priors.py

import logging
import math
from copy import deepcopy
from typing import Callable, Dict, List, Sequence, Tuple, Union

import numpy as np

from .. import C
from .aggregated import AggregatedObjective
from .base import ResultDict
from .function import ObjectiveBase

logger = logging.getLogger(__name__)


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.
    """


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.
    """

    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

    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

    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

    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.'
            )

    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

    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

    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

    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

    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]
        )

    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


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