import numpy as np
from typing import Callable, Dict, List, Sequence, Tuple
from copy import deepcopy
from .function import ObjectiveBase
from .aggregated import AggregatedObjective
from .constants import MODE_FUN, MODE_RES, FVAL, GRAD, HESS
from .base import ResultDict
[docs]class NegLogPriors(AggregatedObjective):
"""
Aggregates different forms of negative log-prior distributions.
Allows to distinguish priors from the likelihood by testing the type of
an objective.
Consists basically of a list of individual negative log-priors,
given in self.objectives.
"""
[docs]class NegLogParameterPriors(ObjectiveBase):
"""
This class implements Negative Log Priors on Parameters.
Contains a list of prior dictionaries for the individual parameters
of the format
{'index': [int],
'density_fun': [Callable],
'density_dx': [Callable],
'density_ddx': [Callable]}
A prior instance can be added to e.g. an objective, that gives the
likelihood, by an AggregatedObjective.
Notes
-----
All callables should correspond to log-densities. That is, they return
log-densities and their corresponding derivatives.
Internally, values are multiplied by -1, since pyPESTO expects the
Objective function to be of a negative log-density type.
"""
[docs] def __init__(self,
prior_list: List[Dict],
x_names: Sequence[str] = None):
"""
Constructor
Parameters
----------
prior_list:
List of dicts containing the individual parameter priors.
Format see above.
x_names:
Sequence of parameter names (optional).
"""
self.prior_list = prior_list
super().__init__(x_names)
def __deepcopy__(self, memodict=None):
other = NegLogParameterPriors(deepcopy(self.prior_list))
return other
[docs] def call_unprocessed(
self,
x: np.ndarray,
sensi_orders: Tuple[int, ...],
mode: str
) -> ResultDict:
res = {}
res[FVAL] = self.neg_log_density(x)
for order in sensi_orders:
if order == 0:
continue
elif order == 1:
res[GRAD] = self.gradient_neg_log_density(x)
elif order == 2:
res[HESS] = self.hessian_neg_log_density(x)
else:
raise ValueError(f'Invalid sensi order {order}.')
return res
[docs] def check_sensi_orders(self,
sensi_orders: Tuple[int, ...],
mode: str) -> bool:
for order in sensi_orders:
if not (0 <= order <= 2):
return False
return True
[docs] def check_mode(self, mode) -> bool:
if mode == MODE_FUN:
return True
elif mode == MODE_RES:
return False
else:
raise ValueError(f'Invalid input: Expected mode {MODE_FUN} or'
f' {MODE_RES}, received {mode} instead.')
[docs] def neg_log_density(self, x):
"""
Computes the negative log-density for a parameter
vector x.
"""
density_val = 0
for prior in self.prior_list:
density_val -= prior['density_fun'](x[prior['index']])
return density_val
[docs] def gradient_neg_log_density(self, x):
"""
Computes the gradient of the negative log-density for a parameter
vector x.
"""
grad = np.zeros_like(x)
for prior in self.prior_list:
grad[prior['index']] -= prior['density_dx'](x[prior['index']])
return grad
[docs] def hessian_neg_log_density(self, x):
"""
Computes the hessian of the negative log-density for a parameter
vector x.
"""
hessian = np.zeros((len(x), len(x)))
for prior in self.prior_list:
hessian[prior['index'], prior['index']] -= \
prior['density_ddx'](x[prior['index']])
return hessian
[docs] def hessian_vp_neg_log_density(self, x, p):
"""
Computes the hessian vector product of the hessian of the
negative log-density for a parameter vector x with a vector p.
"""
h_dot_p = np.zeros_like(p)
for prior in self.prior_list:
h_dot_p[prior['index']] -= \
prior['density_ddx'](x[prior['index']]) * p[prior['index']]
return h_dot_p
def get_parameter_prior_dict(index: int,
prior_type: str,
prior_parameters: list,
parameter_scale: str = 'lin'):
"""
Returns the prior dict used to define priors for some default priors.
index:
index of the parameter in x_full
prior_type:
Prior is defined in LINEAR=untransformed parameter space,
unless it starts with "parameterScale". prior_type
can be any of {"uniform", "normal", "laplace", "logNormal",
"parameterScaleUniform", "parameterScaleNormal",
"parameterScaleLaplace"}
prior_parameters:
Parameters of the priors. Parameters are defined in linear scale.
parameter_scale:
scale in which the parameter is defined (since a parameter can be
log-transformed, while the prior is always defined in the linear
space, unless it starts with "parameterScale")
"""
log_f, d_log_f_dx, dd_log_f_ddx = \
_prior_densities(prior_type, prior_parameters)
if parameter_scale == 'lin' or prior_type.startswith('parameterScale'):
return {'index': index,
'density_fun': log_f,
'density_dx': d_log_f_dx,
'density_ddx': dd_log_f_ddx}
elif parameter_scale == 'log':
def log_f_log(x_log):
"""log-prior for log-parameters"""
return log_f(np.exp(x_log))
def d_log_f_log(x_log):
"""derivative of log-prior w.r.t. log-parameters"""
return d_log_f_dx(np.exp(x_log)) * np.exp(x_log)
def dd_log_f_log(x_log):
"""second derivative of log-prior w.r.t. log-parameters"""
return np.exp(x_log) * \
(d_log_f_dx(np.exp(x_log)) +
np.exp(x_log) * dd_log_f_ddx(np.exp(x_log)))
return {'index': index,
'density_fun': log_f_log,
'density_dx': d_log_f_log,
'density_ddx': dd_log_f_log}
elif parameter_scale == 'log10':
log10 = np.log(10)
def log_f_log10(x_log10):
"""log-prior for log10-parameters"""
return log_f(10**x_log10)
def d_log_f_log10(x_log10):
"""derivative of log-prior w.r.t. log10-parameters"""
return d_log_f_dx(10**x_log10) * log10 * 10**x_log10
def dd_log_f_log10(x_log10):
"""second derivative of log-prior w.r.t. log10-parameters"""
return log10**2 * 10**x_log10 * \
(dd_log_f_ddx(10**x_log10) * 10**x_log10
+ d_log_f_dx(10**x_log10))
return {'index': index,
'density_fun': log_f_log10,
'density_dx': d_log_f_log10,
'density_ddx': dd_log_f_log10}
else:
raise ValueError(f"NegLogPriors in parameters in scale "
f"{parameter_scale} are currently not supported.")
def _prior_densities(prior_type: str,
prior_parameters: np.array) -> [Callable,
Callable,
Callable]:
"""
Returns a tuple of Callables of the (log-)density (in untransformed =
linear scale), unless prior_types starts with "parameterScale",
together with their first + second derivative (= sensis) w.r.t.
the parameters.
Currently the following distributions are supported:
Parameters
----------
prior_type:
string identifier indicating the distribution to be used. Here
"transformed" parameter scale refers to the scale in which
optimization is performed. For example, for parameters with scale
"log", "parameterScaleNormal" will apply a normally distributed prior
to logarithmic parameters, while "normal" will apply a normally
distributed prior to linear parameters. For parameters with scale
"lin", "parameterScaleNormal" and "norma" are equivalent.
* uniform:
Uniform distribution on transformed parameter scale.
* parameterScaleUniform:
Uniform distribution on original parameter scale.
* normal:
Normal distribution on transformed parameter scale.
* parameterScaleNormal:
Normal distribution on original parameter scale.
* laplace:
Laplace distribution on transformed parameter scale
* parameterScaleLaplace:
Laplace distribution on original parameter scale.
* logNormal:
LogNormal distribution on transformed parameter scale
Currently not supported, but eventually planned are the
following distributions:
* logUniform
* logLaplace
prior_parameters:
parameters for the distribution
* uniform/parameterScaleUniform:
- prior_parameters[0]: lower bound
- prior_parameters[1]: upper bound
* laplace/parameterScaleLaplace:
- prior_parameters[0]: location parameter
- prior_parameters[1]: scale parameter
* normal/parameterScaleNormal:
- prior_parameters[0]: mean
- prior_parameters[1]: standard deviation
* logNormal:
- prior_parameters[0]: mean of log-parameters
- prior_parameters[1]: standard deviation of log-parameters
"""
if prior_type in ['uniform', 'parameterScaleUniform']:
def log_f(x):
if prior_parameters[0] <= x <= prior_parameters[1]:
return - np.log(prior_parameters[1] - prior_parameters[0])
else:
return 0
d_log_f_dx = _get_constant_function(0)
dd_log_f_ddx = _get_constant_function(0)
return log_f, d_log_f_dx, dd_log_f_ddx
elif prior_type in ['normal', 'parameterScaleNormal']:
mean = prior_parameters[0]
sigma2 = prior_parameters[1]**2
def log_f(x):
return -np.log(2*np.pi*sigma2)/2 - \
(x-mean)**2/(2*sigma2)
d_log_f_dx = _get_linear_function(-1/sigma2,
mean/sigma2)
dd_log_f_ddx = _get_constant_function(-1/sigma2)
return log_f, d_log_f_dx, dd_log_f_ddx
elif prior_type in ['laplace', 'parameterScaleLaplace']:
mean = prior_parameters[0]
scale = prior_parameters[1]
log_2_sigma = np.log(2*prior_parameters[1])
def log_f(x):
return -log_2_sigma -\
abs(x-mean)/scale
def d_log_f_dx(x):
if x > prior_parameters[0]:
return -1/scale
else:
return 1/scale
dd_log_f_ddx = _get_constant_function(0)
return log_f, d_log_f_dx, dd_log_f_ddx
elif prior_type == 'logUniform':
# when implementing: add to tests
raise NotImplementedError
elif prior_type == 'logNormal':
# TODO check again :)
mean = prior_parameters[0]
sigma = prior_parameters[1]
sqrt2_pi = np.sqrt(2*np.pi)
def log_f(x):
return - np.log(sqrt2_pi * sigma * x) \
- (np.log(x) - mean)**2/(2*sigma**2)
def d_log_f_dx(x):
return - 1/x - (np.log(x) - mean)/(sigma**2 * x)
def dd_log_f_ddx(x):
return 1/(x**2) \
- (1 - np.log(x) + mean)/(sigma**2 * x**2)
return log_f, d_log_f_dx, dd_log_f_ddx
elif prior_type == 'logLaplace':
# when implementing: add to tests
raise NotImplementedError
else:
raise ValueError(f'NegLogPriors of type {prior_type} are currently '
'not supported')
def _get_linear_function(slope: float,
intercept: float = 0):
"""
Returns a linear function
"""
def function(x):
return slope * x + intercept
return function
def _get_constant_function(constant: float):
"""
Defines a callable, that returns the constant, regardless of the input.
"""
def function(x):
return constant
return function