1D integrator Stan model very slow

Hi, I’m new to Stan and Bayesian Inference in general. I’m running a model on a relatively small dataset - vector of 40 elements and a 40x40 covariance matrix. My model involves doing a 1D integration, though. And its been running very slow. I would appreciate suggestions on how I can improve the model performance.

Gradient evaluation took 2.34406 seconds
1000 transitions using 10 leapfrog steps per transition would take 23440.6 seconds.

IIUC, this means it should take about 10 hours to run with a sampling of 1500 iterations (500 of which are warmup). But the code is still running for close to 24 hours now.

I have attached text files containing the stan model and data.
cov.txt (39.9 KB)
data.txt (2.0 KB)
lcdm.stan (2.4 KB)

Link to jupyter notebook to load the data and run the model. The notebook also describes the math to make it easier to understand what the model is doing. - Google Colab

Note that the notebook was run locally on my Mac. The cell outputs are from the local run.

Also posting the code here for convenience

functions {
   real e_func(real z, real Om, real Ode, real Ok) {
      real zp1, x;
      zp1 = 1 + z;
      x = zp1 * zp1 * (Ok + Om*zp1) + Ode;
     return sqrt(x); 

   //integrand function
   real integrand_fz(real z, real zc, array[] real theta,
                     array[] real z_r, array[] int z_i) {
    real ef;      
	ef = e_func((zc > 0 && zc < z) ? zc : z, theta[1], theta[2], theta[3]);
 return 1.0/ef;

   //f(z; params)
   real f_func(real z, real H, real Om, real Ode, real Ok, real c, data array[] real x_r, data array[] int x_i) {
	return (c/H)*integrate_1d(integrand_fz, 0.0, z, {Om, Ode, Ok}, x_r, x_i, 1.0e-8);

   //r(z; params) - comoving distance for a given redshift and model params
   real r_z(real z, real H, real Om, real Ode, real Ok, real c, data array[] real x_r, data array[] int x_i) {
      real fz;
      fz = f_func(z, H, Om, Ode, Ok, c, x_r, x_i);

      return Ok > 0 ? sin(fz) : (Ok < 0 ? sinh(fz): fz);

   //d_L(z; params) - luminosity distance for a given redshift and model params
   real dl_z(real z, real H, real Om, real Ode, real Ok, real c, data array[] real x_r, data array[] int x_i) {
      return (1 + z) * r_z(z, H, Om, Ode, Ok, c, x_r, x_i);

   //mu(z; params) - predicted apparent magnitude for a given redshift and model params
   real model_apparent_mag(real z, real H, real Om, real Ode, real M, real Ok, real c, data array[] real x_r, data array[] int x_i) {
      real dlz, dist_mod;

      dlz = dl_z(z, H, Om, Ode, Ok, c, x_r, x_i);
      dist_mod = 5.0 * log10(dlz) - 5.0;

      return dist_mod + M;

data {
   int N; //number of data points
   vector[N] z; //redshifts
   vector[N] m; //apparent magnitudes
   matrix[N,N] C; //covariance matrix

transformed data {
   array[0] real x_r;
   array[0] int x_i;

parameters {
   real H0; //hubble paramater today
   real Om0; //matter density today
   real Ode0; //real dark energy density today
   real M0; //absolute supernova magnitude

transformed parameters {
   real Ok0; #curvature density today
   Ok0 = 1.0 - Om0 - Ode0;

model {
   vector[N] mu; //model predicted apparent magnitudes
   real c; //speed of light
   c = 299792.458;

   H0 ~ uniform(0.0, 100.0);
   Om0 ~ uniform(0.0, 1.0);
   Ode0 ~ uniform(0.0, 1.0);
   M0 ~ uniform(-50.0, 30.0);

   for(n in 1:N){	
      mu[n] = model_apparent_mag(z[n], H0, Om0, Ode0, M0, Ok0, c, x_r, x_i);

   m ~ multi_normal(mu, C);

Couple of pointers about the code which may provide useful

  1. I use the argument xc in my integrated in place of x, when 0 < xc < x. Otherwise I was getting a lot of errors like Error in function tanh_sinh<double>::integrate: The tanh_sinh quadrature evaluated your function at a singular point and got nan. With this change, I still get the errors but they are less often.

  2. I set a relative tolerance of 1e-8 because otherwise I was getting errors like integration error of 2.3e-9 is less than relative tolerance times the norm of the integral.

Operating System: macOS Catalina 10.15.6
Interface Version: Pystan 3.4.0. This is the version I got when I installed via pip last week.

Happy to provide any more information. Any suggestions would be appreciated. Thanks in advance!

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FYI, updated stan code to fix a couple of bugs. The updated model is available here (sharing a link so in case I fix more bugs I don’t have to repost everytime)

I verified that the integration yields the expected results. But the sampler is still just as slow.

You have several uniform priors, but no corresponding constraints in the parameters declarations, which causes the posterior to be nasty near the bounds which can cause also slow sampling. If the bounds are true physical constraints, then you should add those constraints in the parameter declarations, too. If the bounds are just describing likely range, it would be better to use non-bounded prior with decreasing tails presenting the likely range smoothly.

In addition to fixing the priors or adding constraints, it would be useful to know whether the sloweness is due to the many log density evaluations or due to each integral being slow. Look at the HMC diagnostics and average n_leapfrog per iteration. If that is high then the posterior has a difficult shape, and it is worth thinking if that can be improved. You can make the 1D integration faster by making the error threshold bigger and using the diagnostics presented in [2205.09059] An importance sampling approach for reliable and efficient inference in Bayesian ordinary differential equation models

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Thanks for the suggestions. Increasing the error threshold for the integration (from 1e-8 to 1e-4) did the trick.

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