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.

Interested in the application of R in insurance? Join us at the 3rd R in Insurance conference in Amsterdam, 29 June 2015.

### 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