def inference(self, kern_r, kern_c, Xr, Xc, Zr, Zc, likelihood, Y, qU_mean ,qU_var_r, qU_var_c, indexD, output_dim):
        """
        The SVI-VarDTC inference
        """

        N, D, Mr, Mc, Qr, Qc = Y.shape[0], output_dim,Zr.shape[0], Zc.shape[0], Zr.shape[1], Zc.shape[1]

        uncertain_inputs_r = isinstance(Xr, VariationalPosterior)
        uncertain_inputs_c = isinstance(Xc, VariationalPosterior)
        uncertain_outputs = isinstance(Y, VariationalPosterior)

        grad_dict = self._init_grad_dict(N,D,Mr,Mc)

        beta = 1./likelihood.variance
        if len(beta)==1:
            beta = np.zeros(D)+beta

        psi0_r, psi1_r, psi2_r = self.gatherPsiStat(kern_r, Xr, Zr, uncertain_inputs_r)
        psi0_c, psi1_c, psi2_c = self.gatherPsiStat(kern_c, Xc, Zc, uncertain_inputs_c)

        #======================================================================
        # Compute Common Components
        #======================================================================

        Kuu_r = kern_r.K(Zr).copy()
        diag.add(Kuu_r, self.const_jitter)
        Lr = jitchol(Kuu_r)

        Kuu_c = kern_c.K(Zc).copy()
        diag.add(Kuu_c, self.const_jitter)
        Lc = jitchol(Kuu_c)

        mu, Sr, Sc = qU_mean, qU_var_r, qU_var_c
        LSr = jitchol(Sr)
        LSc = jitchol(Sc)

        LcInvMLrInvT = dtrtrs(Lc,dtrtrs(Lr,mu.T)[0].T)[0]
        LcInvLSc = dtrtrs(Lc, LSc)[0]
        LrInvLSr = dtrtrs(Lr, LSr)[0]
        LcInvScLcInvT = tdot(LcInvLSc)
        LrInvSrLrInvT = tdot(LrInvLSr)
        tr_LrInvSrLrInvT = np.square(LrInvLSr).sum()
        tr_LcInvScLcInvT = np.square(LcInvLSc).sum()

        mid_res = {
            'psi0_r': psi0_r,
            'psi1_r': psi1_r,
            'psi2_r': psi2_r,
            'psi0_c': psi0_c,
            'psi1_c': psi1_c,
            'psi2_c': psi2_c,
            'Lr':Lr,
            'Lc':Lc,
            'LcInvMLrInvT': LcInvMLrInvT,
            'LcInvScLcInvT': LcInvScLcInvT,
            'LrInvSrLrInvT': LrInvSrLrInvT,
        }

        #======================================================================
        # Compute log-likelihood
        #======================================================================

        logL = 0.
        for d in range(D):
            logL += self.inference_d(d, beta, Y, indexD, grad_dict, mid_res, uncertain_inputs_r, uncertain_inputs_c, Mr, Mc)

        logL += -Mc * (np.log(np.diag(Lr)).sum()-np.log(np.diag(LSr)).sum())  -Mr * (np.log(np.diag(Lc)).sum()-np.log(np.diag(LSc)).sum()) \
               - np.square(LcInvMLrInvT).sum()/2. - tr_LrInvSrLrInvT * tr_LcInvScLcInvT/2. + Mr*Mc/2.

        #======================================================================
        # Compute dL_dKuu
        #======================================================================

        tmp =  tdot(LcInvMLrInvT)/2. + tr_LrInvSrLrInvT/2. * LcInvScLcInvT - Mr/2.*np.eye(Mc)

        dL_dKuu_c = backsub_both_sides(Lc, tmp, 'left')
        dL_dKuu_c += dL_dKuu_c.T
        dL_dKuu_c *= 0.5

        tmp =  tdot(LcInvMLrInvT.T)/2. + tr_LcInvScLcInvT/2. * LrInvSrLrInvT - Mc/2.*np.eye(Mr)

        dL_dKuu_r = backsub_both_sides(Lr, tmp, 'left')
        dL_dKuu_r += dL_dKuu_r.T
        dL_dKuu_r *= 0.5

        #======================================================================
        # Compute dL_dqU
        #======================================================================

        tmp = - LcInvMLrInvT
        dL_dqU_mean = dtrtrs(Lc, dtrtrs(Lr, tmp.T, trans=1)[0].T, trans=1)[0]

        LScInv = dtrtri(LSc)
        tmp = -tr_LrInvSrLrInvT/2.*np.eye(Mc)
        dL_dqU_var_c = backsub_both_sides(Lc, tmp, 'left') + tdot(LScInv.T) * Mr/2.

        LSrInv = dtrtri(LSr)
        tmp =  -tr_LcInvScLcInvT/2.*np.eye(Mr)
        dL_dqU_var_r = backsub_both_sides(Lr, tmp, 'left') + tdot(LSrInv.T) * Mc/2.

        #======================================================================
        # Compute the Posterior distribution of inducing points p(u|Y)
        #======================================================================

        post = PosteriorMultioutput(LcInvMLrInvT=LcInvMLrInvT, LcInvScLcInvT=LcInvScLcInvT,
                LrInvSrLrInvT=LrInvSrLrInvT, Lr=Lr, Lc=Lc, kern_r=kern_r, Xr=Xr, Zr=Zr)

        #======================================================================
        # Compute dL_dpsi
        #======================================================================

        grad_dict['dL_dqU_mean'] += dL_dqU_mean
        grad_dict['dL_dqU_var_c'] += dL_dqU_var_c
        grad_dict['dL_dqU_var_r'] += dL_dqU_var_r
        grad_dict['dL_dKuu_c'] += dL_dKuu_c
        grad_dict['dL_dKuu_r'] += dL_dKuu_r

        if not uncertain_inputs_c:
            grad_dict['dL_dKdiag_c'] = grad_dict['dL_dpsi0_c']
            grad_dict['dL_dKfu_c'] = grad_dict['dL_dpsi1_c']

        if not uncertain_inputs_r:
            grad_dict['dL_dKdiag_r'] = grad_dict['dL_dpsi0_r']
            grad_dict['dL_dKfu_r'] = grad_dict['dL_dpsi1_r']

        return post, logL, grad_dict