# Copyright (c) 2012-2015 The GPy authors (see AUTHORS.txt)
# Licensed under the BSD 3-clause license (see LICENSE.txt)

import numpy as np
from scipy import stats,special
import scipy as sp
from . import link_functions
from ..util.misc import chain_1, chain_2, chain_3, blockify_dhess_dtheta, blockify_third, blockify_hessian, safe_exp
from scipy.integrate import quad
import warnings
from ..core.parameterization import Parameterized

class Likelihood(Parameterized):
    """
    Likelihood base class, used to defing p(y|f).

    All instances use _inverse_ link functions, which can be swapped out. It is
    expected that inheriting classes define a default inverse link function

    To use this class, inherit and define missing functionality.

    Inheriting classes *must* implement:
       pdf_link : a bound method which turns the output of the link function into the pdf
       logpdf_link : the logarithm of the above

    To enable use with EP, inheriting classes *must* define:
       TODO: a suitable derivative function for any parameters of the class
    It is also desirable to define:
       moments_match_ep : a function to compute the EP moments If this isn't defined, the moments will be computed using 1D quadrature.

    To enable use with Laplace approximation, inheriting classes *must* define:
       Some derivative functions *AS TODO*

    For exact Gaussian inference, define *JH TODO*

    """
    def __init__(self, gp_link, name):
        super(Likelihood, self).__init__(name)
        assert isinstance(gp_link,link_functions.GPTransformation), "gp_link is not a valid GPTransformation."
        self.gp_link = gp_link
        self.log_concave = False
        self.not_block_really = False

    def request_num_latent_functions(self, Y):
        """
        The likelihood should infer how many latent functions are needed for the likelihood

        Default is the number of outputs
        """
        return Y.shape[1]

    def _gradients(self,partial):
        return np.zeros(0)

    def update_gradients(self, partial):
        if self.size > 0:
            raise NotImplementedError('Must be implemented for likelihoods with parameters to be optimized')

    def _preprocess_values(self,Y):
        """
        In case it is needed, this function assess the output values or makes any pertinent transformation on them.

        :param Y: observed output
        :type Y: Nx1 numpy.darray

        """
        return Y

    def conditional_mean(self, gp):
        """
        The mean of the random variable conditioned on one value of the GP
        """
        raise NotImplementedError

    def conditional_variance(self, gp):
        """
        The variance of the random variable conditioned on one value of the GP
        """
        raise NotImplementedError

    def log_predictive_density(self, y_test, mu_star, var_star, Y_metadata=None):
        """
        Calculation of the log predictive density

        .. math:
            p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*})

        :param y_test: test observations (y_{*})
        :type y_test: (Nx1) array
        :param mu_star: predictive mean of gaussian p(f_{*}|mu_{*}, var_{*})
        :type mu_star: (Nx1) array
        :param var_star: predictive variance of gaussian p(f_{*}|mu_{*}, var_{*})
        :type var_star: (Nx1) array
        """
        assert y_test.shape==mu_star.shape
        assert y_test.shape==var_star.shape
        assert y_test.shape[1] == 1

        flat_y_test = y_test.flatten()
        flat_mu_star = mu_star.flatten()
        flat_var_star = var_star.flatten()

        if Y_metadata is not None:
            #Need to zip individual elements of Y_metadata aswell
            Y_metadata_flat = {}
            if Y_metadata is not None:
                for key, val in Y_metadata.items():
                    Y_metadata_flat[key] = np.atleast_1d(val).reshape(-1,1)

            zipped_values = []

            for i in range(y_test.shape[0]):
                y_m = {}
                for key, val in Y_metadata_flat.items():
                    if np.isscalar(val) or val.shape[0] == 1:
                        y_m[key] = val
                    else:
                        #Won't broadcast yet
                        y_m[key] = val[i]
                zipped_values.append((flat_y_test[i], flat_mu_star[i], flat_var_star[i], y_m))
        else:
            #Otherwise just pass along None's
            zipped_values = zip(flat_y_test, flat_mu_star, flat_var_star, [None]*y_test.shape[0])

        def integral_generator(yi, mi, vi, yi_m):
            """Generate a function which can be integrated
            to give p(Y*|Y) = int p(Y*|f*)p(f*|Y) df*"""
            def f(fi_star):
                #exponent = np.exp(-(1./(2*vi))*np.square(mi-fi_star))
                #from GPy.util.misc import safe_exp
                #exponent = safe_exp(exponent)
                #res = safe_exp(self.logpdf(fi_star, yi, yi_m))*exponent

                #More stable in the log space
                res = np.exp(self.logpdf(fi_star, yi, yi_m)
                              - 0.5*np.log(2*np.pi*vi)
                              - 0.5*np.square(fi_star-mi)/vi)
                if not np.isfinite(res):
                    import ipdb; ipdb.set_trace()  # XXX BREAKPOINT
                return res

            return f

        p_ystar, _ = zip(*[quad(integral_generator(yi, mi, vi, yi_m), -np.inf, np.inf)
                           for yi, mi, vi, yi_m in zipped_values])
        p_ystar = np.array(p_ystar).reshape(*y_test.shape)
        return np.log(p_ystar)

    def log_predictive_density_sampling(self, y_test, mu_star, var_star, Y_metadata=None, num_samples=1000):
        """
        Calculation of the log predictive density via sampling

        .. math:
            log p(y_{*}|D) = log 1/num_samples prod^{S}_{s=1} p(y_{*}|f_{*s})
            f_{*s} ~ p(f_{*}|\mu_{*}\\sigma^{2}_{*})

        :param y_test: test observations (y_{*})
        :type y_test: (Nx1) array
        :param mu_star: predictive mean of gaussian p(f_{*}|mu_{*}, var_{*})
        :type mu_star: (Nx1) array
        :param var_star: predictive variance of gaussian p(f_{*}|mu_{*}, var_{*})
        :type var_star: (Nx1) array
        :param num_samples: num samples of p(f_{*}|mu_{*}, var_{*}) to take
        :type num_samples: int
        """
        assert y_test.shape==mu_star.shape
        assert y_test.shape==var_star.shape
        assert y_test.shape[1] == 1

        #Take samples of p(f*|y)
        #fi_samples = np.random.randn(num_samples)*np.sqrt(var_star) + mu_star
        fi_samples = np.random.normal(mu_star, np.sqrt(var_star), size=(mu_star.shape[0], num_samples))

        from scipy.misc import logsumexp
        log_p_ystar = -np.log(num_samples) + logsumexp(self.logpdf(fi_samples, y_test, Y_metadata=Y_metadata), axis=1)
        log_p_ystar = np.array(log_p_ystar).reshape(*y_test.shape)
        return log_p_ystar


    def _moments_match_ep(self,obs,tau,v):
        """
        Calculation of moments using quadrature

        :param obs: observed output
        :param tau: cavity distribution 1st natural parameter (precision)
        :param v: cavity distribution 2nd natural paramenter (mu*precision)
        """
        #Compute first integral for zeroth moment.
        #NOTE constant np.sqrt(2*pi/tau) added at the end of the function
        mu = v/tau
        def int_1(f):
            return self.pdf(f, obs)*np.exp(-0.5*tau*np.square(mu-f))
        z_scaled, accuracy = quad(int_1, -np.inf, np.inf)

        #Compute second integral for first moment
        def int_2(f):
            return f*self.pdf(f, obs)*np.exp(-0.5*tau*np.square(mu-f))
        mean, accuracy = quad(int_2, -np.inf, np.inf)
        mean /= z_scaled

        #Compute integral for variance
        def int_3(f):
            return (f**2)*self.pdf(f, obs)*np.exp(-0.5*tau*np.square(mu-f))
        Ef2, accuracy = quad(int_3, -np.inf, np.inf)
        Ef2 /= z_scaled
        variance = Ef2 - mean**2

        #Add constant to the zeroth moment
        #NOTE: this constant is not needed in the other moments because it cancells out.
        z = z_scaled/np.sqrt(2*np.pi/tau)

        return z, mean, variance

    #only compute gh points if required
    __gh_points = None
    def _gh_points(self, T=20):
        if self.__gh_points is None:
            self.__gh_points = np.polynomial.hermite.hermgauss(T)
        return self.__gh_points

    def variational_expectations(self, Y, m, v, gh_points=None, Y_metadata=None):
        """
        Use Gauss-Hermite Quadrature to compute

           E_p(f) [ log p(y|f) ]
           d/dm E_p(f) [ log p(y|f) ]
           d/dv E_p(f) [ log p(y|f) ]

        where p(f) is a Gaussian with mean m and variance v. The shapes of Y, m and v should match.

        if no gh_points are passed, we construct them using defualt options
        """

        if gh_points is None:
            gh_x, gh_w = self._gh_points()
        else:
            gh_x, gh_w = gh_points

        shape = m.shape
        m,v,Y = m.flatten(), v.flatten(), Y.flatten()

        #make a grid of points
        X = gh_x[None,:]*np.sqrt(2.*v[:,None]) + m[:,None]

        #evaluate the likelhood for the grid. First ax indexes the data (and mu, var) and the second indexes the grid.
        # broadcast needs to be handled carefully.
        logp = self.logpdf(X,Y[:,None], Y_metadata=Y_metadata)
        dlogp_dx = self.dlogpdf_df(X, Y[:,None], Y_metadata=Y_metadata)
        d2logp_dx2 = self.d2logpdf_df2(X, Y[:,None], Y_metadata=Y_metadata)

        #clipping for numerical stability
        #logp = np.clip(logp,-1e9,1e9)
        #dlogp_dx = np.clip(dlogp_dx,-1e9,1e9)
        #d2logp_dx2 = np.clip(d2logp_dx2,-1e9,1e9)

        #average over the gird to get derivatives of the Gaussian's parameters
        #division by pi comes from fact that for each quadrature we need to scale by 1/sqrt(pi)
        F = np.dot(logp, gh_w)/np.sqrt(np.pi)
        dF_dm = np.dot(dlogp_dx, gh_w)/np.sqrt(np.pi)
        dF_dv = np.dot(d2logp_dx2, gh_w)/np.sqrt(np.pi)
        dF_dv /= 2.

        if np.any(np.isnan(dF_dv)) or np.any(np.isinf(dF_dv)):
            stop
        if np.any(np.isnan(dF_dm)) or np.any(np.isinf(dF_dm)):
            stop

        if self.size:
            dF_dtheta = self.dlogpdf_dtheta(X, Y[:,None], Y_metadata=Y_metadata) # Ntheta x (orig size) x N_{quad_points}
            dF_dtheta = np.dot(dF_dtheta, gh_w)/np.sqrt(np.pi)
            dF_dtheta = dF_dtheta.reshape(self.size, shape[0], shape[1])
        else:
            dF_dtheta = None # Not yet implemented
        return F.reshape(*shape), dF_dm.reshape(*shape), dF_dv.reshape(*shape), dF_dtheta

    def predictive_mean(self, mu, variance, Y_metadata=None):
        """
        Quadrature calculation of the predictive mean: E(Y_star|Y) = E( E(Y_star|f_star, Y) )

        :param mu: mean of posterior
        :param sigma: standard deviation of posterior

        """
        #conditional_mean: the edpected value of y given some f, under this likelihood
        fmin = -np.inf
        fmax = np.inf
        def int_mean(f,m,v):
            exponent = -(0.5/v)*np.square(f - m)
            #If exponent is under -30 then exp(exponent) will be very small, so don't exp it!)
            #If p is zero then conditional_mean will overflow
            assert v.all() > 0
            p = safe_exp(exponent)

            #If p is zero then conditional_variance will overflow
            if p < 1e-10:
                return 0.
            else:
                return self.conditional_mean(f)*p
        scaled_mean = [quad(int_mean, fmin, fmax,args=(mj,s2j))[0] for mj,s2j in zip(mu,variance)]
        mean = np.array(scaled_mean)[:,None] / np.sqrt(2*np.pi*(variance))
        return mean

    def predictive_variance(self, mu,variance, predictive_mean=None, Y_metadata=None):
        """
        Approximation to the predictive variance: V(Y_star)

        The following variance decomposition is used:
        V(Y_star) = E( V(Y_star|f_star)**2 ) + V( E(Y_star|f_star) )**2

        :param mu: mean of posterior
        :param sigma: standard deviation of posterior
        :predictive_mean: output's predictive mean, if None _predictive_mean function will be called.

        """
        #sigma2 = sigma**2
        normalizer = np.sqrt(2*np.pi*variance)

        fmin_v = -np.inf
        fmin_m = np.inf
        fmin = -np.inf
        fmax = np.inf

        from ..util.misc import safe_exp
        # E( V(Y_star|f_star) )
        def int_var(f,m,v):
            exponent = -(0.5/v)*np.square(f - m)
            p = safe_exp(exponent)
            #If p is zero then conditional_variance will overflow
            if p < 1e-10:
                return 0.
            else:
                return self.conditional_variance(f)*p
        scaled_exp_variance = [quad(int_var, fmin_v, fmax,args=(mj,s2j))[0] for mj,s2j in zip(mu,variance)]
        exp_var = np.array(scaled_exp_variance)[:,None] / normalizer

        #V( E(Y_star|f_star) ) =  E( E(Y_star|f_star)**2 ) - E( E(Y_star|f_star) )**2

        #E( E(Y_star|f_star) )**2
        if predictive_mean is None:
            predictive_mean = self.predictive_mean(mu,variance)
        predictive_mean_sq = predictive_mean**2

        #E( E(Y_star|f_star)**2 )
        def int_pred_mean_sq(f,m,v,predictive_mean_sq):
            exponent = -(0.5/v)*np.square(f - m)
            p = np.exp(exponent)
            #If p is zero then conditional_mean**2 will overflow
            if p < 1e-10:
                return 0.
            else:
                return self.conditional_mean(f)**2*p

        scaled_exp_exp2 = [quad(int_pred_mean_sq, fmin_m, fmax,args=(mj,s2j,pm2j))[0] for mj,s2j,pm2j in zip(mu,variance,predictive_mean_sq)]
        exp_exp2 = np.array(scaled_exp_exp2)[:,None] / normalizer

        var_exp = exp_exp2 - predictive_mean_sq

        # V(Y_star) = E[ V(Y_star|f_star) ] + V[ E(Y_star|f_star) ]
        # V(Y_star) = E[ V(Y_star|f_star) ] + E(Y_star**2|f_star) - E[Y_star|f_star]**2
        return exp_var + var_exp

    def pdf_link(self, inv_link_f, y, Y_metadata=None):
        raise NotImplementedError

    def logpdf_link(self, inv_link_f, y, Y_metadata=None):
        raise NotImplementedError

    def dlogpdf_dlink(self, inv_link_f, y, Y_metadata=None):
        raise NotImplementedError

    def d2logpdf_dlink2(self, inv_link_f, y, Y_metadata=None):
        raise NotImplementedError

    def d3logpdf_dlink3(self, inv_link_f, y, Y_metadata=None):
        raise NotImplementedError

    def dlogpdf_link_dtheta(self, inv_link_f, y, Y_metadata=None):
        raise NotImplementedError

    def dlogpdf_dlink_dtheta(self, inv_link_f, y, Y_metadata=None):
        raise NotImplementedError

    def d2logpdf_dlink2_dtheta(self, inv_link_f, y, Y_metadata=None):
        raise NotImplementedError

    def pdf(self, f, y, Y_metadata=None):
        """
        Evaluates the link function link(f) then computes the likelihood (pdf) using it

        .. math:
            p(y|\\lambda(f))

        :param f: latent variables f
        :type f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param Y_metadata: Y_metadata which is not used in student t distribution - not used
        :returns: likelihood evaluated for this point
        :rtype: float
        """
        if isinstance(self.gp_link, link_functions.Identity):
            return self.pdf_link(f, y, Y_metadata=Y_metadata)
        else:
            inv_link_f = self.gp_link.transf(f)
            return self.pdf_link(inv_link_f, y, Y_metadata=Y_metadata)

    def logpdf_sum(self, f, y, Y_metadata=None):
        """
        Convenience function that can overridden for functions where this could
        be computed more efficiently
        """
        return np.sum(self.logpdf(f, y, Y_metadata=Y_metadata))

    def logpdf(self, f, y, Y_metadata=None):
        """
        Evaluates the link function link(f) then computes the log likelihood (log pdf) using it

        .. math:
            \\log p(y|\\lambda(f))

        :param f: latent variables f
        :type f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param Y_metadata: Y_metadata which is not used in student t distribution - not used
        :returns: log likelihood evaluated for this point
        :rtype: float
        """
        if isinstance(self.gp_link, link_functions.Identity):
            return self.logpdf_link(f, y, Y_metadata=Y_metadata)
        else:
            inv_link_f = self.gp_link.transf(f)
            return self.logpdf_link(inv_link_f, y, Y_metadata=Y_metadata)

    def dlogpdf_df(self, f, y, Y_metadata=None):
        """
        Evaluates the link function link(f) then computes the derivative of log likelihood using it
        Uses the Faa di Bruno's formula for the chain rule

        .. math::
            \\frac{d\\log p(y|\\lambda(f))}{df} = \\frac{d\\log p(y|\\lambda(f))}{d\\lambda(f)}\\frac{d\\lambda(f)}{df}

        :param f: latent variables f
        :type f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param Y_metadata: Y_metadata which is not used in student t distribution - not used
        :returns: derivative of log likelihood evaluated for this point
        :rtype: 1xN array
        """
        if isinstance(self.gp_link, link_functions.Identity):
            return self.dlogpdf_dlink(f, y, Y_metadata=Y_metadata)
        else:
            inv_link_f = self.gp_link.transf(f)
            dlogpdf_dlink = self.dlogpdf_dlink(inv_link_f, y, Y_metadata=Y_metadata)
            dlink_df = self.gp_link.dtransf_df(f)
            return chain_1(dlogpdf_dlink, dlink_df)

    @blockify_hessian
    def d2logpdf_df2(self, f, y, Y_metadata=None):
        """
        Evaluates the link function link(f) then computes the second derivative of log likelihood using it
        Uses the Faa di Bruno's formula for the chain rule

        .. math::
            \\frac{d^{2}\\log p(y|\\lambda(f))}{df^{2}} = \\frac{d^{2}\\log p(y|\\lambda(f))}{d^{2}\\lambda(f)}\\left(\\frac{d\\lambda(f)}{df}\\right)^{2} + \\frac{d\\log p(y|\\lambda(f))}{d\\lambda(f)}\\frac{d^{2}\\lambda(f)}{df^{2}}

        :param f: latent variables f
        :type f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param Y_metadata: Y_metadata which is not used in student t distribution - not used
        :returns: second derivative of log likelihood evaluated for this point (diagonal only)
        :rtype: 1xN array
        """
        if isinstance(self.gp_link, link_functions.Identity):
            d2logpdf_df2 = self.d2logpdf_dlink2(f, y, Y_metadata=Y_metadata)
        else:
            inv_link_f = self.gp_link.transf(f)
            d2logpdf_dlink2 = self.d2logpdf_dlink2(inv_link_f, y, Y_metadata=Y_metadata)
            dlink_df = self.gp_link.dtransf_df(f)
            dlogpdf_dlink = self.dlogpdf_dlink(inv_link_f, y, Y_metadata=Y_metadata)
            d2link_df2 = self.gp_link.d2transf_df2(f)
            d2logpdf_df2 = chain_2(d2logpdf_dlink2, dlink_df, dlogpdf_dlink, d2link_df2)
        return d2logpdf_df2

    @blockify_third
    def d3logpdf_df3(self, f, y, Y_metadata=None):
        """
        Evaluates the link function link(f) then computes the third derivative of log likelihood using it
        Uses the Faa di Bruno's formula for the chain rule

        .. math::
            \\frac{d^{3}\\log p(y|\\lambda(f))}{df^{3}} = \\frac{d^{3}\\log p(y|\\lambda(f)}{d\\lambda(f)^{3}}\\left(\\frac{d\\lambda(f)}{df}\\right)^{3} + 3\\frac{d^{2}\\log p(y|\\lambda(f)}{d\\lambda(f)^{2}}\\frac{d\\lambda(f)}{df}\\frac{d^{2}\\lambda(f)}{df^{2}} + \\frac{d\\log p(y|\\lambda(f)}{d\\lambda(f)}\\frac{d^{3}\\lambda(f)}{df^{3}}

        :param f: latent variables f
        :type f: Nx1 array
        :param y: data
        :type y: Nx1 array
        :param Y_metadata: Y_metadata which is not used in student t distribution - not used
        :returns: third derivative of log likelihood evaluated for this point
        :rtype: float
        """
        if isinstance(self.gp_link, link_functions.Identity):
            d3logpdf_df3 = self.d3logpdf_dlink3(f, y, Y_metadata=Y_metadata)
        else:
            inv_link_f = self.gp_link.transf(f)
            d3logpdf_dlink3 = self.d3logpdf_dlink3(inv_link_f, y, Y_metadata=Y_metadata)
            dlink_df = self.gp_link.dtransf_df(f)
            d2logpdf_dlink2 = self.d2logpdf_dlink2(inv_link_f, y, Y_metadata=Y_metadata)
            d2link_df2 = self.gp_link.d2transf_df2(f)
            dlogpdf_dlink = self.dlogpdf_dlink(inv_link_f, y, Y_metadata=Y_metadata)
            d3link_df3 = self.gp_link.d3transf_df3(f)
            d3logpdf_df3 = chain_3(d3logpdf_dlink3, dlink_df, d2logpdf_dlink2, d2link_df2, dlogpdf_dlink, d3link_df3)
        return d3logpdf_df3


    def dlogpdf_dtheta(self, f, y, Y_metadata=None):
        """
        TODO: Doc strings
        """
        if self.size > 0:
            if self.not_block_really:
                raise NotImplementedError("Need to make a decorator for this!")
            if isinstance(self.gp_link, link_functions.Identity):
                return self.dlogpdf_link_dtheta(f, y, Y_metadata=Y_metadata)
            else:
                inv_link_f = self.gp_link.transf(f)
                return self.dlogpdf_link_dtheta(inv_link_f, y, Y_metadata=Y_metadata)
        else:
            # There are no parameters so return an empty array for derivatives
            return np.zeros((0, f.shape[0], f.shape[1]))

    def dlogpdf_df_dtheta(self, f, y, Y_metadata=None):
        """
        TODO: Doc strings
        """
        if self.size > 0:
            if self.not_block_really:
                raise NotImplementedError("Need to make a decorator for this!")
            if isinstance(self.gp_link, link_functions.Identity):
                return self.dlogpdf_dlink_dtheta(f, y, Y_metadata=Y_metadata)
            else:
                inv_link_f = self.gp_link.transf(f)
                dlink_df = self.gp_link.dtransf_df(f)
                dlogpdf_dlink_dtheta = self.dlogpdf_dlink_dtheta(inv_link_f, y, Y_metadata=Y_metadata)

                dlogpdf_df_dtheta = np.zeros((self.size, f.shape[0], f.shape[1]))
                #Chain each parameter of hte likelihood seperately
                for p in range(self.size):
                    dlogpdf_df_dtheta[p, :, :] = chain_1(dlogpdf_dlink_dtheta[p,:,:], dlink_df)
                return dlogpdf_df_dtheta
                #return chain_1(dlogpdf_dlink_dtheta, dlink_df)
        else:
            # There are no parameters so return an empty array for derivatives
            return np.zeros((0, f.shape[0], f.shape[1]))

    def d2logpdf_df2_dtheta(self, f, y, Y_metadata=None):
        """
        TODO: Doc strings
        """
        if self.size > 0:
            if self.not_block_really:
                raise NotImplementedError("Need to make a decorator for this!")
            if isinstance(self.gp_link, link_functions.Identity):
                return self.d2logpdf_dlink2_dtheta(f, y, Y_metadata=Y_metadata)
            else:
                inv_link_f = self.gp_link.transf(f)
                dlink_df = self.gp_link.dtransf_df(f)
                d2link_df2 = self.gp_link.d2transf_df2(f)
                d2logpdf_dlink2_dtheta = self.d2logpdf_dlink2_dtheta(inv_link_f, y, Y_metadata=Y_metadata)
                dlogpdf_dlink_dtheta = self.dlogpdf_dlink_dtheta(inv_link_f, y, Y_metadata=Y_metadata)

                d2logpdf_df2_dtheta = np.zeros((self.size, f.shape[0], f.shape[1]))
                #Chain each parameter of hte likelihood seperately
                for p in range(self.size):
                    d2logpdf_df2_dtheta[p, :, :] = chain_2(d2logpdf_dlink2_dtheta[p,:,:], dlink_df, dlogpdf_dlink_dtheta[p,:,:], d2link_df2)
                return d2logpdf_df2_dtheta
                #return chain_2(d2logpdf_dlink2_dtheta, dlink_df, dlogpdf_dlink_dtheta, d2link_df2)
        else:
            # There are no parameters so return an empty array for derivatives
            return np.zeros((0, f.shape[0], f.shape[1]))

    def _laplace_gradients(self, f, y, Y_metadata=None):
        dlogpdf_dtheta = self.dlogpdf_dtheta(f, y, Y_metadata=Y_metadata)
        dlogpdf_df_dtheta = self.dlogpdf_df_dtheta(f, y, Y_metadata=Y_metadata)
        d2logpdf_df2_dtheta = self.d2logpdf_df2_dtheta(f, y, Y_metadata=Y_metadata)

        #Parameters are stacked vertically. Must be listed in same order as 'get_param_names'
        # ensure we have gradients for every parameter we want to optimize
        assert dlogpdf_dtheta.shape[0] == self.size #num_param array x f, d
        assert dlogpdf_df_dtheta.shape[0] == self.size #num_param x f x d x matrix or just num_param x f
        assert d2logpdf_df2_dtheta.shape[0] == self.size #num_param x f matrix or num_param x f x d x matrix, num_param x f x f or num_param x f x f x d

        return dlogpdf_dtheta, dlogpdf_df_dtheta, d2logpdf_df2_dtheta

    def predictive_values(self, mu, var, full_cov=False, Y_metadata=None):
        """
        Compute  mean, variance of the  predictive distibution.

        :param mu: mean of the latent variable, f, of posterior
        :param var: variance of the latent variable, f, of posterior
        :param full_cov: whether to use the full covariance or just the diagonal
        :type full_cov: Boolean
        """
        try:
            pred_mean = self.predictive_mean(mu, var, Y_metadata=Y_metadata)
            pred_var = self.predictive_variance(mu, var, pred_mean, Y_metadata=Y_metadata)
        except NotImplementedError:
            print("Finding predictive mean and variance via sampling rather than quadrature")
            Nf_samp = 300
            Ny_samp = 1
            s = np.random.randn(mu.shape[0], Nf_samp)*np.sqrt(var) + mu
            ss_y = self.samples(s, Y_metadata, samples=Ny_samp)
            pred_mean = np.mean(ss_y, axis=1)[:, None]
            pred_var = np.var(ss_y, axis=1)[:, None]

        return pred_mean, pred_var

    def predictive_quantiles(self, mu, var, quantiles, Y_metadata=None):
        #compute the quantiles by sampling!!!
        Nf_samp = 300
        Ny_samp = 1
        s = np.random.randn(mu.shape[0], Nf_samp)*np.sqrt(var) + mu
        ss_y = self.samples(s, Y_metadata)#, samples=Ny_samp)
        #ss_y = ss_y.reshape(mu.shape[0], mu.shape[1], Nf_samp*Ny_samp)

        pred_quantiles = [np.percentile(ss_y, q, axis=1)[:,None] for q in quantiles]
        return pred_quantiles

    def samples(self, gp, Y_metadata=None, samples=1):
        """
        Returns a set of samples of observations based on a given value of the latent variable.

        :param gp: latent variable
        :param samples: number of samples to take for each f location
        """
        raise NotImplementedError("""May be possible to use MCMC with user-tuning, see
                                  MCMC_pdf_samples in likelihood.py and write samples function
                                  using this, beware this is a simple implementation
                                  of Metropolis and will not work well for all likelihoods""")

    def MCMC_pdf_samples(self, fNew, num_samples=1000, starting_loc=None, stepsize=0.1, burn_in=1000, Y_metadata=None):
        """
        Simple implementation of Metropolis sampling algorithm

        Will run a parallel chain for each input dimension (treats each f independently)
        Thus assumes f*_1 independant of f*_2 etc.

        :param num_samples: Number of samples to take
        :param fNew: f at which to sample around
        :param starting_loc: Starting locations of the independant chains (usually will be conditional_mean of likelihood), often link_f
        :param stepsize: Stepsize for the normal proposal distribution (will need modifying)
        :param burnin: number of samples to use for burnin (will need modifying)
        :param Y_metadata: Y_metadata for pdf
        """
        print("Warning, using MCMC for sampling y*, needs to be tuned!")
        if starting_loc is None:
            starting_loc = fNew
        from functools import partial
        logpdf = partial(self.logpdf, f=fNew, Y_metadata=Y_metadata)
        pdf = lambda y_star: np.exp(logpdf(y=y_star[:, None]))
        #Should be the link function of f is a good starting point
        #(i.e. the point before you corrupt it with the likelihood)
        par_chains = starting_loc.shape[0]
        chain_values = np.zeros((par_chains, num_samples))
        chain_values[:, 0][:,None] = starting_loc
        #Use same stepsize for all par_chains
        stepsize = np.ones(par_chains)*stepsize
        accepted = np.zeros((par_chains, num_samples+burn_in))
        accept_ratio = np.zeros(num_samples+burn_in)
        #Whilst burning in, only need to keep the previous lot
        burnin_cache = np.zeros(par_chains)
        burnin_cache[:] = starting_loc.flatten()
        burning_in = True
        for i in xrange(burn_in+num_samples):
            next_ind = i-burn_in
            if burning_in:
                old_y = burnin_cache
            else:
                old_y = chain_values[:,next_ind-1]

            old_lik = pdf(old_y)
            #Propose new y from Gaussian proposal
            new_y = np.random.normal(loc=old_y, scale=stepsize)
            new_lik = pdf(new_y)
            #Accept using Metropolis (not hastings) acceptance
            #Always accepts if new_lik > old_lik
            accept_probability = np.minimum(1, new_lik/old_lik)
            u = np.random.uniform(0,1,par_chains)
            #print "Accept prob: ", accept_probability
            accepts = u < accept_probability
            if burning_in:
                burnin_cache[accepts] = new_y[accepts]
                burnin_cache[~accepts] = old_y[~accepts]
                if i == burn_in:
                    burning_in = False
                    chain_values[:,0] = burnin_cache
            else:
                #If it was accepted then new_y becomes the latest sample
                chain_values[accepts, next_ind] = new_y[accepts]
                #Otherwise use old y as the sample
                chain_values[~accepts, next_ind] = old_y[~accepts]

            accepted[~accepts, i] = 0
            accepted[accepts, i] = 1
            accept_ratio[i] = np.sum(accepted[:,i])/float(par_chains)

            #Show progress
            if i % int((burn_in+num_samples)*0.1) == 0:
                print("{}% of samples taken ({})".format((i/int((burn_in+num_samples)*0.1)*10), i))
                print("Last run accept ratio: ", accept_ratio[i])

        print("Average accept ratio: ", np.mean(accept_ratio))
        return chain_values
