Example of an rhat of 6e+13

General
1

This post is based on @avehtari asking for examples of high rhat values on twitter. Turns out my simulation study is also producing those :)
This is not a question for how to make the model work!
This is just a fun showcase of really bad rhat values for Aki (and everyone else interested.)

Here is the brms fit object for you to play around with :)
fit_object.rdata (1.6 MB)

Dataset

The dataset of size 100 is generated from the following model:
y \sim lognormal(\mu_y, 2)
\mu_y = 0.25x + 0.5z_1 + 0.8z_2
x \sim N(\mu_x, 0.5)
\mu_x = 0.5z_1 + 0.8z_3
z_1, z_2, z_3 \sim N(0,0.5)
z4 \sim N(\mu_{z4}, 0.5)
\mu_{z4} = y + 0.5x

The DAG looks like this:

DAG-1
DAG-11344×960 16.7 KB

Model

The dataset is then fit with the following model:
y \sim Gompertz(\mu, \beta)
log(\mu) = 1 + x + z1 + z2 + z4
With flat priors on everything, 2 chains, iter = 2500, warmup = 500

Software

I am using R (4.2), cmdstan(r) (2.29.2 / 0.5.2) and brms (2.17.4) and my own Gompertz custom family you can find here.

Fit

summary(fit)
Family: gompertz 
  Links: mu = log; beta = identity 
Formula: y ~ x + z1 + z2 + z4 
   Data: dataset (Number of observations: 100) 
  Draws: 2 chains, each with iter = 2500; warmup = 500; thin = 1;
         total post-warmup draws = 4000

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI              Rhat Bulk_ESS Tail_ESS
Intercept    -1.47      0.56    -2.03    -0.91                NA        2       NA
x            -1.27      0.57    -1.84    -0.70 66142364047444.67        2       NA
z1           -0.11      0.33    -0.44     0.22                NA        2       NA
z2            0.36      1.54    -1.18     1.91                NA        2       NA
z4            0.02      0.00     0.02     0.03 66142364047444.67        2       NA

Family Specific Parameters: 
     Estimate Est.Error l-95% CI u-95% CI              Rhat Bulk_ESS Tail_ESS
beta     0.54      0.12     0.42     0.66 66142364047444.67        2       NA

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
Warning messages:
1: Parts of the model have not converged (some Rhats are > 1.05). Be careful when analysing the results! We recommend running more iterations and/or setting stronger priors. 
2: There were 1029 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
3 Likes
2

rhat should be NA for all parameters. There are 2 chains and both of them are stuck, and each chain is just a constant. It’s strange that the code is ending up estimating within-chain-variance to be non-zero. Need to check the code that constant chains are detected.