Actuarial Outpost How to know when to use posterior vs. predictive for premium?
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#1
06-24-2018, 04:26 PM
 alwaysseamus Member CAS SOA Join Date: Jun 2013 College: Clemson University Posts: 171
How to know when to use posterior vs. predictive for premium?

It seems like it's arbitrary whether to use posterior or predictive for the premium. I do notice that if the average of the distribution is value of the parameter that varies, than it makes sense for the posterior to be what we look at. For example, if x is exponential with theta, where theta is distributed as pareto.. Then we know that the expected value for theta will with the same as the expected value for x. Other than that, I'm a chicken with his head cut off, no idea.

Is there a resource out there that explains it well?
#2
06-24-2018, 06:32 PM
 pershing Member SOA Join Date: Jul 2012 College: Starfleet Academy Posts: 49

Let's say the loss amount $X$ follows an exponential distribution, given $\theta$. The prior distribution of $\theta$ is single parameter Pareto($\alpha, \beta$). 3 losses are observed, $x_1, x_2$, and $x_3$.

$f(X| \theta)=\frac{1}{\theta}e^{-\frac{x}{\theta}}$
$E(X| \theta)=\theta$
$V(X| \theta)=\theta^{2}$

$\pi(\theta)=\frac{\alpha \beta^{\alpha}}{\theta^{\alpha+1}}$, $\theta > \beta$

The experience is:

$f(experience | \theta)=\frac{1}{\theta^3}e^{-\frac{\sum{x_i}}{\theta}}$

The posterior distribution of $\theta$ is:

$f(\theta|experience)=\frac{\frac{1}{\theta^3}e^{-\frac{\sum{x_i}}{\theta}} \frac{\alpha \beta^{\alpha}}{\theta^{\alpha+1}}}{\int_{\beta}^{ \infty} {\frac{1}{\theta^3}e^{-\frac{\sum{x_i}}{\theta}} \frac{\alpha \beta^{\alpha}}{\theta^{\alpha+1}} d {\theta}}} = c \cdot {\frac{1}{\theta^3}e^{-\frac{\sum{x_i}}{\theta}} \frac{\alpha \beta^{\alpha}}{\theta^{\alpha+1}}} \sim \theta^{-(\alpha+4)}e^{-\frac{\sum{x_i}}{\theta}}$

Looks like an inverse Gamma to me, with $\alpha^{*}=\alpha+3$ and $\theta^{*}=\sum{x_i}$.

If you need posterior mean of $\theta$, calculate $E[\theta|experience]$:

$E[\theta|experience]=\frac{\theta^{*}}{\alpha^{*}-1}$

If you need expected value of next claim, use the predictive distribution or double expectation, whichever is easier. In this case, the double expectation is much easier:

$E[X]=E[E(X|\theta)] =E[\theta|experience]=\frac{\theta^{*}}{\alpha^{*}-1}$.

Hope this helps,
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C
#3
06-24-2018, 09:49 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 7,790

If the likelihood is exponential, normal, Poisson, or bernoulli, there is a single parameter equal to the expected value. The mean of the posterior will be the mean of the predictive in these cases. If all you are interested in is the pure premium, you could use either mean. If you wanted some measure of risk you would need the predictive.