// Not a general pseudowishpdfln (because L*Diff*L' at a single locus has rank 1
double pseudowishpdfln(const MatrixXd &X, const MatrixXd &Sigma, const int df) {
int rank = 1;
double ldX = pseudologdet(X,rank);
double ldS = logdet(Sigma);
int n = X.rows();
int q = (df<n) ? df : n;
return (0.5*(-df*ldS - Sigma.selfadjointView<Lower>().llt().solve(X).trace() +
(df-n-1.0)*ldX - df*n*log_2 - df*(n-q)*log_pi) - mvgammaln(0.5*df,q));
}
/*
This function implements the computations described in the Section S1.3 in the Supplementary Information,
"Computing the Wishart log likelihood l(k, m, q, sigma2)", and I have tried to used similar notation
For example, MatrixXd X = lu.solve(T) is equation (S20)
since lu is the decomposition of (B*C - W) and T is B * inv(W)
Returns wishpdfln( -L*D*L' ; - (sigma2/df) * L*Delta(m,q)*L' , df )
*/
double EEMS::EEMS_wishpdfln(const MatrixXd &Binv, const VectorXd &W, const double sigma2, const double df,
double &triDeltaQD, double &ll_atfixdf) const {
double df_2 = 0.5 * df;
MatrixXd iDelta = Binv.topLeftCorner(o,o);
MatrixXd DiDDi = iDelta * JtDobsJ * iDelta;
double oDinvo = iDelta.sum(); // oDinvo = 1'*inv(Delta)*1
double oDiDDi = DiDDi.sum(); // oDiDDi = 1'*inv(Delta)*D*inv(D)*1
MatrixXd Q = MatrixXd::Constant(o,o,-1.0/oDinvo);
Q *= iDelta;
Q += cvec.asDiagonal();
MatrixXd iDeltaQ = iDelta*Q;
double ldetDinvQ = pseudologdet(-1.0 * iDeltaQ,o-1); // ldetDinvQ = logDet(-inv(Delta)*Q)
triDeltaQD = (iDeltaQ*JtDobsJ).trace(); // triDeltaQD = tr(inv(Delta)*Q*D)
ll_atfixdf = ldetDinvQ - triDeltaQD/sigma2 - nmin1*log(sigma2) - ldDiQ;
return (df_2*ll_atfixdf + nmin1*df_2*log(df_2) - mvgammaln(df_2,nmin1) - n_2*ldLDLt);
}
