import logging
import math
from collections.abc import Sequence
from copy import deepcopy
from typing import Callable, Union
import numpy as np
from .. import C
from .aggregated import AggregatedObjective
from .base import ResultDict
from .function import ObjectiveBase
logger = logging.getLogger(__name__)
[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):
"""
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,
):
"""
Initialize.
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):
"""Create deepcopy of object."""
other = NegLogParameterPriors(deepcopy(self.prior_list))
return other
[docs]
def call_unprocessed(
self,
x: np.ndarray,
sensi_orders: tuple[int, ...],
mode: C.ModeType,
**kwargs,
) -> ResultDict:
"""
Call objective function without pre- or post-processing and formatting.
Returns
-------
result:
A dict containing the results.
"""
res = {}
res[C.FVAL] = self.neg_log_density(x)
if mode == C.MODE_FUN:
for order in sensi_orders:
if order == 0:
continue
elif order == 1:
res[C.GRAD] = self.gradient_neg_log_density(x)
elif order == 2:
res[C.HESS] = self.hessian_neg_log_density(x)
else:
raise ValueError(f"Invalid sensi order {order}.")
if mode == C.MODE_RES:
for order in sensi_orders:
if order == 0:
res[C.RES] = self.residual(x)
elif order == 1:
res[C.SRES] = self.residual_jacobian(x)
else:
raise ValueError(f"Invalid sensi order {order}.")
return res
[docs]
def check_sensi_orders(
self,
sensi_orders: tuple[int, ...],
mode: C.ModeType,
) -> bool:
"""See `ObjectiveBase` documentation."""
if mode == C.MODE_FUN:
for order in sensi_orders:
if not (0 <= order <= 2):
return False
elif mode == C.MODE_RES:
for order in sensi_orders:
if order == 0:
return all(
prior.get("residual", None) is not None
for prior in self.prior_list
)
elif order == 1:
return all(
prior.get("residual_dx", None) is not None
for prior in self.prior_list
)
else:
return False
else:
raise ValueError(
f"Invalid input: Expected mode {C.MODE_FUN} or "
f"{C.MODE_RES}, received {mode} instead."
)
return True
[docs]
def check_mode(self, mode: C.ModeType) -> bool:
"""See `ObjectiveBase` documentation."""
if mode == C.MODE_FUN:
return True
elif mode == C.MODE_RES:
return all(
prior.get("residual", None) is not None
for prior in self.prior_list
)
else:
raise ValueError(
f"Invalid input: Expected mode {C.MODE_FUN} or "
f"{C.MODE_RES}, received {mode} instead."
)
[docs]
def neg_log_density(self, x):
"""Evaluate the negative log-density at 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):
"""Evaluate the gradient of the negative log-density at 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):
"""Evaluate the hessian of the negative log-density at 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):
"""Compute vector product of the hessian at 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
[docs]
def residual(self, x):
"""Evaluate the residual representation of the prior at x."""
return np.asarray(
[prior["residual"](x[prior["index"]]) for prior in self.prior_list]
)
[docs]
def residual_jacobian(self, x):
"""
Evaluate residual Jacobian.
Evaluate the Jacobian of the residual representation of the prior
for a parameter vector x w.r.t. x, if available.
"""
sres = np.zeros((len(self.prior_list), len(x)))
for iprior, prior in enumerate(self.prior_list):
sres[iprior, prior["index"]] = prior["residual_dx"](
x[prior["index"]]
)
return sres
[docs]
def get_parameter_prior_dict(
index: int,
prior_type: str,
prior_parameters: list,
parameter_scale: str = C.LIN,
):
"""
Return 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, res, d_res_dx = _prior_densities(
prior_type, prior_parameters
)
if parameter_scale == C.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,
"residual": res,
"residual_dx": d_res_dx,
}
elif parameter_scale == C.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):
"""First 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))
)
if res is not None:
def res_log(x_log):
"""Residual-prior for log-parameters."""
return res(np.exp(x_log))
else:
res_log = None
if d_res_dx is not None:
def d_res_log(x_log):
"""Residual-prior for log-parameters."""
return d_res_dx(np.exp(x_log)) * np.exp(x_log)
else:
d_res_log = None
return {
"index": index,
"density_fun": log_f_log,
"density_dx": d_log_f_log,
"density_ddx": dd_log_f_log,
"residual": res_log,
"residual_dx": d_res_log,
}
elif parameter_scale == C.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):
"""Rerivative 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)
)
)
res_log = None
if res is not None:
def res_log(x_log10):
"""Residual-prior for log10-parameters."""
return res(10**x_log10)
d_res_log = None
if d_res_dx is not None:
def d_res_log(x_log10):
"""Residual-prior for log10-parameters."""
return d_res_dx(10**x_log10) * log10 * 10**x_log10
return {
"index": index,
"density_fun": log_f_log10,
"density_dx": d_log_f_log10,
"density_ddx": dd_log_f_log10,
"residual": res_log,
"residual_dx": d_res_log,
}
else:
raise ValueError(
"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,
Union[Callable, None],
Union[Callable, None],
]:
"""
Create prior density functions.
Return 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. If possible, a residual representation and its first
derivative w.r.t. the parameters is included as 4th and 5th element of
the vector. If a reformulation as residual is not possible, the respective
entries will be `None`.
Currently the following distributions are supported:
* 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
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 "normal" are equivalent.
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
Returns
-------
log_f, d_log_f_dx, dd_log_f_ddx, res, d_res_dx:
Log density, first and second derivative, and if possible a residual
representation and its first derivative.
"""
if prior_type in [C.UNIFORM, C.PARAMETER_SCALE_UNIFORM]:
def log_f(x):
if prior_parameters[0] <= x <= prior_parameters[1]:
return -np.log(prior_parameters[1] - prior_parameters[0])
else:
return -np.inf
d_log_f_dx = _get_constant_function(0)
dd_log_f_ddx = _get_constant_function(0)
def res(x):
if prior_parameters[0] <= x <= prior_parameters[1]:
return 0
else:
return np.inf
d_res_dx = _get_constant_function(0)
return log_f, d_log_f_dx, dd_log_f_ddx, res, d_res_dx
elif prior_type in [C.NORMAL, C.PARAMETER_SCALE_NORMAL]:
mean = prior_parameters[0]
sigma = prior_parameters[1]
sigma2 = sigma**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)
def res(x):
return (x - mean) / (np.sqrt(2) * sigma)
d_res_dx = _get_constant_function(1 / (np.sqrt(2) * sigma))
return log_f, d_log_f_dx, dd_log_f_ddx, res, d_res_dx
elif prior_type in [C.LAPLACE, C.PARAMETER_SCALE_LAPLACE]:
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 > mean:
return -1 / scale
else:
return 1 / scale
dd_log_f_ddx = _get_constant_function(0)
def res(x):
return np.sqrt(abs(x - mean) / scale)
def d_res_dx(x):
if x == mean:
logger.warning(
"x == mean in d_res_dx of Laplace prior. Returning NaN."
)
return math.nan
return 1 / 2 * (x - mean) / np.sqrt(scale * abs(x - mean) ** 3)
return log_f, d_log_f_dx, dd_log_f_ddx, res, d_res_dx
elif prior_type == C.LOG_UNIFORM:
# when implementing: add to tests
raise NotImplementedError
elif prior_type == C.LOG_NORMAL:
# 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, None, None
elif prior_type == C.LOG_LAPLACE:
# 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 = 0.0,
):
"""Return a linear function."""
def function(x):
return slope * x + intercept
return function
def _get_constant_function(constant: float):
"""Define a callable returning the constant, regardless of the input."""
def function(x):
return constant
return function