Consider a simple normal-normal hierarchical model for meta-analysis (i.e. a “random-effects meta-analysis”). This is basically the model for the “8 schools” example, but I will use it in its marginal form (i.e. with the true study-specific effects integrated out) and with other (proper) priors:

```
data {
int<lower=1> K; // Number of studies
vector[K] hat_theta_k; // Study-specific effect estimates
vector<lower=0>[K] hat_sesq_k; // Study-specific squared standard error estimates (treated as known, i.e. constants)
}
parameters {
real theta; // Overall treatment effect
real<lower=0> tau; // Between-study standard deviation
}
model {
// Prior:
theta ~ cauchy(0, 5);
tau ~ cauchy(0, 0.5);
// Likelihood:
hat_theta_k ~ normal(theta, sqrt(square(tau) + hat_sesq_k));
}
```

Now I first sampled from the posterior of this model in RStan without thinning, i.e. using the defaults:

```
MA_fit <- stan(
file = path_to_stancode_above,
data = MA_dat,
seed = 46996L
)
```

with `MA_dat`

as follows:

```
$K
[1] 2
$hat_theta_k
[1] 0.5099547 0.7532247
$hat_sesq_k
[1] 0.01134995 0.10782436
```

I then tried to assess if the posterior draws may be used for inference. I don’t get divergent transitions or hits of the maximum treedepth. \hat{R} and the new \hat{R} (implemented in `rstan::Rhat()`

) look fine, too (maximum \hat{R}: 1.007, maximum new \hat{R}: 1.006). However, the trace plots look suspicious (for `theta`

, `tau`

, as well as `lp__`

):

For me, those peaks indicate that the parameters occasionally drift out to the tails where the draws get highly correlated. This is also reflected by the autocorrelation which is quite high (maximum absolute value: 0.73). Of course, the heavy-tailed Cauchy priors require that the tails are also explored and it seems natural that in doing so, autocorrelation is high. Nevertheless, for a finite sample from the posterior that we are unfortunately forced to use, the peaks in those trace plots made me doubt that convergence to the true posterior is guaranteed. [Note: The effective sample size (ESS) ratio is always above 0.1, though (minimum ESS ratio: 0.13). The same holds for the bulk-ESS ratio (minimum: 0.18) and the tail-ESS ratio (minimum: 0.12). So perhaps – if my deductions so far are correct – the common threshold of 0.1 should be discussed. The same holds for the Monte Carlo standard error (MCSE) with a maximum value of 0.04 here (the common threshold being 0.1).]

Because of the diagnostics from above, I then thinned the Markov chains by steps of 4:

```
MA_fit <- stan(
file = path_to_stancode_above,
data = MA_dat,
seed = 46996L,
iter = 5000, # Increased so that the total number of posterior draws remains the same.
warmup = 1000,
thin = 4
)
```

Now the diagnostic metrics improve (no divergent transitions, no hits of maximum treedepth, maximum \hat{R}: 1.0016, maximum new \hat{R}: 1.0007, maximum absolute autocorrelation: 0.47, minimum ESS ratio: 0.46, minimum ESS-bulk ratio: 0.58, minimum ESS-tail ratio: 0.52, MCSEs at about 0.02) and the trace plots, too:

Nevertheless, I am still not sure whether the remaining peaks indicate that the posterior draws should not be used for inference, keeping in mind that this is a finite sample from the posterior. Could maybe someone with more experience tell me if she or he would regard those posterior draws as valid for inference? If not, I guess I will have to switch to less heavy-tailed priors like a Student-t with 7 degrees of freedom?

Session info:

```
R version 3.6.2 (2019-12-12)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18363)
Matrix products: default
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] rstan_2.19.2 ggplot2_3.2.1 StanHeaders_2.19.0
loaded via a namespace (and not attached):
[1] Rcpp_1.0.3 rstudioapi_0.10 magrittr_1.5 tidyselect_0.2.5 munsell_0.5.0 colorspace_1.4-1
[7] R6_2.4.1 rlang_0.4.2 fansi_0.4.0 dplyr_0.8.3 tools_3.6.2 parallel_3.6.2
[13] pkgbuild_1.0.6 grid_3.6.2 gtable_0.3.0 loo_2.2.0 cli_2.0.0 withr_2.1.2
[19] matrixStats_0.55.0 lazyeval_0.2.2 assertthat_0.2.1 tibble_2.1.3 lifecycle_0.1.0 crayon_1.3.4
[25] processx_3.4.1 gridExtra_2.3 purrr_0.3.3 callr_3.4.0 ps_1.3.0 inline_0.3.15
[31] glue_1.3.1 compiler_3.6.2 pillar_1.4.3 prettyunits_1.0.2 scales_1.1.0 stats4_3.6.2
[37] pkgconfig_2.0.3
```