def compute_perrakis_estimate(marginal_sample,
lnlikefunc,
lnpriorfunc,
lnlikeargs=(),
lnpriorargs=(),
densityestimation='histogram',
**kwargs):
"""
Computes the Perrakis estimate of the bayesian evidence.
The estimation is based on n marginal posterior samples
(indexed by s, with s = 0, ..., n-1).
:param array marginal_sample:
A sample from the parameter marginal posterior distribution.
Dimensions are (n x k), where k is the number of parameters.
:param callable lnlikefunc:
Function to compute ln(likelihood) on the marginal samples.
:param callable lnpriorfunc:
Function to compute ln(prior density) on the marginal samples.
:param tuple lnlikeargs:
Extra arguments passed to the likelihood function.
:param tuple lnpriorargs:
Extra arguments passed to the lnprior function.
:param str densityestimation:
The method used to estimate the marginal posterior density of each
model parameter ("normal", "kde", or "histogram").
Other parameters
----------------
:param kwargs:
Additional arguments passed to estimate_density function.
:return:
References
----------
Perrakis et al. (2014; arXiv:1311.0674)
"""
if not isinstance(marginal_sample, np.ndarray):
marginal_sample = np.array(marginal_sample)
number_parameters = marginal_sample.shape[1]
##
# Estimate marginal posterior density for each parameter.
marginal_posterior_density = np.zeros(marginal_sample.shape)
for parameter_index in range(number_parameters):
# Extract samples for this parameter.
x = marginal_sample[:, parameter_index]
# Estimate density with method "densityestimation".
marginal_posterior_density[:, parameter_index] = \
estimate_density(x, method=densityestimation, **kwargs)
# Compute produt of marginal posterior densities for all parameters
prod_marginal_densities = marginal_posterior_density.prod(axis=1)
##
##
# Compute lnprior and likelihood in marginal sample.
log_prior = lnpriorfunc(marginal_sample, *lnpriorargs)
log_likelihood = lnlikefunc(marginal_sample, *lnlikeargs)
##
# Mask values with zero likelihood (a problem in lnlike)
cond = log_likelihood != 0
# Use identity for summation
# http://en.wikipedia.org/wiki/List_of_logarithmic_identities#Summation.2Fsubtraction
# ln(sum(x)) = ln(x[0]) + ln(1 + sum( exp( ln(x[1:]) - ln(x[0]) ) ) )
# log_summands = log_likelihood[cond] + np.log(prior_probability[cond])
# - np.log(prod_marginal_densities[cond])
log_summands = (log_likelihood[cond] + log_prior[cond] -
np.log(prod_marginal_densities[cond]))
perr = lib.log_sum(log_summands) - log(len(log_summands))
return perr
def compute_perrakis_estimate(marginal_sample, lnlikefunc, lnpriorfunc,
lnlikeargs=(), lnpriorargs=(),
densityestimation='histogram', **kwargs):
"""
Computes the Perrakis estimate of the bayesian evidence.
The estimation is based on n marginal posterior samples
(indexed by s, with s = 0, ..., n-1).
:param array marginal_sample:
A sample from the parameter marginal posterior distribution.
Dimensions are (n x k), where k is the number of parameters.
:param callable lnlikefunc:
Function to compute ln(likelihood) on the marginal samples.
:param callable lnpriorfunc:
Function to compute ln(prior density) on the marginal samples.
:param tuple lnlikeargs:
Extra arguments passed to the likelihood function.
:param tuple lnpriorargs:
Extra arguments passed to the lnprior function.
:param str densityestimation:
The method used to estimate the marginal posterior density of each
model parameter ("normal", "kde", or "histogram").
Other parameters
----------------
:param kwargs:
Additional arguments passed to estimate_density function.
:return:
References
----------
Perrakis et al. (2014; arXiv:1311.0674)
"""
if not isinstance(marginal_sample, np.ndarray):
marginal_sample = np.array(marginal_sample)
number_parameters = marginal_sample.shape[1]
##
# Estimate marginal posterior density for each parameter.
marginal_posterior_density = np.zeros(marginal_sample.shape)
for parameter_index in range(number_parameters):
# Extract samples for this parameter.
x = marginal_sample[:, parameter_index]
# Estimate density with method "densityestimation".
marginal_posterior_density[:, parameter_index] = \
estimate_density(x, method=densityestimation, **kwargs)
# Compute produt of marginal posterior densities for all parameters
prod_marginal_densities = marginal_posterior_density.prod(axis=1)
##
##
# Compute lnprior and likelihood in marginal sample.
log_prior = lnpriorfunc(marginal_sample, *lnpriorargs)
log_likelihood = lnlikefunc(marginal_sample, *lnlikeargs)
##
# Mask values with zero likelihood (a problem in lnlike)
cond = log_likelihood != 0
# Use identity for summation
# http://en.wikipedia.org/wiki/List_of_logarithmic_identities#Summation.2Fsubtraction
# ln(sum(x)) = ln(x[0]) + ln(1 + sum( exp( ln(x[1:]) - ln(x[0]) ) ) )
# log_summands = log_likelihood[cond] + np.log(prior_probability[cond])
# - np.log(prod_marginal_densities[cond])
log_summands = (log_likelihood[cond] + log_prior[cond] -
np.log(prod_marginal_densities[cond])
)
perr = lib.log_sum(log_summands) - log(len(log_summands))
return perr
def compute_harmonicmean(lnlike_post, posterior_sample=None, lnlikefunc=None,
lnlikeargs=(), **kwargs):
"""
Computes the harmonic mean estimate of the marginal likelihood.
The estimation is based on n posterior samples
(indexed by s, with s = 0, ..., n-1), but can be done directly if the
log(likelihood) in this sample is passed.
:param array lnlike_post:
log(likelihood) computed over a posterior sample. 1-D array of length n.
If an emply array is given, then compute from posterior sample.
:param array posterior_sample:
A sample from the parameter posterior distribution.
Dimensions are (n x k), where k is the number of parameters. If None
the computation is done using the log(likelihood) obtained from the
posterior sample.
:param callable lnlikefunc:
Function to compute ln(likelihood) on the marginal samples.
:param tuple lnlikeargs:
Extra arguments passed to the likelihood function.
Other parameters
----------------
:param int size:
Size of sample to use for computation. If none is given, use size of
given array or posterior sample.
References
----------
Kass & Raftery (1995), JASA vol. 90, N. 430, pp. 773-795
"""
if len(lnlike_post) == 0 and posterior_sample is not None:
samplesize = kwargs.pop('size', len(posterior_sample))
if samplesize < len(posterior_sample):
posterior_subsample = numpy.random.choice(posterior_sample,
size=samplesize,
replace=False)
else:
posterior_subsample = posterior_sample.copy()
# Compute log likelihood in posterior sample.
log_likelihood = lnlikefunc(posterior_subsample, *lnlikeargs)
elif len(lnlike_post) > 0:
samplesize = kwargs.pop('size', len(lnlike_post))
log_likelihood = numpy.random.choice(lnlike_post, size=samplesize,
replace=False)
# Use identity for summation
# http://en.wikipedia.org/wiki/List_of_logarithmic_identities#Summation.2Fsubtraction
# ln(sum(x)) = ln(x[0]) + ln(1 + sum( exp( ln(x[1:]) - ln(x[0]) ) ) )
hme = -lib.log_sum(-log_likelihood) + log(len(log_likelihood))
return hme
def compute_harmonicmean(lnlike_post,
posterior_sample=None,
lnlikefunc=None,
lnlikeargs=(),
**kwargs):
"""
Computes the harmonic mean estimate of the marginal likelihood.
The estimation is based on n posterior samples
(indexed by s, with s = 0, ..., n-1), but can be done directly if the
log(likelihood) in this sample is passed.
:param array lnlike_post:
log(likelihood) computed over a posterior sample. 1-D array of length n.
If an emply array is given, then compute from posterior sample.
:param array posterior_sample:
A sample from the parameter posterior distribution.
Dimensions are (n x k), where k is the number of parameters. If None
the computation is done using the log(likelihood) obtained from the
posterior sample.
:param callable lnlikefunc:
Function to compute ln(likelihood) on the marginal samples.
:param tuple lnlikeargs:
Extra arguments passed to the likelihood function.
Other parameters
----------------
:param int size:
Size of sample to use for computation. If none is given, use size of
given array or posterior sample.
References
----------
Kass & Raftery (1995), JASA vol. 90, N. 430, pp. 773-795
"""
if len(lnlike_post) == 0 and posterior_sample is not None:
samplesize = kwargs.pop('size', len(posterior_sample))
if samplesize < len(posterior_sample):
posterior_subsample = numpy.random.choice(posterior_sample,
size=samplesize,
replace=False)
else:
posterior_subsample = posterior_sample.copy()
# Compute log likelihood in posterior sample.
log_likelihood = lnlikefunc(posterior_subsample, *lnlikeargs)
elif len(lnlike_post) > 0:
samplesize = kwargs.pop('size', len(lnlike_post))
log_likelihood = numpy.random.choice(lnlike_post,
size=samplesize,
replace=False)
# Use identity for summation
# http://en.wikipedia.org/wiki/List_of_logarithmic_identities#Summation.2Fsubtraction
# ln(sum(x)) = ln(x[0]) + ln(1 + sum( exp( ln(x[1:]) - ln(x[0]) ) ) )
hme = -lib.log_sum(-log_likelihood) + log(len(log_likelihood))
return hme