# Matrix transformations of parameters (Stan math library)

#1

I am trying to build a model in which my parameter vector is transformed in various ways before calculating the likelihood. Eventually, I will want to calculate the gradient and Hessian of the likelihood. One or more of the transformations of (parts of) the input vector will involve matrix-matrix or matrix-vector multiplication, where the matrices and vectors involved will either contain doubles, or (transformations of) parameter elements.

As a simple example, here is some code which calculates a linear combination of parameters for the mean parameter vector of a normal distribution. In the function prop_to_mean, I show 3 alternatives (two of which must be commented out before compilation), of which only the third (which only uses element-by-element multiplication) works. The first alternatives generate a host of compiler errors, which I also list below (but that I do not understand).

C++ code (placed in functions.cpp)
``````#include <Rcpp.h>
#include <RcppEigen.h>
#include <vector>

#include <stan/math.hpp>
#include <stan/math/prim/scal.hpp>
#include <stan/math/mix/mat.hpp>

#include <boost/math/tools/promotion.hpp>

using namespace Rcpp;
using namespace RcppEigen;
// [[Rcpp::depends(RcppEigen)]]

// Transform proportions together with endmember mean hyperpars to obs. means
template <typename T_P, int P_rows, int P_cols>
Eigen::Matrix<typename boost::math::tools::promote_args<T_P>::type, P_rows, 1>
prop_to_mean(const Eigen::Matrix<T_P,    P_rows, P_cols>& P,
const Eigen::Matrix<double, P_rows, P_cols>& M) {
Eigen::Matrix<typename boost::math::tools::promote_args<T_P>::type,
P_rows, 1> mu(P.rows());

// Alternative 1 (does not work)
for (int i = 0; i < P.rows(); ++i) {
mu(i) = P.row(i) * M.row(i).transpose();
}

// Alternative 2 (does not work)
Eigen::Matrix<T_P, P_rows, P_cols> Mcasted(M.rows(), M.cols());
Mcasted = M.template cast<T_P>();
for (int i = 0; i < P.rows(); ++i) {
mu(i) = P.row(i) * M.row(i).transpose();
}

// Alternative 3 (works)
for (int i = 0; i < P.rows(); ++i) {
mu(i) = 0.0;
for (int j = 0; j < P.cols(); ++j) {
mu(i) += P(i, j) * M(i, j);
}
}
return mu;
}

struct test_lpdf {
const Eigen::VectorXd y_;
const Eigen::MatrixXd M_;
test_lpdf(const Eigen::VectorXd& y,
const Eigen::MatrixXd& M) : y_(y), M_(M) { }

template <typename T>
T operator()(const Eigen::Matrix<T, Eigen::Dynamic, 1>& theta) const {

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> X(y_.size(), 2);
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> P(y_.size(), 3);
for (int i = 0; i < y_.size(); ++i) {
P(i, 0) = theta(i);
P(i, 1) = theta(i + y_.size());
P(i, 2) = theta(i + 2 * y_.size());
}
Eigen::Matrix<T, Eigen::Dynamic, 1> mu(y_.size());
mu = prop_to_mean(P, M_);
T lp = stan::math::normal_lpdf(y_, mu, 1.0);
return lp;
}
};

Rcpp::List test_calc(const Eigen::VectorXd& y,
const Eigen::MatrixXd& M,
const Eigen::VectorXd& x) {
test_lpdf f(y, M);

double fx;
Eigen::MatrixXd H;
stan::math::hessian(f, x, fx, grad_fx, H);

return Rcpp::List::create(Rcpp::Named("lp") = fx,
Rcpp::Named("hessian") = H);
}

// [[Rcpp::export]]
Rcpp::List R_test_calc(const Eigen::Map<Eigen::VectorXd>& y,
const Eigen::Map<Eigen::MatrixXd>& M,
const Eigen::Map<Eigen::VectorXd>& x) {
return test_calc(y, M, x);
}
``````
R code (compiles functions.cpp)
``````library(Rcpp)
lib_path <- .libPaths()[1]
includes <- paste0('-I\"', lib_path, "/",
c("Rcpp", "RcppEigen", "StanHeaders", "BH"),
'/include\"')
Sys.setenv("PKG_CXXFLAGS" = paste(includes, collapse = " "))

y <- rnorm(5, 8:12)
M <- matrix(1:15 + 0, 5, 3)
V <- compositions::ilrBase(D = 3)
X <- matrix(runif(10), 5, 2)
P <- matrix(c(0.1354226, 0.2546265, 0.6099509,
0.1870608, 0.3919542, 0.4209849,
0.1797275, 0.3958411, 0.4244314,
0.2167451, 0.4061172, 0.3771377,
0.2102265, 0.3311246, 0.4586489), ncol = 3, byrow = TRUE)
mu <- rowSums(P * M)
lp <- sum(dnorm(y, mu, 1, log = TRUE))

sourceCpp("functions.cpp")
R_test_calc(y, M, as.vector(P)+ runif(15, -0.01, 0.01))
``````

What do I need to do in order to make matrix operations possible in this scenario (and generally)?

As a side question, I noted that for the functioning Alternative 3 above, the ()-operator of the lpdf function seems to be used as many times as the number of parameters (the vector theta that the ()-operator takes as input). Is this to be expected?

#2

https://cran.r-project.org/web/packages/rstan/vignettes/external.html
It is easiest to write the function (or at least its signature) in Stan code, parse it to C++, tweak the C++, and `#include` it back in.

#3

Following up on what @bgoodri said, hacking around in the generated code isn’t so bad. stanc (the thing that turns .stan files into .cpp files) is super handy.

Here’s some exceedingly hacky code for computing Hessians of a compiled model using the autodiff: https://github.com/bbbales2/hessian

It’s super untested and will almost definitely fail in a variety of unpredictable ways but it might help you get where you’re going :P.

#4

Thanks, I’ll give that a shot!

#5

You might have better luck with our operations like `multiply()` — we never figured out how to overload all the C++ operators to work with matrix autodiff.

If this is all you’re doing and you’re not computing any analytic gradients, you can just write that function in Stan. There’s no efficiency gain from writing it in C++.

#6

Thank you, I may try that.

The model I am ultimately working with would be possible to implement in Stan if it weren’t for this: I have two (latent) multivariate Gaussian variables whose covariance (or precision) matrices depend on a (uniform) randomly indexed row of another matrix. Thus, I have a discrete parameter that cannot be integrated out. This random index also needs to be saved in each iteration.

#7

If the log likelihood for item `n` depends on row `z[n]` of matrix `x`, you can code that this way in general:

``````matrix[M, K] x;                  // matrix on which likelihood depends
int<lower = 1, upper = M> z[N];  // indicators
simplex[M] theta[N];             // probability item n uses row m
for (n in 1:N) {
z[n] ~ categorical(theta[n]);    // choose row for item
target += foo_lpdf(y[n] | x[z[n]], ...);  // log likelihood
}
``````

and then marginalize to get

``````matrix[M, K] x;
simplex[M] theta[N] = ...;
for (n in 1:N) {
vector[M] lp;
for (m in 1:M)
lp[m] = log(theta[n, m]) + foo_lpdf(y[n] | x[m], ...);  // log likelihood
target += log_sum_exp(lp);
}
``````