Log density with proportionality in ADVI drops more than constant

Summary:

Log density with proportionality in ADVI drops more than constant

Description:

According to stan C++ guidance (https://github.com/stan-dev/stan/wiki/Model-Concept), the log probability of a stan model, with or without proportionality, can be calculated from

template <bool propto, bool jacobian_adjust_transforms, typename T>
T 
log_prob(std::vector<T>& params_r,
         std::vector<int>& params_i,
         std::ostream* msgs = 0) 
const;

But I find the difference between the exact and proportional log density in Stan ADVI code is not a constant.

Reproducible Steps:

In order to calculate both the exact log density and proportional log density in the unconstrained space, and to print their difference, add these lines before
https://github.com/stan-dev/stan/blob/12f031565df6265356bdc4c6b191d8df619151f5/src/stan/variational/advi.hpp#L540

            double log_p_exact=0;
            double log_p_prop=0;
            log_p_exact = model_.template log_prob<false, true>(cont_params_);
            log_p_prop = model_.template log_prob<true, true>(cont_params_);
            std::stringstream ss;
            ss << "constant dropped="<<log_p_exact-log_p_prop<<"======";
            std::cout<<ss.str();

Then we can make cmdstan and run the bernoulli example:

examples/bernoulli/bernoulli variational algorithm=meanfield iter=1000 output_samples=30 data file=examples/bernoulli/bernoulli.data.R

Current Output:

We should expect the output being a constant for all posterior draws, which stands for the log constant term in densities. But it is not. For instance here is the first 6 lines of the output:

constant dropped==-5.24628 
constant dropped==-5.20678
constant dropped==-5.78415
constant dropped==-5.37678
constant dropped==-5.02283
constant dropped==-5.59698

I check that the exact log density ( model_.template log_prob<false, true> ) does return the log density plus jacobian.

Current Version:

v2.17.1

Start in src/stan/variational/advi.hpp around the referenced line and reproduce the discrepancy with the bernoulli CmdStan variational command. Compare model_.log_prob<false, true> and model_.log_prob<true, true> across posterior draws; done means determining why their difference is not constant and correcting the ADVI behavior or documenting the confirmed explanation.