# Hello Stan!

In my previous post I discussed how Longley-Cook, an actuary at an insurance company in the 1950’s, used Bayesian reasoning to estimate the probability for a mid-air collision of two planes.

Here I will use the same model to get started with Stan/RStan, a probabilistic programming language for Bayesian inference.

Last week my prior was given as a Beta distribution with parameters \(\alpha=1, \beta=1\) and the likelihood was assumed to be a Bernoulli distribution with parameter \(\theta\): \[\begin{aligned} \theta & \sim \mbox{Beta}(1, 1)\\\\\\ y_i & \sim \mbox{Bernoulli}(\theta), \;\forall i \in N \end{aligned}\]For the previous five years no mid-air collision were observed, \(x=\{0, 0, 0, 0, 0\}\). That’s my data.

In this case the posterior distributions can be derived analytically. The posterior hyper-parameters are \(\alpha'=\alpha + \sum_{i=1}^n x_i,\, \beta'=\beta + n - \sum_{i=1}^n x_i\) and with that I get the posterior parameter for the predictive distribution, which is a Bernoulli distribution again: \(\theta' = \alpha'/(\alpha'+\beta')=\frac{1}{7} \approx 14.3\)%.

Still, I can use Stan and MCMC simulations to come to the same answers (of course I am using a sledgehammer here to crack a nut).

In the first code block the model is written in Stan’s modelling language. The next section calls `stan`

and finally the results can be analysed. The answers are very much the same as the analytical approach in my previous post.

### Session Info

```
R version 3.2.0 (2015-04-16)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.3 (Yosemite)
locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] rstan_2.6.0 inline_0.3.14 Rcpp_0.11.6
loaded via a namespace (and not attached):
[1] tools_3.2.0 codetools_0.2-11 stats4_3.2.0
```