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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’)=1/7approx14.3%).

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