"""
Linear Regression with interval-censored observations
=====================================================
"""

# %%
# 1. Model formulation
# --------------------
#
# We consider the following linear model:
#
# .. math::
#
#    \vect{Y} = \mat{X} \vect{\theta} + \vect{\varepsilon},
#
# where the observation vector :math:`\vect{Y} \in \mathbb R^{n}` is modeled as
# the sum of:
#
# - a linear part, with an :math:`n \times p` design matrix :math:`\mat{X}`
#   and unknown regression coefficients :math:`\vect{\theta} \in\mathbb R^p`,
# - a Gaussian error term
#   :math:`\vect{\varepsilon} \sim \mathcal N_n(\vect{0}, \tau^{-1} \mat{I}_n+\mat{Q}^{-1})`,
#
# where :math:`\mat{Q}` represents a known *precision* (inverse variance) matrix
# for measurement errors,
# and :math:`\tau` an unknown precision parameter
# quantifying the variability of the observed phenomenon.
#
# 1.1. Likelihood of the linear regression model
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# The above linear regression model can thus be written:
#
# .. math::
#
#    \vect{Y} | \vect{\theta}, \tau
#    {\sim} \mathcal N_{n}(\mat{X} \vect{\theta} ; \tau^{-1} \mat{I}_n +\mat{Q}^{-1}),
#
# We then have the following likelihood:
#
# .. math::
#
#    \mathcal L(\vect{Y}|\vect{\theta}, \tau) =
#    (2\pi)^{-\frac{n}{2}}
#    {\rm det}(\tau^{-1} \mat{I}_n+\mat{Q}^{-1})^{-\frac{1}{2}}
#    e^{-\frac{1}{2}
#    ||\vect{Y} - \mat{X} \vect{\theta} ||_{\tau^{-1} \mat{I}_n +\mat{Q}^{-1}}^2
#    }
#
# where :math:`||\vect{X}-\vect{Y}||_{\mat{V}}^2` is the Mahalanobis distance between
# :math:`\vect{X}` and :math:`\vect{Y}` with covariance matrice :math:`\mat{V}`:
#
# .. math::
#
#    ||\vect{X}-\vect{Y}||_{\mat{V}}^2 = (\vect{X}-\vect{Y})^\top\ \mat{V}^{-1}\ (\vect{X}-\vect{Y}).
#
# 1.2. Interval Censorship
# ~~~~~~~~~~~~~~~~~~~~~~~~
#
# We now assume that, instead of observing directly the
# :math:`Y_i, i=1,\ldots,n` as described above, we only have access to
# discretized values :math:`Y_i^{obs} \in \delta \times \mathbb N`, where
# :math:`\delta > 0` is a grid length and :math:`Y_i^{obs} = \delta \times k` means that
# :math:`Y_i \in \delta \times \left[ k - \frac{1}{2}; k + \frac{1}{2} \right[`.
#
# 1.3. Remarks
# ~~~~~~~~~~~~
#
# -  The presence of a composite matrix :math:`\tau^{-1} \mat{I}_n+\mat{Q}^{-1}` makes
#    estimation more complex than with a spherical one
#    :math:`\tau^{-1}\mat{I}_n,` since we would then have explicit (closed-form)
#    maximum likelihood estimators, and also conjugate priors leading to
#    explicit full joint posterior distributions.
# -  Another difficulty is the presence of censored data, since the
#    likelihood is no more available in closed-form. As we will see, this
#    can be overcome thanks to Bayesian inference.
# -  Heteroscedastic linear modeling under interval censorship as
#    formulated above was originally motivated by an industrial case-study
#    in seismology, wherein the :math:`Y_i` correspond to the observed
#    intensity of an earthquake in a distant site, and explanatory
#    variables :math:`\vect{X}_{i}` are derived from the epicentral distance to the
#    earthquake's source as well as its characteristics (magnitude,
#    depth). But it can also arise in many different contexts, as soon as
#    observations are available with known statistical precisions (hence
#    the heteroscedasticity) and limited numerical accuracy (hence the
#    censorship).

# %%
# 1.4 Simulate the dataset
# ~~~~~~~~~~~~~~~~~~~~~~~~
import openturns as ot
from openturns.viewer import View
import numpy as np

ot.Log.Show(ot.Log.NONE)
ot.RandomGenerator.SetSeed(1)

n = 10
delta = 0.5

# %%
# Build the design matrix :math:`\mat{X}`
X = ot.Sample([[1.0]] * n)
X.stack(ot.Normal(0.0, 10.0).getSample(n))

# %%
# Make the precision matrix :math:`\mat{Q}` a diagonal matrix and sample
# its diagonal coefficients from an :class:`~openturns.Exponential` distribution.
Q = np.ones((n, 1)) + ot.Exponential().getSample(n)

# %%
# Choose values for the parameters :math:`\vect{\theta}` and :math:`\tau`.
theta_true = np.array([[0.0], [1.0]])
p = len(theta_true)
tau_true = 0.1

# %%
# First sample uncensored, and then censored observation data.
mean_true = np.dot(X, theta_true).ravel()
std_true = np.sqrt(1.0 / tau_true + 1.0 / Q).ravel()

Y_sim = mean_true + [x[0] for x in ot.Normal().getSample(n)] * std_true
Yobs_sim = np.round(Y_sim / delta) * delta

# %%
# Plot the simulated dataset.
graph = ot.Graph("Simulated data", "$X_1$", "$Y$", True, "topleft", 16)
cloud_obs = ot.Cloud(X[:, 1].asPoint(), Yobs_sim)
cloud_obs.setPointStyle("bullet")
cloud_sim = ot.Cloud(X[:, 1].asPoint(), Y_sim)
cloud_sim.setPointStyle("bullet")
curve = ot.Curve(X[:, 1].asPoint(), mean_true)
curve.setLineWidth(1.5)
graph.add(curve)
graph.add(cloud_sim)
graph.add(cloud_obs)
graph.setLegends(["Trend", "$Y^{sim}$", "$Y^{obs}$"])
graph.setColors(ot.Drawable.BuildDefaultPalette(3))
_ = View(graph)


# %%
# 2. Bayesian Inference
# ---------------------
#
# 2.1. Choice of a prior law
# ~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# We use the standard Normal-Gamma prior for :math:`\vect{\theta}` and
# :math:`\tau`:
#
# .. math::
#
#    \vect{\theta} | \tau \sim \mathcal N_{p}(\vect{\mu}_0; \tau^{-1}\mat{\Sigma}_0) \\
#    \tau \sim \mathcal G(a_0, b_0),
#
# where all parameters are assumed *a priori* independent if not stated
# otherwise.
#
# Furthermore, a default choice for the hyperparameters consists in having
# all prior variances go to infinity, equivalent to the degenerate case:
#
# .. math::
#
#    \vect{\mu}_0, \mat{\Sigma}_0^{-1}, a_0, b_0 \equiv 0
#
# But the resulting prior is improper. Hence, posterior propriety needs to
# be proven first.
#
# A simpler solution is to ensure prior (hence posterior) propriety by
# imposing bounds :math:`\vect{\theta}_{\min}, \vect{\theta}_{\max}, \tau_{\min}, \tau_{\max}`
# on all parameters following:
#
# .. math::
#
#    \vect{\theta} | \tau \sim \mathcal N_{p}(\vect{\mu}_0; \tau^{-1}\mat{\Sigma}_0) {\bf 1}_{\{\vect{\theta}_{\min} \leq \vect{\theta} \leq \vect{\theta}_{\max}\}} ;\\
#    \tau \sim \mathcal G(a_0, b_0){\bf 1}_{\{\tau_{\min} \leq \tau \leq \tau_{\max}\}},
#
# where inequalites are taken componentwise. When all hyperparameters go to
# :math:`0` as described above, the prior converges to a product of
# uniform distributions.
#
# We will use this product of univariate uniforms as a prior in the
# following. As discussed above, there is no simple way to obtain the
# posterior distribution, justifying the use of Monte-Carlo Markov chain
# techniques, as described hereafter.

lower = (Yobs_sim.ravel() - delta).tolist()
upper = (Yobs_sim.ravel() + delta).tolist()

# Global support of the joint distribution: theta, tau, outputs
support = ot.Interval([-2.0] * p + [1e-4] + lower, [2.0] * p + [1e1] + upper)

prior = ot.ComposedDistribution(
    [ot.Uniform(-2.0, 2.0), ot.Uniform(-2.0, 2.0), ot.Uniform(1e-4, 1e1)]
)

# Initialize to true value
initial_state = theta_true[:, 0].tolist() + [tau_true] + Y_sim.ravel().tolist()
initial_state = ot.Point(initial_state)

# %%
# 2.2. Posterior sampling
# ~~~~~~~~~~~~~~~~~~~~~~~
#
# The solution we advocate consists in sampling from the joint posterior
# distribution of all uncertain parameters, including the vector of
# continuous intensities :math:`\vect{Y},` seen here as a latent variable.
# Indeed, adding :math:`\vect{Y}` to the vector of sampled variables yields a
# posterior density which is available in closed form, up to an unkown
# multiplicative factor
#
# .. math::
#
#    \begin{array}{ll}
#    \pi(\vect{Y},\vect{\theta},\tau|\vect{Y}^{obs})
#        & \propto \pi(\vect{\theta},\tau)\times\mathcal L(\vect{Y}|\vect{\theta},\tau)\mathcal L(\vect{Y}^{obs}|\vect{Y})\\
#        & \propto
#        {\bf 1}_{\{\vect{\theta}_{\min} \leq \vect{\theta} \leq \vect{\theta}_{\max}\}}\times
#        {\bf 1}_{\{\tau_{\min} \leq \tau \leq \tau_{\max}\}}\times
#        \mathcal N_n (\vect{Y}|\mat{X} \vect{\theta};\tau^{-1} \mat{I}_n + \mat{Q}^{-1}) \times
#        {\bf 1}_{\left[\vect{Y}^{obs}-\frac{\delta}{2}; \vect{Y}^{obs}+\frac{\delta}{2}\right[}(\vect{Y}).
#        \end{array}
#
# This allows to perform the following Metropolis within Gibbs sampling
# scheme, wherein the pre-defined blocks of variables
# :math:`(\vect{Y},\vect{\theta},\tau)` are updated in turn, according to their
# conditional posterior density, or to a Markov kernel targeting it, as
# described in the following.
#
# 2.2.1. Updating :math:`\vect{Y}`
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# :math:`\pi(\vect{Y}|\vect{\theta},\tau,\vect{Y}^{obs})\propto\prod_{i=1}^{n}\mathcal N(Y_{i}|\vect{X}_{i} \vect{\theta}, \tau^{-1}+q_{i}^{-1}) {\bf 1}_{\left[Y_{i}^{obs}-\frac{\delta}{2}; Y_{i}^{obs}+\frac{\delta}{2}\right[}(Y_{i})`,
# hence the latent variables :math:`Y_i` are updated by simply simulating
# independent univariate truncated normals.
#

# %%
# Step 1 : Create associated :class:`~openturns.RandomVector`
marginals_trunc = [
    ot.TruncatedNormal(Yobs_sim[i], 1.0, lower[i], upper[i]) for i in range(len(X))
]

trunc_cond_Y = ot.ComposedDistribution(marginals_trunc)
RV_Y = ot.RandomVector(trunc_cond_Y)


# %%
# Step 2 : Link function, giving the parameters of the univariate truncated normals in function of the chain's current state

gen_params = np.array(trunc_cond_Y.getParameter())


def py_link_function_y(x):
    """
    link function for Y conditional density

    Input
        x: vector with length (p + 1 + n), containing the current state of (theta, tau, Y)

    Output
        params: vector with length 4*n, corresponding to mean, std, a and b, for each component of Y

    Notes
        a and b represent the upper and lower bounds for the truncated normals
    """
    theta = [x[i] for i in range(p)]
    tau = x[p]
    # compute conditional mean and standard deviates
    mean = np.dot(X, theta)
    std = np.sqrt(1.0 / tau + 1.0 / Q)
    # inject values in blueprint
    params = gen_params.copy()
    params[::4] = mean
    params[1::4] = std.ravel()
    return params


# %%
# 2.2.2. Updating :math:`\vect{\theta}`
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# :math:`\pi(\vect{\theta}|\vect{Y}, \tau, \vect{Y}^{obs})=\pi(\vect{\theta}|\vect{Y},\tau).` Due to the
# partial conjugacy of the conditional prior :math:`\pi(\vect{\theta}|\tau),`
# this is explicit, and given by the following box-constrained
# multivariate normal:
#
# .. math::
#
#     \vect{\theta}|\vect{Y},\tau \sim \mathcal N_4( \vect{\mu}_n ; \mat{\Sigma}_n ){\bf 1}_{\{\vect{\theta}_{\min} \leq \vect{\theta} \leq \vect{\theta}_{\max}\}},
#
# with
#
# .. math::
#
#     \vect{\mu}_n = \vect{\mu}_0 + \mat{\Sigma}_0X^\top(\mat{X} \mat{\Sigma}_0 \mat{X}^\top + \mat{I}_n + \tau \mat{Q}^{-1})^{-1}(\vect{Y} - \mat{X} \vect{\mu}_0) \\
#     \mat{\Sigma}_n = \tau^{-1}\mat{\Sigma}_0 - \tau^{-1}\mat{\Sigma}_0 \mat{X}^\top (\mat{X} \mat{\Sigma}_0 \mat{X}^\top + \mat{I}_n + \tau \mat{Q}^{-1})^{-1} \mat{X} \mat{\Sigma}_0,
#
# or equivalently, thanks to the matrix Woodsbury identity:
#
# .. math::
#
#     \vect{\mu}_n =  (\mat{\Sigma}_0^{-1} + \mat{X}^\top (\mat{I}_n + \tau \mat{Q}^{-1})^{-1} \mat{X})^{-1}(\mat{\Sigma}_0^{-1} \vect{\mu}_0 + \mat{X}^\top (\mat{I}_n + \tau \mat{Q}^{-1})^{-1} \vect{Y})\\
#     \mat{\Sigma}_n = \tau^{-1}(\mat{\Sigma}_0^{-1} + \mat{X}^\top (\mat{I}_n + \tau \mat{Q}^{-1})^{-1} \mat{X})^{-1}.
#
# By having all hyperparameters go to :math:`0,` we obtain the following
# simplified form:
#
# .. math::
#
#     \vect{\mu}_n =  (\mat{X}^\top (\mat{I}_n + \tau \mat{Q}^{-1})^{-1} \mat{X})^{-1}(\mat{X}^\top (\mat{I}_n + \tau \mat{Q}^{-1})^{-1} \vect{Y})\\
#     \mat{\Sigma}_n = \tau^{-1}(\mat{X}^\top (\mat{I}_n + \tau \mat{Q}^{-1})^{-1} \mat{X})^{-1}.
#
# Explicit simulation from a box-constrained multivariate normal can be
# done with a simple rejection sampling scheme:

# %%
# Step 1 : Create associated :class:`~openturns.RandomVector`


class BoxConstrainedNormal(ot.PythonRandomVector):
    """
    Multivariate normal distribution
    under box constraints
    """

    def __init__(
        self, d=2, mu=np.zeros(2), Sigma=np.eye(2), r=np.zeros(2), s=np.ones(2)
    ):
        for x in mu, Sigma, r, s:
            if len(x) != d:
                print("expectation or bound does not have size %s" % d)
                raise ValueError
        if Sigma.shape != (d, d):
            print("covariance matrix not have dimensions (%s,%s)" % (d, d))
            raise ValueError
        super(BoxConstrainedNormal, self).__init__(d)
        self.mu = mu
        self.Sigma = Sigma
        self.r = r
        self.s = s
        self.interval = ot.Interval(r, s)

    def setParameter(self, parameter):
        d = self.getDimension()
        self.mu = np.array(parameter[:d])
        self.Sigma = np.array(parameter[d: d + d * d]).reshape(d, d)
        self.r = np.array(parameter[-2 * d: -d])
        self.s = np.array(parameter[-d:])
        self.interval = ot.Interval(self.r, self.s)

    def getParameter(self):
        return np.concatenate(
            [self.mu.ravel(), self.Sigma.ravel(), self.r.ravel(), self.s.ravel()]
        )

    def getRealization(self):
        accept = False
        proposaldist = ot.Normal(self.mu, ot.CovarianceMatrix(self.Sigma))
        while not accept:
            proposal = proposaldist.getRealization()
            accept = self.interval.contains(proposal)
        return proposal


RV_theta = ot.RandomVector(BoxConstrainedNormal())


# %%
# Step 2 : Link function, giving the parameters of the box-constrained normal in function of :math:`\tau` and :math:`\vect{Y}` values


def py_link_function_theta(x):
    tau = x[p]
    Y = [x[i] for i in range(p + 1, len(x))]
    # conditional mean and variance
    # for diagonal Q
    Itilde_inv = 1.0 / (1.0 + tau / Q)
    Xtilde = Itilde_inv * X
    Sigma_n = np.linalg.inv(np.dot(Xtilde.T, X)) / tau
    mu_n = np.dot(Xtilde.T, Y)
    mu_n = tau * np.dot(Sigma_n, mu_n)
    # extract parameters in correct order (coherent with getParameter() method of RV_theta)
    return np.concatenate(
        [
            mu_n.ravel(),
            Sigma_n.ravel(),
            support.getLowerBound()[:p],
            support.getUpperBound()[:p],
        ]
    )


# %%
# 2.2.3. Updating :math:`\tau`
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# :math:`\pi(\tau|\vect{Y},\vect{\theta})` is proportional to:
#
# .. math::
#
#
#    \pi(\tau|\vect{Y},\vect{\theta})
#    \propto
#    {\bf 1}_{\{\tau_{\min} \leq \tau \leq \tau_{\max}\}}\times
#    \mathcal N_n(\vect{Y}|\mat{X} \vect{\theta};\tau^{-1} \mat{I}_n + \mat{Q}^{-1}).
#
# Hence, :math:`\tau` can be updated using the Random-walk
# Metropolis-Hastings algorithm.


# %%
# Step 1 : compute tau's conditional posterior density, up to a multiplicative factor


def marginals_Y(theta, tau):
    mu = np.dot(X, theta).ravel()
    sigma = np.sqrt(1.0 / tau + 1.0 / Q).ravel()
    return [ot.Normal(mu[i], sigma[i]) for i in range(len(X))]


def py_log_density(x):
    theta = [x[i] for i in range(p)]
    tau = x[p]
    Y = [x[p + 1 + i] for i in range(len(X))]
    ld = ot.ComposedDistribution(marginals_Y(theta, tau)).computeLogPDF(Y)
    return [ld]


# %%
# Step 2 : define the proposal distribution

proposal_tau = ot.Normal(0.0, 1e-1)

# %%
# 2.3. Initialization
# ~~~~~~~~~~~~~~~~~~~
#
# To avoid all numerical problems, it is better to provide the algorithm
# with a starting point not too far from the posterior mode. To this end,
# we propose to set :math:`\vect{Y}=\vect{Y}^{obs}` for simplicity, then solve the
# following optimization problem
#
# .. math::
#
#     \widehat\theta,\widehat\tau = \arg\max_{\vect{\theta},\tau} \pi(\vect{\theta},\tau|\vect{Y}^{obs},\vect{Y}=\vect{Y}^{obs})
#     = \arg\max_{\theta\in[\vect{\theta}_{\min};\vect{\theta}_{\max}]
#     \tau\in[\tau_{\min};\tau_{\max}]} \mathcal N(\vect{Y}^{obs}|\vect{\theta} \mat{X}; \tau^{-1} \mat{I}_n + \mat{Q}^{-1}).
#
# Note that the unconstrained optimization over :math:`\vect{\theta}` for fixed
# :math:`\tau` is explicit, since:
#
# .. math::
#
#     \arg\max_{\vect{\theta}} \pi(\vect{\theta},\tau|\vect{Y}^{obs},\vect{Y}=\vect{Y}^{obs})
#    =\arg\max_{\vect{\theta}} \pi(\vect{\theta}|\vect{Y}^{obs},\vect{Y}=\vect{Y}^{obs},\tau).
#
# But we have shown that the conditional posterior density of
# :math:`\vect{\theta}` is Gaussian. Hence the unconstrained conditional
# posterior mode (and mean) is given by:
#
# .. math::
#
#     \arg\max_{\vect{\theta}} \pi(\vect{\theta}|\vect{Y}^{obs},\vect{Y}=\vect{Y}^{obs},\tau) = (\mat{X}^\top (\mat{I}_n + \tau \mat{Q}^{-1})^{-1} \mat{X})^{-1}(\mat{X}^\top (\mat{I}_n + \tau \mat{Q}^{-1})^{-1} \vect{Y}):= \vect{\mu}_n(\tau).
#
# If this point does not respect the constraints, then we simply project
# each component unto the constrained space.
#
# Hence the following 1D problem remains to be solved:
#
# .. math::
#
#     \widehat\tau = \arg\max_{\tau\in[\tau_{\min};\tau_{\max}]} \mathcal N(\vect{Y}^{obs}|\vect{\mu}_n(\tau) \mat{X}; \tau^{-1} \mat{I}_n + \mat{Q}^{-1}).
#
# The optimal value of :math:`\vect{\theta}` is then given by:
# :math:`\widehat{\vect{\theta}} = \vect{\mu}_n(\widehat\tau).`


def mu_n(tau):
    x = ot.Point(initial_state)
    x[p] = tau
    # trick: posterior conditional mean is computed by the link function
    return py_link_function_theta(x)[:p]


# optimization criterion
def log_cond_tau_post(tau):
    x = ot.Point(initial_state)
    x[p] = tau
    # replace theta by its conditional posterior mean
    x[:p] = py_link_function_theta(x)[:p]
    # compute log conditional posterior of tau
    ll = py_log_density(x)[0]
    return ll


# %%
# 1D optimization
def func(X):
    return [-log_cond_tau_post(X[0])]


problem = ot.OptimizationProblem(ot.PythonFunction(1, 1, func))
problem.setBounds(ot.Interval([1e-4], [1e4]))
solver = ot.TNC(problem)
solver.setStartingPoint([1.0])
solver.run()
tauhat = solver.getResult().getOptimalPoint()[0]
print("tauhat =", tauhat)

# inject result in initialState vector
x = ot.Point(initial_state)
x[:p] = mu_n(tauhat)
x[p] = tauhat


# %%
# Final step : create a :class:`~openturns.MetropolisHastings` object for each block

initialState = x

mi_Y = [i for i in range(p + 1, p + 1 + n)]
link_function_y = ot.PythonFunction(len(x), 4 * n, py_link_function_y)
rvmh_Y = ot.RandomVectorMetropolisHastings(RV_Y, initialState, mi_Y, link_function_y)

mi_theta = [i for i in range(p)]
link_function_theta = ot.PythonFunction(len(x), p + (p + 2) * p, py_link_function_theta)
rvmh_theta = ot.RandomVectorMetropolisHastings(
    RV_theta, initialState, mi_theta, link_function_theta
)

log_pdf_tau = ot.PythonFunction(len(x), 1, py_log_density)
rwmh_tau = ot.RandomWalkMetropolisHastings(
    log_pdf_tau, support, initialState, proposal_tau, [p]
)

# Now, assemble the blocks to create a Gibbs algorithm:
gibbs = ot.Gibbs([rvmh_Y, rvmh_theta, rwmh_tau])

# %%
# Launch Algorithm
sampleSize = 1000

# Joint posterior density sample
sample = gibbs.getSample(sampleSize)

# compute acceptance rate
tau_post = np.array(sample[:, p]).ravel()
acc_rate = (tau_post[1:] != tau_post[:-1]).mean()
print("Acceptance rate: %s" % acc_rate)

# %%
# Plot posterior distribution marginals

# extract interest parameters
post_sample = sample.getMarginal([i for i in range(p + 1)])
post_sample.setDescription(["$\\theta_0$", "$\\theta_1$", "$\\tau$"])

posterior = ot.KernelSmoothing().build(post_sample)
posterior = ot.TruncatedDistribution(posterior, prior.getRange())

grid = ot.GridLayout(1, 3)
grid.setTitle("Bayesian inference")
xlabs = [r"$\theta_0$", r"$\theta_1$", r"$\tau$"]
p_true = [theta_true[0][0], theta_true[1][0], tau_true]
for parameter_index in range(3):
    graph = posterior.getMarginal(parameter_index).drawPDF()
    bbox = graph.getBoundingBox()
    bound = bbox.getUpperBound()[1]
    prior_pdf = prior.getMarginal(parameter_index).drawPDF()
    graph.add(prior_pdf)
    curve_true = ot.Curve([p_true[parameter_index]] * 2, [0.0, bound])
    graph.add(curve_true)
    graph.setColors(ot.Drawable.BuildDefaultPalette(3))
    graph.setLegends(["Posterior", "Prior", "True value"])
    graph.setXTitle(xlabs[parameter_index])
    grid.setGraph(0, parameter_index, graph)
_ = View(grid)


# %%
# Draw pairplots of the posterior sample.

# sphinx_gallery_thumbnail_number = 3
grid = ot.VisualTest.DrawPairs(post_sample)

graph = grid.getGraph(0, 0)
pt = ot.Cloud(theta_true.T, "#2ca02c", "circle", "True value")
graph.add(pt)
grid.setGraph(0, 0, graph)

graph = grid.getGraph(1, 0)
pt = ot.Cloud([[theta_true[0, 0], tau_true]], "#2ca02c", "circle", "True value")
graph.add(pt)
grid.setGraph(1, 0, graph)

graph = grid.getGraph(1, 1)
pt = ot.Cloud([[theta_true[1, 0], tau_true]], "#2ca02c", "circle", "True value")
graph.add(pt)
grid.setGraph(1, 1, graph)

_ = View(grid)


View.ShowAll()
