**Summary** I have a model that is recovering parameters, has no divergences and good \hat{R}, but very low (<0.1) E-BFMI. Increasing warmup iterations moved E-BFMI up, but it is still < 0.1. I am looking for suggestions to reparametrize or otherwise address this, and/or advice on how to proceed. Thanks for taking a look!

# The model

I have some response time data, and I want to model each response time as a reaction to an unobserved event. The probability of the unobserved event happening at time t is proportional to a normal cumulative density function. I am modeling the reaction time (i.e. how much time passes between the unobserved event and when the button is actually pressed) as lognormal.

where

I have medium-strong expectations about reasonable values for the mus and sigmas based on typical reaction time distributions and the construction of my experiment:

\mu_a \sim \mathcal{N}(1.35, 0.4)

\sigma_a \sim \mathcal{N}^+(0, 1)

\mu_r \sim \mathcal{N}(log(0.35), 0.2)

\sigma_r \sim \mathcal{N}^+(0, 0.2)

Iām simulating about 900 observations on rt, which is about how much data I have.

Code for the model (it uses reduce_sum so maybe isnāt the most readable) and a generator are attached:

antic_only_analysis.stan (4.6 KB)

antic_only_gen.stan (1.4 KB)

# The Problem

I managed to simulate and get fairly quick and trouble-free fitting on simpler models. In one, I just simulated E and recover \mu_a and \sigma_a. Similarly, I can simulate d_{rt} and recover \mu_r and \sigma_r. When I fit their combination (as described above), I get low E-BFMI. I looked for guidance in Brief Guide to Stanās Warnings, and at A Conceptual Introduction to Hamiltonian Monte Carlo. This helped me understand what was going on, but I am not sure of the best approach to fixing it.

Hereās the arviz.plot_energy() output:

And a pairs plot of energy__ and the parameters of the model:

lmu_react and lsig_react are \mu_r and \sigma_r, the parameters of the lognormal, and mu_antic, sig_antic are \mu_a, \sigma_a, respectively.

There is a strong correlation between \mu_a and \sigma_a as well as the correlation between energy__ and \sigma_r. The advice I have found suggests I should consider reparameterizing something to do with \sigma_r, but I am not sure where to begin with that.

Do I need to build a non-centered parameterization? Something like

```
...
parameters {
vector[N] raw_log_drt;
...
} transformed parameters {
vector[N] drt = exp(raw_log_drt*sigma + mu);
...
} model {
raw_log_drt ~ std_normal();
...
}
```

I also considered using numeric integration to marginalize out the unobserved E, but that feels like a pretty terrible idea that might not address the BFMI problem anyway.

Having written all this down, I guess itās not hard to try the non-centered approach I just described, so Iāll go ahead any try that to see if it helps. Maybe thereās some other problem lurking here, and perhaps you experts will find it. Thanks again for reading!

Edit: This solution is not as straightforward as I thought, because E[i] + d_rt[i] must add up to rt[i], so I canāt just do that proposed transformation.