Not only verbs but also believes can be conjugated

Following on from last week, where I presented a simple example of a Bayesian network with discrete probabilities to predict the number of claims for a motor insurance customer, I will look at continuous probability distributions today. Here I follow example 16.17 in Loss Models: From Data to Decisions [1].

Suppose there is a class of risks that incurs random losses following an exponential distribution (density \(f(x) = \Theta {e}^{- \Theta x}\)) with mean \(1/\Theta\). Further, I believe that \(\Theta\) varies according to a gamma distribution (density \(f(x)= \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha \,-\, 1} e^{- \beta x } \)) with shape \(\alpha=4\) and rate \(\beta=1000\).

In the same way as I had good and bad driver in my previous post, here I have clients with different characteristics, reflected by the gamma distribution.

The textbook tells me that the unconditional mixed distribution of an exponential distribution with parameter \(\Theta\), whereby \(\Theta\) has a gamma distribution, is a Pareto II distribution (density \(f(x) = \frac{\alpha \beta^\alpha}{(x+\beta)^{\alpha+1}}\)) with parameters \(\alpha,\, \beta\). Its k-th moment is given in the general case by
\[
E[X^k] = \frac{\beta^k\Gamma(k+1)\Gamma(\alpha - k)}{\Gamma(\alpha)},\; -1 < k < \alpha. \] Thus, I can calculate the prior expected loss (\(k=1\)) as \(\frac{\beta}{\alpha-1}=\,\)333.33.
Now suppose I have three independent observations, namely losses of $100, $950 and $450 over the last 3 years. The mean loss is $500, which is higher than the $333.33 of my model.

Question: How should I update my belief about the client's risk profile to predict the expected loss cost for year 4 given those 3 observations?

Visually I can regard this scenario as a graph, with evidence set for years 1 to 3 that I want to propagate through to year 4.



It turns out that in this case I can solve this problem analytically as the prior and posterior parameter distributions have a conjugate relationship. It means that the posterior parameter distribution is of the same distribution family as the prior, here a gamma, with updated posterior hyper-parameters.

Skipping the maths, I have the following posterior hyper-parameters for my given data (\(n\)=number of data points, \(c\)=claims in year \(i\))
\[
(\alpha +n,\, \beta + \sum_i c_i)
\]

The posterior predictive distribution is a Pareto II distribution as mentioned above, with the derived posterior hyper-parameters. I can calculate the posterior predictive expected claims amount as (1000+1500)/(4+3-1)=2500/(7-1)=416.67, which is higher than the prior $333.33, but still less than the actual average loss of $500.


Ok, let's visualise this. The following chart shows the prior and posterior parameter and predictive distributions. It shows nicely how the distributions shift based on the observed data.



Note that I can calculate the posterior predictive expected loss from the parameters and data directly:
\[
\frac{\beta + \sum c_i}{\alpha + n - 1} = \frac{\alpha - 1}{\alpha+n-1}\frac{\beta}{\alpha-1} + \frac{n}{\alpha + n - 1}\bar{c}
\]
That's the weighted sum of the posterior predictive expected loss \(\mu:=\frac{\beta}{\alpha-1}\) and the sample mean \(\bar{c}\). And as the number of data points increases, the sample mean gains weight or in other words credibility.

Indeed, suppose I set \(Z_n:=\frac{n}{\alpha+n-1}\) then \((1-Z_n)=\frac{\alpha-1}{\alpha+n-1}\) and hence I can write my formula as
\[
(1-Z_n)\,\mu + Z_n \,\bar{c},
\]with \(Z_n\) as my credibility factor.

References

[1] Klugman, S. A., Panjer, H. H. & Willmot, G. E. (2004), Loss Models: From Data to Decisions, Wiley Series in Probability and Statistics.

Session Info

R version 3.0.2 (2013-09-25)
Platform: x86_64-apple-darwin10.8.0 (64-bit)

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] grid      stats     graphics  grDevices utils     datasets  methods  
[8] base     
 
other attached packages:
[1] lattice_0.20-24 actuar_1.1-6    Rgraphviz_2.6.0 igraph_0.6.6   
[5] graph_1.40.0    knitr_1.5      
 
loaded via a namespace (and not attached):
[1] BiocGenerics_0.8.0 evaluate_0.5.1     formatR_0.10      
[4] parallel_3.0.2     stats4_3.0.2       stringr_0.6.2     
[7] tools_3.0.2