"""
Posterior sampling using a PythonDistribution
=============================================
"""
# %%
# In this example we are going to show how to do Bayesian inference using the :class:`~openturns.RandomWalkMetropolisHastings` algorithm
# in a statistical model defined through a :class:`~openturns.PythonDistribution`.
#
# This method is illustrated on a simple lifetime study test-case, which involves censored data, as described hereafter.
#
# In the following, we assume that the lifetime :math:`T_i` of an industrial component follows the Weibull distribution :math:`\mathcal W(\alpha, \beta)`,
# with CDF given by :math:`F(t|\alpha,\beta)= 1 - e^{-\left( \frac{t}{\beta} \right)^\alpha}`.
#
# Our goal is to estimate the model parameters :math:`\alpha, \beta` based on
# a dataset of recorded failures :math:`(t_1, \ldots, t_n),` some of which
# correspond to actual failures, and the remaining are right-censored.
# Let :math:`(f_1, \ldots, f_n) \in \{0,1\}^n` represent the nature of each
# datum, :math:`f_i=1` if :math:`t_i` corresponds to an actual failure,
# :math:`f_i=0` if it is right-censored.
#
# Note that the likelihood of each recorded failure is given by the Weibull density:
#
# .. math::
#   \mathcal L(t_i | f_i=1, \alpha, \beta) = \frac{\alpha}{\beta}\left( \frac{t_i}{\beta} \right)^{\alpha-1} e^{-\left( \frac{t_i}{\beta} \right)^\alpha}.
#
# On the other hand, the likelihood of each right-censored observation is given by:
#
# .. math::
#   \mathcal L(t_i | f_i=0, \alpha, \beta) = e^{-\left( \frac{t_i}{\beta} \right)^\alpha}.
#
# Furthermore, assume that the prior information available on :math:`\alpha, \beta` is represented by independent prior laws,
# whose respective densities are denoted by :math:`\pi(\alpha)` and :math:`\pi(\beta).`
#
# The posterior distribution of :math:`(\alpha, \beta)` represents the update of the prior information on :math:`(\alpha, \beta)` given the dataset.
# Its PDF is known up to a multiplicative constant:
#
#
# .. math::
#   \pi(\alpha, \beta | (t_1, f_1), \ldots, (t_n, f_n) ) \propto \pi(\alpha)\pi(\beta) \left(\frac{\alpha}{\beta}\right)^{\sum_i f_i}
#   \left(\prod_{f_i = 1} \frac{t_i}{\beta}\right)^{\alpha-1} \exp\left[-\sum_{i=1}^n\left(\frac{t_i}{\beta}\right)^\alpha\right].
#
# The :class:`~openturns.RandomWalkMetropolisHastings` class can be used to sample from the posterior distribution. It relies on the following objects:
#
# - The prior probability density :math:`\pi(\vect{\theta})` reflects beliefs about the possible values
#   of :math:`\vect{\theta} = (\alpha, \beta)` before the experimental data are considered.
# - Initial values :math:`\vect{\theta}_0` of the parameters.
# - An proposal distribution used to update parameters.
#
# Additionally we want to define the likelihood term defined by these objects:
#
# - The conditional density :math:`p(t_{1:n}|f_{1:n}, \alpha, \beta)` will be defined as a :class:`~openturns.PythonDistribution`.
# - The sample of observations acting as the parameters of the conditional density
#
#
#
# Set up the PythonDistribution
# -----------------------------
# The censured Weibull likelihood is outside the usual catalog of probability distributions,
# hence we need to define it using the :class:`~openturns.PythonDistribution` class.


# %%
import numpy as np
import openturns as ot
from openturns.viewer import View

ot.Log.Show(ot.Log.NONE)
ot.RandomGenerator.SetSeed(123)

# %%
# The following methods must be defined:
#
# - `getRange`: required for conversion to the :class:`~openturns.Distribution` format
# - `computeLogPDF`: used by :class:`~openturns.RandomWalkMetropolisHastings` to evaluate the posterior density
# - `setParameter` used by :class:`~openturns.RandomWalkMetropolisHastings` to test new parameter values
#
# .. note::
#    We formally define a bivariate distribution on the :math:`(t_i, f_i)` couple, even though :math:`f_i` has no distribution (it is simply a covariate).
#    This is not an issue, since the sole purpose of this :class:`~openturns.PythonDistribution` object is to pass
#    the likelihood calculation over to :class:`~openturns.RandomWalkMetropolisHastings`.

# %%


class CensoredWeibull(ot.PythonDistribution):
    """
    Right-censored Weibull log-PDF calculation
    Each data point x is assumed 2D, with:
        x[0]: observed functioning time
        x[1]: nature of x[0]:
            if x[1]=0: x[0] is a censoring time
            if x[1]=1: x[0] is a time-to failure
    """

    def __init__(self, beta=5000.0, alpha=2.0):
        super(CensoredWeibull, self).__init__(2)
        self.beta = beta
        self.alpha = alpha

    def getRange(self):
        return ot.Interval([0, 0], [1, 1], [True] * 2, [False, True])

    def computeLogPDF(self, x):
        if not (self.alpha > 0.0 and self.beta > 0.0):
            return -np.inf
        log_pdf = -((x[0] / self.beta) ** self.alpha)
        log_pdf += (self.alpha - 1) * np.log(x[0] / self.beta) * x[1]
        log_pdf += np.log(self.alpha / self.beta) * x[1]
        return log_pdf

    def setParameter(self, parameter):
        self.beta = parameter[0]
        self.alpha = parameter[1]

    def getParameter(self):
        return [self.beta, self.alpha]


# %%
# Convert to :class:`~openturns.Distribution`

# %%


conditional = ot.Distribution(CensoredWeibull())


# %%
# Observations, prior, initial point and proposal distributions
# -------------------------------------------------------------
#
# Define the observations


# %%

Tobs = np.array([4380, 1791, 1611, 1291, 6132, 5694, 5296, 4818, 4818, 4380])
fail = np.array([True] * 4 + [False] * 6)
x = ot.Sample(np.vstack((Tobs, fail)).T)


# %%
# Define a uniform prior distribution for :math:`\alpha` and a Gamma prior distribution for :math:`\beta`.
#

# %%

alpha_min, alpha_max = 0.5, 3.8
a_beta, b_beta = 2, 2e-4

priorCopula = ot.IndependentCopula(2)  # prior independence
priorMarginals = []  # prior marginals
priorMarginals.append(ot.Gamma(a_beta, b_beta))  # Gamma prior for beta
priorMarginals.append(ot.Uniform(alpha_min, alpha_max))  # uniform prior for alpha
prior = ot.JointDistribution(priorMarginals, priorCopula)
prior.setDescription(["beta", "alpha"])


# %%
# We select prior means as the initial point of the Metropolis-Hastings algorithm.
#

# %%

initialState = [a_beta / b_beta, 0.5 * (alpha_max - alpha_min)]

# %%
# For our random walk proposal distributions, we choose normal steps, with standard deviation equal to roughly :math:`10\%` of the prior range (for the uniform prior)
# or standard deviation (for the normal prior).
#

# %%

proposal = []
proposal.append(ot.Normal(0.0, 0.1 * np.sqrt(a_beta / b_beta**2)))
proposal.append(ot.Normal(0.0, 0.1 * (alpha_max - alpha_min)))
proposal = ot.JointDistribution(proposal)

# %%
# Sample from the posterior distribution
# --------------------------------------

# %%

sampler = ot.RandomWalkMetropolisHastings(prior, initialState, proposal)
sampler.setLikelihood(conditional, x)
sampleSize = 10000
sample = sampler.getSample(sampleSize)
# compute acceptance rate
print("Acceptance rate: %s" % (sampler.getAcceptanceRate()))

# %%
# Plot prior to posterior marginal plots.
#

# %%
kernel = ot.KernelSmoothing()
posterior = kernel.build(sample)
grid = ot.GridLayout(1, 2)
grid.setTitle("Bayesian inference")
for parameter_index in range(2):
    graph = posterior.getMarginal(parameter_index).drawPDF()
    priorGraph = prior.getMarginal(parameter_index).drawPDF()
    graph.add(priorGraph)
    graph.setColors(ot.Drawable.BuildDefaultPalette(2))
    graph.setLegends(["Posterior", "Prior"])
    grid.setGraph(0, parameter_index, graph)
_ = View(grid)

# %%
# Define an improper prior
# --------------------------

# %%
# Now, define an improper prior:
#
# .. math::
#   \mathcal \pi(\beta, \alpha) \propto \frac{1}{\beta}.
#

logpdf = ot.SymbolicFunction(["beta", "alpha"], ["-log(beta)"])
support = ot.Interval([0] * 2, [1] * 2)
support.setFiniteUpperBound([False] * 2)

# %%
# Sample from the posterior distribution

sampler2 = ot.RandomWalkMetropolisHastings(logpdf, support, initialState, proposal)
sampler2.setLikelihood(conditional, x)
sample2 = sampler2.getSample(1000)
print("Acceptance rate: %s" % (sampler2.getAcceptanceRate()))

# %%
# Plot posterior marginal plots only as prior cannot be drawn meaningfully.
#

# %%
kernel = ot.KernelSmoothing()
posterior = kernel.build(sample)
grid = ot.GridLayout(1, 2)
grid.setTitle("Bayesian inference (with log-pdf)")
for parameter_index in range(2):
    graph = posterior.getMarginal(parameter_index).drawPDF()
    graph.setLegends(["Posterior"])
    grid.setGraph(0, parameter_index, graph)
_ = View(grid)

View.ShowAll()
