Thanks. Here’s the Stan code:

```
transformed data {
real r = 10;
real A = 6;
real omega = 6;
}
parameters {
vector[2] x;
}
model {
target += -0.5 * (hypot(x[1], x[2]) + r + A * cos(omega * atan2(x[2], x[1])));
}
```

This is probably another good case for convergence diagnoistics. Here’s the RStan call:

```
fit <- sampling(model,
control = list(stepsize = 0.01, adapt_delta = 0.999, max_treedepth=11),
chains = 4, iter = 50000)
```

There was still one iteration that exceed max treedepth 11 and one divergent iteration among the 100K iterations.

```
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
x[1] 0.15 0.15 3.35 -7.06 -1.32 0.03 1.79 7.22 499 1.01
x[2] -0.09 0.15 3.42 -7.15 -2.10 -0.17 1.95 6.78 518 1.01
lp__ -4.53 0.02 1.60 -8.54 -5.35 -4.20 -3.35 -2.41 7576 1.00
```

Nevetheless, it looks like the true posterior means (which should be zero) are within standard error, but that `n_eff`

is terrible. This is 100K draws! We start to get nervous when the effective sample size rate is 200 draws. Stan’s definitely not exploring this posterior well, though I get the feeling it’d get there with more draws and ideally more chains.

Because of the low n_eff, the standard error for the means is pretty bad. I’m not sure what the true posterior sd is supposed to be. Although it appears to be roughly symmetric for `x[1]`

and `x[2]`

, it may not be getting far enough into the tails.

Here’s the code for the plot:

```
> post_x <- extract(fit)$x
> post_x1 <- post_x[ , 1]
> post_x2 <- post_x[ , 2]
> df <- data.frame(x1 = post_x1, x2 = post_x2)
> plot(post_x1, post_x2)
> library(ggplot2)
> ggplot(df, aes(x1, x2)) + geom_scatterplot()
```

Which is why we odn’t evaluate these things by eye:

So I tried 16 chains with some thinning so as not to kill my machine:

```
fit <- sampling(model, chains=16, iter=50000, thin = 25,
control=list(adapt_delta=0.999,
stepsize=0.999,
max_treedepth=11))
```

It takes about 20s/chain on my Mac notebook (6-year old i7). This produces much better results, as one might expect, with a total of 16K saved posterior draws.

```
> print(fit)
Inference for Stan model: flower.
16 chains, each with iter=50000; warmup=25000; thin=25;
post-warmup draws per chain=1000, total post-warmup draws=16000.
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
x[1] -0.04 0.07 3.45 -7.37 -1.71 -0.02 1.61 7.35 2322 1.01
x[2] -0.05 0.07 3.44 -7.04 -2.06 -0.19 2.00 6.79 2271 1.01
lp__ -4.56 0.01 1.62 -8.60 -5.39 -4.21 -3.37 -2.44 15039 1.00
```

The sd are even a little higher. As you can see, 2.5% and 97.5% intervals are terrible. This may be indicating poor tail exploration, but they’re still within MCMC standard error here because 2.5% tail probabiliites only use about 1/40 of the draws, meaning 1/sqrt(40) the effective sample size, which is still decent here at around 300.