`suppressPackageStartupMessages(require(rstan))`

# Challenge questions

Not for grades!

These are not essential for learning the material and can be skipped without affecting your grade. If you successfully solve one set of problem, a week of participation activity will be waived (it does not have to be the same week you submit the challenge question). Submit your answer at any time. I will not post solutions for the challenge questions.

Consider the following regression model:

```
data {
matrix[3,2] design_matrix; // number of successes
vector[3] observations;
}
parameters {
vector[2] coefficients;
}
model {
1] ~ cauchy(0, 1);
coefficients[2] ~ cauchy(0, 1);
coefficients[
for (i in 1:3) {
1);
observations[i] ~ normal(design_matrix[i] * coefficients,
} }
```

We now run it on a simple synthetic dataset (note the zeros in the design matrix):

```
= rbind(
matrix c(0.1, 0),
c(-0.3, 0),
c(0, 0)
)= c(0.2, 0.1, -0.4)
obs = sampling(
fit
cauchy,seed = 1,
refresh = 0,
data = list(design_matrix = matrix, observations = obs),
iter = 10000
)
```

```
Warning: There were 71 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
https://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded
```

```
Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
https://mc-stan.org/misc/warnings.html#bfmi-low
```

`Warning: Examine the pairs() plot to diagnose sampling problems`

…and look at the trace plots…

```
suppressPackageStartupMessages(require(bayesplot))
suppressPackageStartupMessages(require(ggplot2))
mcmc_trace(fit, pars = c("coefficients[1]", "coefficients[2]")) + theme_minimal()
```

What is going on with the second coefficient? Would you get a different behaviour if you used i.i.d. samples?