AIM-PIbd-32-Kurbanova-A-A/aimenv/Lib/site-packages/statsmodels/genmod/bayes_mixed_glm.py

r"""
Bayesian inference for generalized linear mixed models.

Currently only families without additional scale or shape parameters
are supported (binomial and Poisson).

Two estimation approaches are supported: Laplace approximation
('maximum a posteriori'), and variational Bayes (mean field
approximation to the posterior distribution).

All realizations of random effects are modeled to be mutually
independent in this implementation.

The `exog_vc` matrix is the design matrix for the random effects.
Every column of `exog_vc` corresponds to an independent realization of
a random effect.  These random effects have mean zero and an unknown
standard deviation.  The standard deviation parameters are constrained
to be equal within subsets of the columns. When not using formulas,
these subsets are specified through the parameter `ident`.  `ident`
must have the same length as the number of columns of `exog_vc`, and
two columns whose `ident` values are equal have the same standard
deviation.  When formulas are used, the columns of `exog_vc` derived
from a common formula are constrained to have the same standard
deviation.

In many applications, `exog_vc` will be sparse.  A sparse matrix may
be passed when constructing a model class.  If a dense matrix is
passed, it will be converted internally to a sparse matrix.  There
currently is no way to avoid creating a temporary dense version of
`exog_vc` when using formulas.

Model and parameterization
--------------------------
The joint density of data and parameters factors as:

.. math::

    p(y | vc, fep) p(vc | vcp) p(vcp) p(fe)

The terms :math:`p(vcp)` and :math:`p(fe)` are prior distributions
that are taken to be Gaussian (the :math:`vcp` parameters are log
standard deviations so the standard deviations have log-normal
distributions).  The random effects distribution :math:`p(vc | vcp)`
is independent Gaussian (random effect realizations are independent
within and between values of the `ident` array).  The model
:math:`p(y | vc, fep)` depends on the specific GLM being fit.
"""

import numpy as np
from scipy.optimize import minimize
from scipy import sparse
import statsmodels.base.model as base
from statsmodels.iolib import summary2
from statsmodels.genmod import families
import pandas as pd
import warnings
import patsy

# Gauss-Legendre weights
glw = [
    [0.2955242247147529, -0.1488743389816312],
    [0.2955242247147529, 0.1488743389816312],
    [0.2692667193099963, -0.4333953941292472],
    [0.2692667193099963, 0.4333953941292472],
    [0.2190863625159820, -0.6794095682990244],
    [0.2190863625159820, 0.6794095682990244],
    [0.1494513491505806, -0.8650633666889845],
    [0.1494513491505806, 0.8650633666889845],
    [0.0666713443086881, -0.9739065285171717],
    [0.0666713443086881, 0.9739065285171717],
]

_init_doc = r"""
    Generalized Linear Mixed Model with Bayesian estimation

    The class implements the Laplace approximation to the posterior
    distribution (`fit_map`) and a variational Bayes approximation to
    the posterior (`fit_vb`).  See the two fit method docstrings for
    more information about the fitting approaches.

    Parameters
    ----------
    endog : array_like
        Vector of response values.
    exog : array_like
        Array of covariates for the fixed effects part of the mean
        structure.
    exog_vc : array_like
        Array of covariates for the random part of the model.  A
        scipy.sparse array may be provided, or else the passed
        array will be converted to sparse internally.
    ident : array_like
        Array of integer labels showing which random terms (columns
        of `exog_vc`) have a common variance.
    vcp_p : float
        Prior standard deviation for variance component parameters
        (the prior standard deviation of log(s) is vcp_p, where s is
        the standard deviation of a random effect).
    fe_p : float
        Prior standard deviation for fixed effects parameters.
    family : statsmodels.genmod.families instance
        The GLM family.
    fep_names : list[str]
        The names of the fixed effects parameters (corresponding to
        columns of exog).  If None, default names are constructed.
    vcp_names : list[str]
        The names of the variance component parameters (corresponding
        to distinct labels in ident).  If None, default names are
        constructed.
    vc_names : list[str]
        The names of the random effect realizations.

    Returns
    -------
    MixedGLMResults object

    Notes
    -----
    There are three types of values in the posterior distribution:
    fixed effects parameters (fep), corresponding to the columns of
    `exog`, random effects realizations (vc), corresponding to the
    columns of `exog_vc`, and the standard deviations of the random
    effects realizations (vcp), corresponding to the unique integer
    labels in `ident`.

    All random effects are modeled as being independent Gaussian
    values (given the variance structure parameters).  Every column of
    `exog_vc` has a distinct realized random effect that is used to
    form the linear predictors.  The elements of `ident` determine the
    distinct variance structure parameters.  Two random effect
    realizations that have the same value in `ident` have the same
    variance.  When fitting with a formula, `ident` is constructed
    internally (each element of `vc_formulas` yields a distinct label
    in `ident`).

    The random effect standard deviation parameters (`vcp`) have
    log-normal prior distributions with mean 0 and standard deviation
    `vcp_p`.

    Note that for some families, e.g. Binomial, the posterior mode may
    be difficult to find numerically if `vcp_p` is set to too large of
    a value.  Setting `vcp_p` to 0.5 seems to work well.

    The prior for the fixed effects parameters is Gaussian with mean 0
    and standard deviation `fe_p`.  It is recommended that quantitative
    covariates be standardized.

    Examples
    --------{example}


    References
    ----------
    Introduction to generalized linear mixed models:
    https://stats.idre.ucla.edu/other/mult-pkg/introduction-to-generalized-linear-mixed-models

    SAS documentation:
    https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_intromix_a0000000215.htm

    An assessment of estimation methods for generalized linear mixed
    models with binary outcomes
    https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3866838/
    """

# The code in the example should be identical to what appears in
# the test_doc_examples unit test
_logit_example = """
    A binomial (logistic) random effects model with random intercepts
    for villages and random slopes for each year within each village:

    >>> random = {"a": '0 + C(Village)', "b": '0 + C(Village)*year_cen'}
    >>> model = BinomialBayesMixedGLM.from_formula(
                   'y ~ year_cen', random, data)
    >>> result = model.fit_vb()
"""

# The code in the example should be identical to what appears in
# the test_doc_examples unit test
_poisson_example = """
    A Poisson random effects model with random intercepts for villages
    and random slopes for each year within each village:

    >>> random = {"a": '0 + C(Village)', "b": '0 + C(Village)*year_cen'}
    >>> model = PoissonBayesMixedGLM.from_formula(
                    'y ~ year_cen', random, data)
    >>> result = model.fit_vb()
"""


class _BayesMixedGLM(base.Model):
    def __init__(self,
                 endog,
                 exog,
                 exog_vc=None,
                 ident=None,
                 family=None,
                 vcp_p=1,
                 fe_p=2,
                 fep_names=None,
                 vcp_names=None,
                 vc_names=None,
                 **kwargs):

        if exog.ndim == 1:
            if isinstance(exog, np.ndarray):
                exog = exog[:, None]
            else:
                exog = pd.DataFrame(exog)

        if exog.ndim != 2:
            msg = "'exog' must have one or two columns"
            raise ValueError(msg)

        if exog_vc.ndim == 1:
            if isinstance(exog_vc, np.ndarray):
                exog_vc = exog_vc[:, None]
            else:
                exog_vc = pd.DataFrame(exog_vc)

        if exog_vc.ndim != 2:
            msg = "'exog_vc' must have one or two columns"
            raise ValueError(msg)

        ident = np.asarray(ident)
        if ident.ndim != 1:
            msg = "ident must be a one-dimensional array"
            raise ValueError(msg)

        if len(ident) != exog_vc.shape[1]:
            msg = "len(ident) should match the number of columns of exog_vc"
            raise ValueError(msg)

        if not np.issubdtype(ident.dtype, np.integer):
            msg = "ident must have an integer dtype"
            raise ValueError(msg)

        # Get the fixed effects parameter names
        if fep_names is None:
            if hasattr(exog, "columns"):
                fep_names = exog.columns.tolist()
            else:
                fep_names = ["FE_%d" % (k + 1) for k in range(exog.shape[1])]

        # Get the variance parameter names
        if vcp_names is None:
            vcp_names = ["VC_%d" % (k + 1) for k in range(int(max(ident)) + 1)]
        else:
            if len(vcp_names) != len(set(ident)):
                msg = "The lengths of vcp_names and ident should be the same"
                raise ValueError(msg)

        if not sparse.issparse(exog_vc):
            exog_vc = sparse.csr_matrix(exog_vc)

        ident = ident.astype(int)
        vcp_p = float(vcp_p)
        fe_p = float(fe_p)

        # Number of fixed effects parameters
        if exog is None:
            k_fep = 0
        else:
            k_fep = exog.shape[1]

        # Number of variance component structure parameters and
        # variance component realizations.
        if exog_vc is None:
            k_vc = 0
            k_vcp = 0
        else:
            k_vc = exog_vc.shape[1]
            k_vcp = max(ident) + 1

        # power might be better but not available in older scipy
        exog_vc2 = exog_vc.multiply(exog_vc)

        super().__init__(endog, exog, **kwargs)

        self.exog_vc = exog_vc
        self.exog_vc2 = exog_vc2
        self.ident = ident
        self.family = family
        self.k_fep = k_fep
        self.k_vc = k_vc
        self.k_vcp = k_vcp
        self.fep_names = fep_names
        self.vcp_names = vcp_names
        self.vc_names = vc_names
        self.fe_p = fe_p
        self.vcp_p = vcp_p
        self.names = fep_names + vcp_names
        if vc_names is not None:
            self.names += vc_names

    def _unpack(self, vec):

        ii = 0

        # Fixed effects parameters
        fep = vec[:ii + self.k_fep]
        ii += self.k_fep

        # Variance component structure parameters (standard
        # deviations).  These are on the log scale.  The standard
        # deviation for random effect j is exp(vcp[ident[j]]).
        vcp = vec[ii:ii + self.k_vcp]
        ii += self.k_vcp

        # Random effect realizations
        vc = vec[ii:]

        return fep, vcp, vc

    def logposterior(self, params):
        """
        The overall log-density: log p(y, fe, vc, vcp).

        This differs by an additive constant from the log posterior
        log p(fe, vc, vcp | y).
        """

        fep, vcp, vc = self._unpack(params)

        # Contributions from p(y | x, vc)
        lp = 0
        if self.k_fep > 0:
            lp += np.dot(self.exog, fep)
        if self.k_vc > 0:
            lp += self.exog_vc.dot(vc)

        mu = self.family.link.inverse(lp)
        ll = self.family.loglike(self.endog, mu)

        if self.k_vc > 0:

            # Contributions from p(vc | vcp)
            vcp0 = vcp[self.ident]
            s = np.exp(vcp0)
            ll -= 0.5 * np.sum(vc**2 / s**2) + np.sum(vcp0)

            # Contributions from p(vc)
            ll -= 0.5 * np.sum(vcp**2 / self.vcp_p**2)

        # Contributions from p(fep)
        if self.k_fep > 0:
            ll -= 0.5 * np.sum(fep**2 / self.fe_p**2)

        return ll

    def logposterior_grad(self, params):
        """
        The gradient of the log posterior.
        """

        fep, vcp, vc = self._unpack(params)

        lp = 0
        if self.k_fep > 0:
            lp += np.dot(self.exog, fep)
        if self.k_vc > 0:
            lp += self.exog_vc.dot(vc)

        mu = self.family.link.inverse(lp)

        score_factor = (self.endog - mu) / self.family.link.deriv(mu)
        score_factor /= self.family.variance(mu)

        te = [None, None, None]

        # Contributions from p(y | x, z, vc)
        if self.k_fep > 0:
            te[0] = np.dot(score_factor, self.exog)
        if self.k_vc > 0:
            te[2] = self.exog_vc.transpose().dot(score_factor)

        if self.k_vc > 0:
            # Contributions from p(vc | vcp)
            # vcp0 = vcp[self.ident]
            # s = np.exp(vcp0)
            # ll -= 0.5 * np.sum(vc**2 / s**2) + np.sum(vcp0)
            vcp0 = vcp[self.ident]
            s = np.exp(vcp0)
            u = vc**2 / s**2 - 1
            te[1] = np.bincount(self.ident, weights=u)
            te[2] -= vc / s**2

            # Contributions from p(vcp)
            # ll -= 0.5 * np.sum(vcp**2 / self.vcp_p**2)
            te[1] -= vcp / self.vcp_p**2

        # Contributions from p(fep)
        if self.k_fep > 0:
            te[0] -= fep / self.fe_p**2

        te = [x for x in te if x is not None]

        return np.concatenate(te)

    def _get_start(self):
        start_fep = np.zeros(self.k_fep)
        start_vcp = np.ones(self.k_vcp)
        start_vc = np.random.normal(size=self.k_vc)
        start = np.concatenate((start_fep, start_vcp, start_vc))
        return start

    @classmethod
    def from_formula(cls,
                     formula,
                     vc_formulas,
                     data,
                     family=None,
                     vcp_p=1,
                     fe_p=2):
        """
        Fit a BayesMixedGLM using a formula.

        Parameters
        ----------
        formula : str
            Formula for the endog and fixed effects terms (use ~ to
            separate dependent and independent expressions).
        vc_formulas : dictionary
            vc_formulas[name] is a one-sided formula that creates one
            collection of random effects with a common variance
            parameter.  If using categorical (factor) variables to
            produce variance components, note that generally `0 + ...`
            should be used so that an intercept is not included.
        data : data frame
            The data to which the formulas are applied.
        family : genmod.families instance
            A GLM family.
        vcp_p : float
            The prior standard deviation for the logarithms of the standard
            deviations of the random effects.
        fe_p : float
            The prior standard deviation for the fixed effects parameters.
        """

        ident = []
        exog_vc = []
        vcp_names = []
        j = 0
        for na, fml in vc_formulas.items():
            mat = patsy.dmatrix(fml, data, return_type='dataframe')
            exog_vc.append(mat)
            vcp_names.append(na)
            ident.append(j * np.ones(mat.shape[1], dtype=np.int_))
            j += 1
        exog_vc = pd.concat(exog_vc, axis=1)
        vc_names = exog_vc.columns.tolist()

        ident = np.concatenate(ident)

        model = super().from_formula(
            formula,
            data=data,
            family=family,
            subset=None,
            exog_vc=exog_vc,
            ident=ident,
            vc_names=vc_names,
            vcp_names=vcp_names,
            fe_p=fe_p,
            vcp_p=vcp_p)

        return model

    def fit(self, method="BFGS", minim_opts=None):
        """
        fit is equivalent to fit_map.

        See fit_map for parameter information.

        Use `fit_vb` to fit the model using variational Bayes.
        """
        self.fit_map(method, minim_opts)

    def fit_map(self, method="BFGS", minim_opts=None, scale_fe=False):
        """
        Construct the Laplace approximation to the posterior distribution.

        Parameters
        ----------
        method : str
            Optimization method for finding the posterior mode.
        minim_opts : dict
            Options passed to scipy.minimize.
        scale_fe : bool
            If True, the columns of the fixed effects design matrix
            are centered and scaled to unit variance before fitting
            the model.  The results are back-transformed so that the
            results are presented on the original scale.

        Returns
        -------
        BayesMixedGLMResults instance.
        """

        if scale_fe:
            mn = self.exog.mean(0)
            sc = self.exog.std(0)
            self._exog_save = self.exog
            self.exog = self.exog.copy()
            ixs = np.flatnonzero(sc > 1e-8)
            self.exog[:, ixs] -= mn[ixs]
            self.exog[:, ixs] /= sc[ixs]

        def fun(params):
            return -self.logposterior(params)

        def grad(params):
            return -self.logposterior_grad(params)

        start = self._get_start()

        r = minimize(fun, start, method=method, jac=grad, options=minim_opts)
        if not r.success:
            msg = ("Laplace fitting did not converge, |gradient|=%.6f" %
                   np.sqrt(np.sum(r.jac**2)))
            warnings.warn(msg)

        from statsmodels.tools.numdiff import approx_fprime
        hess = approx_fprime(r.x, grad)
        cov = np.linalg.inv(hess)

        params = r.x

        if scale_fe:
            self.exog = self._exog_save
            del self._exog_save
            params[ixs] /= sc[ixs]
            cov[ixs, :][:, ixs] /= np.outer(sc[ixs], sc[ixs])

        return BayesMixedGLMResults(self, params, cov, optim_retvals=r)

    def predict(self, params, exog=None, linear=False):
        """
        Return the fitted mean structure.

        Parameters
        ----------
        params : array_like
            The parameter vector, may be the full parameter vector, or may
            be truncated to include only the mean parameters.
        exog : array_like
            The design matrix for the mean structure.  If omitted, use the
            model's design matrix.
        linear : bool
            If True, return the linear predictor without passing through the
            link function.

        Returns
        -------
        A 1-dimensional array of predicted values
        """

        if exog is None:
            exog = self.exog

        q = exog.shape[1]
        pr = np.dot(exog, params[0:q])

        if not linear:
            pr = self.family.link.inverse(pr)

        return pr


class _VariationalBayesMixedGLM:
    """
    A mixin providing generic (not family-specific) methods for
    variational Bayes mean field fitting.
    """

    # Integration range (from -rng to +rng).  The integrals are with
    # respect to a standard Gaussian distribution so (-5, 5) will be
    # sufficient in many cases.
    rng = 5

    verbose = False

    # Returns the mean and variance of the linear predictor under the
    # given distribution parameters.
    def _lp_stats(self, fep_mean, fep_sd, vc_mean, vc_sd):

        tm = np.dot(self.exog, fep_mean)
        tv = np.dot(self.exog**2, fep_sd**2)
        tm += self.exog_vc.dot(vc_mean)
        tv += self.exog_vc2.dot(vc_sd**2)

        return tm, tv

    def vb_elbo_base(self, h, tm, fep_mean, vcp_mean, vc_mean, fep_sd, vcp_sd,
                     vc_sd):
        """
        Returns the evidence lower bound (ELBO) for the model.

        This function calculates the family-specific ELBO function
        based on information provided from a subclass.

        Parameters
        ----------
        h : function mapping 1d vector to 1d vector
            The contribution of the model to the ELBO function can be
            expressed as y_i*lp_i + Eh_i(z), where y_i and lp_i are
            the response and linear predictor for observation i, and z
            is a standard normal random variable.  This formulation
            can be achieved for any GLM with a canonical link
            function.
        """

        # p(y | vc) contributions
        iv = 0
        for w in glw:
            z = self.rng * w[1]
            iv += w[0] * h(z) * np.exp(-z**2 / 2)
        iv /= np.sqrt(2 * np.pi)
        iv *= self.rng
        iv += self.endog * tm
        iv = iv.sum()

        # p(vc | vcp) * p(vcp) * p(fep) contributions
        iv += self._elbo_common(fep_mean, fep_sd, vcp_mean, vcp_sd, vc_mean,
                                vc_sd)

        r = (iv + np.sum(np.log(fep_sd)) + np.sum(np.log(vcp_sd)) + np.sum(
            np.log(vc_sd)))

        return r

    def vb_elbo_grad_base(self, h, tm, tv, fep_mean, vcp_mean, vc_mean, fep_sd,
                          vcp_sd, vc_sd):
        """
        Return the gradient of the ELBO function.

        See vb_elbo_base for parameters.
        """

        fep_mean_grad = 0.
        fep_sd_grad = 0.
        vcp_mean_grad = 0.
        vcp_sd_grad = 0.
        vc_mean_grad = 0.
        vc_sd_grad = 0.

        # p(y | vc) contributions
        for w in glw:
            z = self.rng * w[1]
            u = h(z) * np.exp(-z**2 / 2) / np.sqrt(2 * np.pi)
            r = u / np.sqrt(tv)
            fep_mean_grad += w[0] * np.dot(u, self.exog)
            vc_mean_grad += w[0] * self.exog_vc.transpose().dot(u)
            fep_sd_grad += w[0] * z * np.dot(r, self.exog**2 * fep_sd)
            v = self.exog_vc2.multiply(vc_sd).transpose().dot(r)
            v = np.squeeze(np.asarray(v))
            vc_sd_grad += w[0] * z * v

        fep_mean_grad *= self.rng
        vc_mean_grad *= self.rng
        fep_sd_grad *= self.rng
        vc_sd_grad *= self.rng
        fep_mean_grad += np.dot(self.endog, self.exog)
        vc_mean_grad += self.exog_vc.transpose().dot(self.endog)

        (fep_mean_grad_i, fep_sd_grad_i, vcp_mean_grad_i, vcp_sd_grad_i,
         vc_mean_grad_i, vc_sd_grad_i) = self._elbo_grad_common(
             fep_mean, fep_sd, vcp_mean, vcp_sd, vc_mean, vc_sd)

        fep_mean_grad += fep_mean_grad_i
        fep_sd_grad += fep_sd_grad_i
        vcp_mean_grad += vcp_mean_grad_i
        vcp_sd_grad += vcp_sd_grad_i
        vc_mean_grad += vc_mean_grad_i
        vc_sd_grad += vc_sd_grad_i

        fep_sd_grad += 1 / fep_sd
        vcp_sd_grad += 1 / vcp_sd
        vc_sd_grad += 1 / vc_sd

        mean_grad = np.concatenate((fep_mean_grad, vcp_mean_grad,
                                    vc_mean_grad))
        sd_grad = np.concatenate((fep_sd_grad, vcp_sd_grad, vc_sd_grad))

        if self.verbose:
            print(
                "|G|=%f" % np.sqrt(np.sum(mean_grad**2) + np.sum(sd_grad**2)))

        return mean_grad, sd_grad

    def fit_vb(self,
               mean=None,
               sd=None,
               fit_method="BFGS",
               minim_opts=None,
               scale_fe=False,
               verbose=False):
        """
        Fit a model using the variational Bayes mean field approximation.

        Parameters
        ----------
        mean : array_like
            Starting value for VB mean vector
        sd : array_like
            Starting value for VB standard deviation vector
        fit_method : str
            Algorithm for scipy.minimize
        minim_opts : dict
            Options passed to scipy.minimize
        scale_fe : bool
            If true, the columns of the fixed effects design matrix
            are centered and scaled to unit variance before fitting
            the model.  The results are back-transformed so that the
            results are presented on the original scale.
        verbose : bool
            If True, print the gradient norm to the screen each time
            it is calculated.

        Notes
        -----
        The goal is to find a factored Gaussian approximation
        q1*q2*...  to the posterior distribution, approximately
        minimizing the KL divergence from the factored approximation
        to the actual posterior.  The KL divergence, or ELBO function
        has the form

            E* log p(y, fe, vcp, vc) - E* log q

        where E* is expectation with respect to the product of qj.

        References
        ----------
        Blei, Kucukelbir, McAuliffe (2017).  Variational Inference: A
        review for Statisticians
        https://arxiv.org/pdf/1601.00670.pdf
        """

        self.verbose = verbose

        if scale_fe:
            mn = self.exog.mean(0)
            sc = self.exog.std(0)
            self._exog_save = self.exog
            self.exog = self.exog.copy()
            ixs = np.flatnonzero(sc > 1e-8)
            self.exog[:, ixs] -= mn[ixs]
            self.exog[:, ixs] /= sc[ixs]

        n = self.k_fep + self.k_vcp + self.k_vc
        ml = self.k_fep + self.k_vcp + self.k_vc
        if mean is None:
            m = np.zeros(n)
        else:
            if len(mean) != ml:
                raise ValueError(
                    "mean has incorrect length, %d != %d" % (len(mean), ml))
            m = mean.copy()
        if sd is None:
            s = -0.5 + 0.1 * np.random.normal(size=n)
        else:
            if len(sd) != ml:
                raise ValueError(
                    "sd has incorrect length, %d != %d" % (len(sd), ml))

            # s is parametrized on the log-scale internally when
            # optimizing the ELBO function (this is transparent to the
            # caller)
            s = np.log(sd)

        # Do not allow the variance parameter starting mean values to
        # be too small.
        i1, i2 = self.k_fep, self.k_fep + self.k_vcp
        m[i1:i2] = np.where(m[i1:i2] < -1, -1, m[i1:i2])

        # Do not allow the posterior standard deviation starting values
        # to be too small.
        s = np.where(s < -1, -1, s)

        def elbo(x):
            n = len(x) // 2
            return -self.vb_elbo(x[:n], np.exp(x[n:]))

        def elbo_grad(x):
            n = len(x) // 2
            gm, gs = self.vb_elbo_grad(x[:n], np.exp(x[n:]))
            gs *= np.exp(x[n:])
            return -np.concatenate((gm, gs))

        start = np.concatenate((m, s))
        mm = minimize(
            elbo, start, jac=elbo_grad, method=fit_method, options=minim_opts)
        if not mm.success:
            warnings.warn("VB fitting did not converge")

        n = len(mm.x) // 2
        params = mm.x[0:n]
        va = np.exp(2 * mm.x[n:])

        if scale_fe:
            self.exog = self._exog_save
            del self._exog_save
            params[ixs] /= sc[ixs]
            va[ixs] /= sc[ixs]**2

        return BayesMixedGLMResults(self, params, va, mm)

    # Handle terms in the ELBO that are common to all models.
    def _elbo_common(self, fep_mean, fep_sd, vcp_mean, vcp_sd, vc_mean, vc_sd):

        iv = 0

        # p(vc | vcp) contributions
        m = vcp_mean[self.ident]
        s = vcp_sd[self.ident]
        iv -= np.sum((vc_mean**2 + vc_sd**2) * np.exp(2 * (s**2 - m))) / 2
        iv -= np.sum(m)

        # p(vcp) contributions
        iv -= 0.5 * (vcp_mean**2 + vcp_sd**2).sum() / self.vcp_p**2

        # p(b) contributions
        iv -= 0.5 * (fep_mean**2 + fep_sd**2).sum() / self.fe_p**2

        return iv

    def _elbo_grad_common(self, fep_mean, fep_sd, vcp_mean, vcp_sd, vc_mean,
                          vc_sd):

        # p(vc | vcp) contributions
        m = vcp_mean[self.ident]
        s = vcp_sd[self.ident]
        u = vc_mean**2 + vc_sd**2
        ve = np.exp(2 * (s**2 - m))
        dm = u * ve - 1
        ds = -2 * u * ve * s
        vcp_mean_grad = np.bincount(self.ident, weights=dm)
        vcp_sd_grad = np.bincount(self.ident, weights=ds)

        vc_mean_grad = -vc_mean.copy() * ve
        vc_sd_grad = -vc_sd.copy() * ve

        # p(vcp) contributions
        vcp_mean_grad -= vcp_mean / self.vcp_p**2
        vcp_sd_grad -= vcp_sd / self.vcp_p**2

        # p(b) contributions
        fep_mean_grad = -fep_mean.copy() / self.fe_p**2
        fep_sd_grad = -fep_sd.copy() / self.fe_p**2

        return (fep_mean_grad, fep_sd_grad, vcp_mean_grad, vcp_sd_grad,
                vc_mean_grad, vc_sd_grad)


class BayesMixedGLMResults:
    """
    Class to hold results from a Bayesian estimation of a Mixed GLM model.

    Attributes
    ----------
    fe_mean : array_like
        Posterior mean of the fixed effects coefficients.
    fe_sd : array_like
        Posterior standard deviation of the fixed effects coefficients
    vcp_mean : array_like
        Posterior mean of the logged variance component standard
        deviations.
    vcp_sd : array_like
        Posterior standard deviation of the logged variance component
        standard deviations.
    vc_mean : array_like
        Posterior mean of the random coefficients
    vc_sd : array_like
        Posterior standard deviation of the random coefficients
    """

    def __init__(self, model, params, cov_params, optim_retvals=None):

        self.model = model
        self.params = params
        self._cov_params = cov_params
        self.optim_retvals = optim_retvals

        self.fe_mean, self.vcp_mean, self.vc_mean = (model._unpack(params))

        if cov_params.ndim == 2:
            cp = np.diag(cov_params)
        else:
            cp = cov_params
        self.fe_sd, self.vcp_sd, self.vc_sd = model._unpack(cp)
        self.fe_sd = np.sqrt(self.fe_sd)
        self.vcp_sd = np.sqrt(self.vcp_sd)
        self.vc_sd = np.sqrt(self.vc_sd)

    def cov_params(self):

        if hasattr(self.model.data, "frame"):
            # Return the covariance matrix as a dataframe or series
            na = (self.model.fep_names + self.model.vcp_names +
                  self.model.vc_names)
            if self._cov_params.ndim == 2:
                return pd.DataFrame(self._cov_params, index=na, columns=na)
            else:
                return pd.Series(self._cov_params, index=na)

        # Return the covariance matrix as a ndarray
        return self._cov_params

    def summary(self):

        df = pd.DataFrame()
        m = self.model.k_fep + self.model.k_vcp
        df["Type"] = (["M" for k in range(self.model.k_fep)] +
                      ["V" for k in range(self.model.k_vcp)])

        df["Post. Mean"] = self.params[0:m]

        if self._cov_params.ndim == 2:
            v = np.diag(self._cov_params)[0:m]
            df["Post. SD"] = np.sqrt(v)
        else:
            df["Post. SD"] = np.sqrt(self._cov_params[0:m])

        # Convert variance parameters to natural scale
        df["SD"] = np.exp(df["Post. Mean"])
        df["SD (LB)"] = np.exp(df["Post. Mean"] - 2 * df["Post. SD"])
        df["SD (UB)"] = np.exp(df["Post. Mean"] + 2 * df["Post. SD"])
        df["SD"] = ["%.3f" % x for x in df.SD]
        df["SD (LB)"] = ["%.3f" % x for x in df["SD (LB)"]]
        df["SD (UB)"] = ["%.3f" % x for x in df["SD (UB)"]]
        df.loc[df.index < self.model.k_fep, "SD"] = ""
        df.loc[df.index < self.model.k_fep, "SD (LB)"] = ""
        df.loc[df.index < self.model.k_fep, "SD (UB)"] = ""

        df.index = self.model.fep_names + self.model.vcp_names

        summ = summary2.Summary()
        summ.add_title(self.model.family.__class__.__name__ +
                       " Mixed GLM Results")
        summ.add_df(df)

        summ.add_text("Parameter types are mean structure (M) and "
                      "variance structure (V)")
        summ.add_text("Variance parameters are modeled as log "
                      "standard deviations")

        return summ

    def random_effects(self, term=None):
        """
        Posterior mean and standard deviation of random effects.

        Parameters
        ----------
        term : int or None
            If None, results for all random effects are returned.  If
            an integer, returns results for a given set of random
            effects.  The value of `term` refers to an element of the
            `ident` vector, or to a position in the `vc_formulas`
            list.

        Returns
        -------
        Data frame of posterior means and posterior standard
        deviations of random effects.
        """

        z = self.vc_mean
        s = self.vc_sd
        na = self.model.vc_names

        if term is not None:
            termix = self.model.vcp_names.index(term)
            ii = np.flatnonzero(self.model.ident == termix)
            z = z[ii]
            s = s[ii]
            na = [na[i] for i in ii]

        x = pd.DataFrame({"Mean": z, "SD": s})

        if na is not None:
            x.index = na

        return x

    def predict(self, exog=None, linear=False):
        """
        Return predicted values for the mean structure.

        Parameters
        ----------
        exog : array_like
            The design matrix for the mean structure.  If None,
            use the model's design matrix.
        linear : bool
            If True, returns the linear predictor, otherwise
            transform the linear predictor using the link function.

        Returns
        -------
        A one-dimensional array of fitted values.
        """

        return self.model.predict(self.params, exog, linear)


class BinomialBayesMixedGLM(_VariationalBayesMixedGLM, _BayesMixedGLM):

    __doc__ = _init_doc.format(example=_logit_example)

    def __init__(self,
                 endog,
                 exog,
                 exog_vc,
                 ident,
                 vcp_p=1,
                 fe_p=2,
                 fep_names=None,
                 vcp_names=None,
                 vc_names=None):

        super().__init__(
            endog,
            exog,
            exog_vc=exog_vc,
            ident=ident,
            vcp_p=vcp_p,
            fe_p=fe_p,
            family=families.Binomial(),
            fep_names=fep_names,
            vcp_names=vcp_names,
            vc_names=vc_names)

        if not np.all(np.unique(endog) == np.r_[0, 1]):
            msg = "endog values must be 0 and 1, and not all identical"
            raise ValueError(msg)

    @classmethod
    def from_formula(cls, formula, vc_formulas, data, vcp_p=1, fe_p=2):

        fam = families.Binomial()
        x = _BayesMixedGLM.from_formula(
            formula, vc_formulas, data, family=fam, vcp_p=vcp_p, fe_p=fe_p)

        # Copy over to the intended class structure
        mod = BinomialBayesMixedGLM(
            x.endog,
            x.exog,
            exog_vc=x.exog_vc,
            ident=x.ident,
            vcp_p=x.vcp_p,
            fe_p=x.fe_p,
            fep_names=x.fep_names,
            vcp_names=x.vcp_names,
            vc_names=x.vc_names)
        mod.data = x.data

        return mod

    def vb_elbo(self, vb_mean, vb_sd):
        """
        Returns the evidence lower bound (ELBO) for the model.
        """

        fep_mean, vcp_mean, vc_mean = self._unpack(vb_mean)
        fep_sd, vcp_sd, vc_sd = self._unpack(vb_sd)
        tm, tv = self._lp_stats(fep_mean, fep_sd, vc_mean, vc_sd)

        def h(z):
            return -np.log(1 + np.exp(tm + np.sqrt(tv) * z))

        return self.vb_elbo_base(h, tm, fep_mean, vcp_mean, vc_mean, fep_sd,
                                 vcp_sd, vc_sd)

    def vb_elbo_grad(self, vb_mean, vb_sd):
        """
        Returns the gradient of the model's evidence lower bound (ELBO).
        """

        fep_mean, vcp_mean, vc_mean = self._unpack(vb_mean)
        fep_sd, vcp_sd, vc_sd = self._unpack(vb_sd)
        tm, tv = self._lp_stats(fep_mean, fep_sd, vc_mean, vc_sd)

        def h(z):
            u = tm + np.sqrt(tv) * z
            x = np.zeros_like(u)
            ii = np.flatnonzero(u > 0)
            uu = u[ii]
            x[ii] = 1 / (1 + np.exp(-uu))
            ii = np.flatnonzero(u <= 0)
            uu = u[ii]
            x[ii] = np.exp(uu) / (1 + np.exp(uu))
            return -x

        return self.vb_elbo_grad_base(h, tm, tv, fep_mean, vcp_mean, vc_mean,
                                      fep_sd, vcp_sd, vc_sd)


class PoissonBayesMixedGLM(_VariationalBayesMixedGLM, _BayesMixedGLM):

    __doc__ = _init_doc.format(example=_poisson_example)

    def __init__(self,
                 endog,
                 exog,
                 exog_vc,
                 ident,
                 vcp_p=1,
                 fe_p=2,
                 fep_names=None,
                 vcp_names=None,
                 vc_names=None):

        super().__init__(
            endog=endog,
            exog=exog,
            exog_vc=exog_vc,
            ident=ident,
            vcp_p=vcp_p,
            fe_p=fe_p,
            family=families.Poisson(),
            fep_names=fep_names,
            vcp_names=vcp_names,
            vc_names=vc_names)

    @classmethod
    def from_formula(cls,
                     formula,
                     vc_formulas,
                     data,
                     vcp_p=1,
                     fe_p=2,
                     vcp_names=None,
                     vc_names=None):

        fam = families.Poisson()
        x = _BayesMixedGLM.from_formula(
            formula,
            vc_formulas,
            data,
            family=fam,
            vcp_p=vcp_p,
            fe_p=fe_p)

        # Copy over to the intended class structure
        mod = PoissonBayesMixedGLM(
            endog=x.endog,
            exog=x.exog,
            exog_vc=x.exog_vc,
            ident=x.ident,
            vcp_p=x.vcp_p,
            fe_p=x.fe_p,
            fep_names=x.fep_names,
            vcp_names=x.vcp_names,
            vc_names=x.vc_names)
        mod.data = x.data

        return mod

    def vb_elbo(self, vb_mean, vb_sd):
        """
        Returns the evidence lower bound (ELBO) for the model.
        """

        fep_mean, vcp_mean, vc_mean = self._unpack(vb_mean)
        fep_sd, vcp_sd, vc_sd = self._unpack(vb_sd)
        tm, tv = self._lp_stats(fep_mean, fep_sd, vc_mean, vc_sd)

        def h(z):
            return -np.exp(tm + np.sqrt(tv) * z)

        return self.vb_elbo_base(h, tm, fep_mean, vcp_mean, vc_mean, fep_sd,
                                 vcp_sd, vc_sd)

    def vb_elbo_grad(self, vb_mean, vb_sd):
        """
        Returns the gradient of the model's evidence lower bound (ELBO).
        """

        fep_mean, vcp_mean, vc_mean = self._unpack(vb_mean)
        fep_sd, vcp_sd, vc_sd = self._unpack(vb_sd)
        tm, tv = self._lp_stats(fep_mean, fep_sd, vc_mean, vc_sd)

        def h(z):
            y = -np.exp(tm + np.sqrt(tv) * z)
            return y

        return self.vb_elbo_grad_base(h, tm, tv, fep_mean, vcp_mean, vc_mean,
                                      fep_sd, vcp_sd, vc_sd)