## May 10, 2011

### Homo Actuarius Bayesianis

Bayesian fallacies are often the most trickiest.....

A classical example of a Bayesian fallacy is the so called "Prosecutor's fallacy" in case of DNA testing...

Multiple DNA testing (Source: Wikipedia)
A crime-scene DNA sample is compared against a database of 20,000 men.

A match is found, the corresponding man is accused and at his trial, it is testified that the probability that two DNA profiles match by chance is only 1 in 10,000.

Sounds logical, doesn't it?
Yes... 'Sounds'... As this does not mean the probability that the suspect is innocent is also 1 in 10,000. Since 20,000 men were tested, there were 20,000 opportunities to find a match by chance.

Even if none of the men in the database left the crime-scene DNA, a match by chance to an innocent is more likely than not. The chance of getting at least one match among the records is in this case:

$\large \fn_jvn \1 - \left(1-\frac{1}{10000}\right)^{20000} \approx 86\%$

So, this evidence alone is an uncompelling data dredging result. If the culprit was in the database then he and one or more other men would probably be matched; in either case, it would be a fallacy to ignore the number of records searched when weighing the evidence. "Cold hits" like this on DNA data-banks are now understood to require careful presentation as trial evidence.

In a similar (Dutch) case, an innocent nurse (Lucia de Berk) was at first wrongly accused (and convicted!) of murdering several of her patients.

Other Bayesian fallacies
Bayesian fallacies can come close to the actuarial profession and even be humorous, as the next two examples show:
1. Pension Fund Management
It turns out that from all pension board members that were involved in a pension fund deficit, only 25% invested more than half in stocks.

Therefore 75% of the pension fund board members with a pension fund deficit invested 50% or less in stocks.

From this we may conclude that pension fund board members should have done en do better by investing more in stocks....

2. The Drunken Driver
It turns out that of from all drivers involved in car crashes 41% were drunk and 59% sober.

Therefore to limit the probability of a car crash it's better to drink...

It's often not easy to recognize the 'Bayesian Monster' in your models. If you doubt, always set up a 2 by 2 contingency table to check the conclusions....

Homo Actuarius
Let's  dive into the historical development of Asset Liability Management (ALM) to illustrate the different stages we as actuaries went through to finally cope with Bayesian stats. We do this by going (far) back to prehistoric actuarial times.

As we all know, the word actuary originated from the Latin word actuarius (the person who occupied this position kept the minutes at the sessions of the Senate in the Ancient Rome). This explains part of the name-giving of our species.

Going back further in time we recognize the following species of actuaries..

1. Homo Actuarius Apriorius
This actuarial creature (we could hardly call him an actuary) establishes the probability of an hypothesis, no matter what data tell.

ALM example: H0: E(return)=4.0%. Contributions, liabilities and investments are all calculated at 4%. What the data tell is uninteresting.

2. Homo Actuarius Pragmaticus
The more developed 'Homo Actuarius Pragamiticus' demonstrates he's only interested in the (results of the) data.
ALM example: In my experiments I found x=4.0%, full stop.
Therefore, let's calculate with this 4.0%.

3. Homo Actuarius Frequentistus
In this stage, the 'Homo Actuarius Frequentistus' measures the probability of the data given a certain hypothesis.

ALM example: If H0: E(return)=4.0%, then the probability to get an observed value more different from the one I observed is given by an opportune expression. Don't ask myself if my observed value is near the true one, I can only tell you that if my observed value(s) is the true one, then the probability of observing data more extreme than mine is given by an opportune expression.
In this stage the so called Monte Carlo Methods was developed...

4. Homo Actuarius Contemplatus
The Homo Actuarius Contemplatus measures the probability of the data and of the hypothesis.

ALM example
:You decide to take over the (divided!) yearly advice of the 'Parameters Committee' to base your ALM on the maximum expected value for the return on fixed-income securities, which is at that moment  4.0%. Every year you measure the (deviation) of the real data as well and start contemplating on how the two might match...... (btw: they don't!)

5. Homo Actuarius Bayesianis
The Homo Actuarius Bayesianis measures the probability of the hypothesis, given the data.  Was the  Frequentistus'  approach about 'modeling mechanisms' in the world, the Bayesian interpretations are more about 'modeling rational reasoning'.

ALM example: Given the data of a certain period we test wetter the value of H0: E(return)=4.0% is true : near 4.0% with a P% (P=99?) confidence level.

Knowledge: All probabilities are conditional
Knowledge is a strange  phenomenon...

 When I was born I knew nothing about everything. When I grew up learned something about some thing. Now I've grown old I know everything about nothing. Joshua Maggid

The moment we become aware that ALL probabilities - even quantum probabilities - are in fact hidden conditional Bayesian probabilities, we (as actuaries) get enlightened (if you don't : don't worry, just fake it and read on)!

Simple Proof: P(A)=P(A|S), where S is the set of all possible outcomes.

From this moment on your probabilistic life will change.

To demonstrate this, examine the next simple example.

Tossing a coin
• When tossing a coin, we all know: P (heads)=0.5
• However, we implicitly assumed a 'fair coin', didn't we?
• So what we in fact stated was: P (heads|fair)=0.5
• Now a small problem appears on the horizon: We all know a fair coin is hypothetical, it doesn't really exist in a real world as every 'real coin' has some physical properties and/or environmental circumstances that makes it more or less biased.
• We can not but conclude that the expression
'P (heads|fair)=0.5'  is theoretical true, but has unfortunately no practical value.
• The only way out is to define fairness in a practical way is by stating something like:  0.4999≥P(heads|fair)≤0.5001
• Conclusion: Defining one point estimates in practice is practically  useless, always define estimate intervals (based on confidence levels).

From this beginners  example, let's move on to something more actuarial:

Estimating Interest Rates: A Multi Economic Approach
• Suppose you base your (ALM) Bond Returns (R) upon:
μ= E(R)=4%
and σ=2%

• Regardless what kind of brilliant interest- generating model (Monte Carlo or whatever) you developed, chances are your model is based upon several implicit assumptions like inflation or unemployment.

The actual Return (Rt) on time (t) depends on many (correlated, mostly exogenous) variables like Inflation (I), Unemployment (U), GDP growth(G), Country (C) and last but not least  (R[t-x]).

A well defined Asset Liability Model should therefore define (Rt) more on basis of a 'Multi Economic Approach'  (MEA) in a form that looks more or less something like: Rt = F(I,U,G,σ,R[t-1],R[t-2],etc.)

• In discussing with the board which economic future scenarios will be most likely and can be used as strategic scenarios, we (actuaries) will be better able to advice with the help of MEA. This approach, based on new technical economic models and intensive discussions with the board, will guarantee  more realistic output and better underpinned decision taking.

I. Stats....
- Make your own car crash query
- Alcohol-Impaired Driving Fatalities (National Statistics)
- D r u n k D r i v i n g Fatalities in America (2009)
- Drunk Driving Facts (2006)

II. Humor, Cartoons, Inspiration...
- Jesse van Muylwijck Cartoons (The Judge)
- PHDCOMICS
- Interference : Evolution inspired by Mike West

III. Bayesian Math....
- New Conceptual Approach of the Interpretation of Clinical Tests (2004)
- The Bayesian logic of frequency-based conjunction fallacies (pdf,2011)
- The Bayesian Fallacy: Distinguishing Four Kinds of Beliefs (2008)
- Resource Material for Promoting the Bayesian View of Everything
- A Constructivist View of the Statistical Quantification of Evidence
- Conditional Probability and Conditional Expectation
- Getting fair results from a biased coin
- INTRODUCTION TO MATHEMATICAL FINANCE