = logpargs
self.loglkwargs = loglkwargs
self.logpkwargs = logpkwargs
self.nwalkers = nwalkers
self.dim = dim
if betas is None:
self._betas = default_beta_ladder(self.dim, ntemps=ntemps, Tmax=Tmax)
else:
self._betas = betas
self.ntemps = self.betas.shape[0]
assert self.nwalkers % 2 == 0, \
"The number of walkers must be even."
assert self.nwalkers >= 2*self.dim, \
"The number of walkers must be greater than 2*dimension."
self._chain = None
self._lnprob = None
self._lnlikelihood = None
self.nswap = np.zeros(self.ntemps, dtype=np.float)
self.nswap_accepted = np.zeros(self.ntemps, dtype=np.float)
self.nprop = np.zeros((self.ntemps, self.nwalkers), dtype=np.float)
self.nprop_accepted = np.zeros((self.ntemps, self.nwalkers),
dtype=np.float)
self.pool = pool
if threads > 1 and pool is None:
self.pool = multi.Pool(threads)
[docs] def reset(self):
"""
Clear the ``chain``, ``lnprobability``, ``lnlikelihood``,
``acceptance_fraction``, ``tswap_acceptance_fraction`` stored
properties.
"""
self.nswap = np.zeros(self.ntemps, dtype=np.float)
self.nswap_accepted = np.zeros(self.ntemps, dtype=np.float)
self.nprop = np.zeros((self.ntemps, self.nwalkers), dtype=np.float)
self.nprop_accepted = np.zeros((self.ntemps, self.nwalkers),
dtype=np.float)
self._chain = None
self._lnprob = None
self._lnlikelihood = None
[docs] def sample(self, p0, lnprob0=None, lnlike0=None, iterations=1,
thin=1, storechain=True):
"""
Advance the chains ``iterations`` steps as a generator.
:param p0:
The initial positions of the walkers. Shape should be
``(ntemps, nwalkers, dim)``.
:param lnprob0: (optional)
The initial posterior values for the ensembles. Shape
``(ntemps, nwalkers)``.
:param lnlike0: (optional)
The initial likelihood values for the ensembles. Shape
``(ntemps, nwalkers)``.
:param iterations: (optional)
The number of iterations to preform.
:param thin: (optional)
The number of iterations to perform between saving the
state to the internal chain.
:param storechain: (optional)
If ``True`` store the iterations in the ``chain``
property.
At each iteration, this generator yields
* ``p``, the current position of the walkers.
* ``lnprob`` the current posterior values for the walkers.
* ``lnlike`` the current likelihood values for the walkers.
"""
p = np.copy(np.array(p0))
# If we have no lnprob or logls compute them
if lnprob0 is None or lnlike0 is None:
fn = PTLikePrior(self.logl, self.logp, self.loglargs,
self.logpargs, self.loglkwargs, self.logpkwargs)
if self.pool is None:
results = list(map(fn, p.reshape((-1, self.dim))))
else:
results = list(self.pool.map(fn, p.reshape((-1, self.dim))))
logls = np.array([r[0] for r in results]).reshape((self.ntemps,
self.nwalkers))
logps = np.array([r[1] for r in results]).reshape((self.ntemps,
self.nwalkers))
lnlike0 = logls
lnprob0 = logls * self.betas.reshape((self.ntemps, 1)) + logps
lnprob = lnprob0
logl = lnlike0
# Expand the chain in advance of the iterations
if storechain:
nsave = iterations // thin
if self._chain is None:
isave = 0
self._chain = np.zeros((self.ntemps, self.nwalkers, nsave,
self.dim))
self._lnprob = np.zeros((self.ntemps, self.nwalkers, nsave))
self._lnlikelihood = np.zeros((self.ntemps, self.nwalkers,
nsave))
else:
isave = self._chain.shape[2]
self._chain = np.concatenate((self._chain,
np.zeros((self.ntemps,
self.nwalkers,
nsave, self.dim))),
axis=2)
self._lnprob = np.concatenate((self._lnprob,
np.zeros((self.ntemps,
self.nwalkers,
nsave))),
axis=2)
self._lnlikelihood = np.concatenate((self._lnlikelihood,
np.zeros((self.ntemps,
self.nwalkers,
nsave))),
axis=2)
for i in range(iterations):
for j in [0, 1]:
jupdate = j
jsample = (j + 1) % 2
pupdate = p[:, jupdate::2, :]
psample = p[:, jsample::2, :]
zs = np.exp(np.random.uniform(low=-np.log(self.a), high=np.log(self.a), size=(self.ntemps, self.nwalkers//2)))
qs = np.zeros((self.ntemps, self.nwalkers//2, self.dim))
for k in range(self.ntemps):
js = np.random.randint(0, high=self.nwalkers // 2,
size=self.nwalkers // 2)
qs[k, :, :] = psample[k, js, :] + zs[k, :].reshape(
(self.nwalkers // 2, 1)) * (pupdate[k, :, :] -
psample[k, js, :])
fn = PTLikePrior(self.logl, self.logp, self.loglargs,
self.logpargs, self.loglkwargs,
self.logpkwargs)
if self.pool is None:
results = list(map(fn, qs.reshape((-1, self.dim))))
else:
results = list(self.pool.map(fn, qs.reshape((-1,
self.dim))))
qslogls = np.array([r[0] for r in results]).reshape(
(self.ntemps, self.nwalkers//2))
qslogps = np.array([r[1] for r in results]).reshape(
(self.ntemps, self.nwalkers//2))
qslnprob = qslogls * self.betas.reshape((self.ntemps, 1)) \
+ qslogps
logpaccept = self.dim*np.log(zs) + qslnprob \
- lnprob[:, jupdate::2]
logrs = np.log(np.random.uniform(low=0.0, high=1.0,
size=(self.ntemps,
self.nwalkers//2)))
accepts = logrs < logpaccept
accepts = accepts.flatten()
pupdate.reshape((-1, self.dim))[accepts, :] = \
qs.reshape((-1, self.dim))[accepts, :]
lnprob[:, jupdate::2].reshape((-1,))[accepts] = \
qslnprob.reshape((-1,))[accepts]
logl[:, jupdate::2].reshape((-1,))[accepts] = \
qslogls.reshape((-1,))[accepts]
accepts = accepts.reshape((self.ntemps, self.nwalkers//2))
self.nprop[:, jupdate::2] += 1.0
self.nprop_accepted[:, jupdate::2] += accepts
p, lnprob, logl = self._temperature_swaps(p, lnprob, logl)
if (i + 1) % thin == 0:
if storechain:
self._chain[:, :, isave, :] = p
self._lnprob[:, :, isave, ] = lnprob
self._lnlikelihood[:, :, isave] = logl
isave += 1
yield p, lnprob, logl
def _temperature_swaps(self, p, lnprob, logl):
"""
Perform parallel-tempering temperature swaps on the state
in ``p`` with associated ``lnprob`` and ``logl``.
"""
ntemps = self.ntemps
for i in range(ntemps - 1, 0, -1):
bi = self.betas[i]
bi1 = self.betas[i - 1]
dbeta = bi1 - bi
iperm = nr.permutation(self.nwalkers)
i1perm = nr.permutation(self.nwalkers)
raccept = np.log(nr.uniform(size=self.nwalkers))
paccept = dbeta * (logl[i, iperm] - logl[i - 1, i1perm])
self.nswap[i] += self.nwalkers
self.nswap[i - 1] += self.nwalkers
asel = (paccept > raccept)
nacc = np.sum(asel)
self.nswap_accepted[i] += nacc
self.nswap_accepted[i - 1] += nacc
ptemp = np.copy(p[i, iperm[asel], :])
ltemp = np.copy(logl[i, iperm[asel]])
prtemp = np.copy(lnprob[i, iperm[asel]])
p[i, iperm[asel], :] = p[i - 1, i1perm[asel], :]
logl[i, iperm[asel]] = logl[i - 1, i1perm[asel]]
lnprob[i, iperm[asel]] = lnprob[i - 1, i1perm[asel]] \
- dbeta * logl[i - 1, i1perm[asel]]
p[i - 1, i1perm[asel], :] = ptemp
logl[i - 1, i1perm[asel]] = ltemp
lnprob[i - 1, i1perm[asel]] = prtemp + dbeta * ltemp
return p, lnprob, logl
[docs] def thermodynamic_integration_log_evidence(self, logls=None, fburnin=0.1):
"""
Thermodynamic integration estimate of the evidence.
:param logls: (optional) The log-likelihoods to use for
computing the thermodynamic evidence. If ``None`` (the
default), use the stored log-likelihoods in the sampler.
Should be of shape ``(Ntemps, Nwalkers, Nsamples)``.
:param fburnin: (optional)
The fraction of the chain to discard as burnin samples; only the
final ``1-fburnin`` fraction of the samples will be used to
compute the evidence; the default is ``fburnin = 0.1``.
:return ``(lnZ, dlnZ)``: Returns an estimate of the
log-evidence and the error associated with the finite
number of temperatures at which the posterior has been
sampled.
The evidence is the integral of the un-normalized posterior
over all of parameter space:
.. math::
Z \\equiv \\int d\\theta \\, l(\\theta) p(\\theta)
Thermodymanic integration is a technique for estimating the
evidence integral using information from the chains at various
temperatures. Let
.. math::
Z(\\beta) = \\int d\\theta \\, l^\\beta(\\theta) p(\\theta)
Then
.. math::
\\frac{d \\ln Z}{d \\beta}
= \\frac{1}{Z(\\beta)} \\int d\\theta l^\\beta p \\ln l
= \\left \\langle \\ln l \\right \\rangle_\\beta
so
.. math::
\\ln Z(\\beta = 1)
= \\int_0^1 d\\beta \\left \\langle \\ln l \\right\\rangle_\\beta
By computing the average of the log-likelihood at the
difference temperatures, the sampler can approximate the above
integral.
"""
if logls is None:
return self.thermodynamic_integration_log_evidence(
logls=self.lnlikelihood, fburnin=fburnin)
else:
betas = np.concatenate((self.betas, np.array([0])))
betas2 = np.concatenate((self.betas[::2], np.array([0])))
istart = int(logls.shape[2] * fburnin + 0.5)
mean_logls = np.mean(np.mean(logls, axis=1)[:, istart:], axis=1)
mean_logls2 = mean_logls[::2]
lnZ = -np.dot(mean_logls, np.diff(betas))
lnZ2 = -np.dot(mean_logls2, np.diff(betas2))
return lnZ, np.abs(lnZ - lnZ2)
@property
def betas(self):
"""
Returns the sequence of inverse temperatures in the ladder.
"""
return self._betas
@property
def chain(self):
"""
Returns the stored chain of samples; shape ``(Ntemps,
Nwalkers, Nsteps, Ndim)``.
"""
return self._chain
@property
def flatchain(self):
"""Returns the stored chain, but flattened along the walker axis, so
of shape ``(Ntemps, Nwalkers*Nsteps, Ndim)``.
"""
s = self.chain.shape
return self._chain.reshape((s[0], -1, s[3]))
@property
def lnprobability(self):
"""
Matrix of lnprobability values; shape ``(Ntemps, Nwalkers, Nsteps)``.
"""
return self._lnprob
@property
def lnlikelihood(self):
"""
Matrix of ln-likelihood values; shape ``(Ntemps, Nwalkers, Nsteps)``.
"""
return self._lnlikelihood
@property
def tswap_acceptance_fraction(self):
"""
Returns an array of accepted temperature swap fractions for
each temperature; shape ``(ntemps, )``.
"""
return self.nswap_accepted / self.nswap
@property
def acceptance_fraction(self):
"""
Matrix of shape ``(Ntemps, Nwalkers)`` detailing the
acceptance fraction for each walker.
"""
return self.nprop_accepted / self.nprop
@property
def acor(self):
"""
Returns a matrix of autocorrelation lengths for each
parameter in each temperature of shape ``(Ntemps, Ndim)``.
"""
return self.get_autocorr_time()
[docs] def get_autocorr_time(self, **kwargs):
"""
Returns a matrix of autocorrelation lengths for each
parameter in each temperature of shape ``(Ntemps, Ndim)``.
Any arguments will be passed to :func:`autocorr.integrate_time`.
"""
acors = np.zeros((self.ntemps, self.dim))
for i in range(self.ntemps):
x = np.mean(self._chain[i, :, :, :], axis=0)
acors[i, :] = autocorr.integrated_time(x, **kwargs)
return acors