An Ode to a Tree

Many Valued Logic

In a short four page essay, Rescher (1993) discussed the use of probability logic as a form of multi-valued logic, where propositions can have truth-values other than True or False.  A few key quotes will serve to distill the basic notion of a probability tree still further:

The leading idea of this probabilistic approach to many-valued logic is that of assigning likelihood values or probabilities to statements.  A measure function Pr is presupposed as given, which assigns some real number value Pr(p) to each and every member p of the domain of statements at issue. 

For present purposes, let us take “statements” to mean “theories”.

Specifically, the following basic conditions are postulated:

(P1) 0 ≤ Pr(p), for any statement p(P2) Pr(p ∨ ¬ p = 1(P3) Pr(p ∨ q) = Pr(p) + Pr(q), provided that p and q are mutually exclusive

So, what does that mean?

(P1) All the alternative theories need to at least be plausible

(P2) The probability of all alternative theories sums to 1

(P3) The alternative theories need to be mutually exclusive, if one is true then the others are not.

For present purposes, however, no particular, specific method for the assignment of probabilities need be assumed.

What he said; any convincing argument will do.

Many-Handed Science

The better part of an editorial that appeared in Science 40 years ago (David, 1975):

A few months ago, Senator Muskie called for "one-armed" scientists. The occasion was a Senate hearing on the health effects of pollutants. Testimony from the National Academy of Sciences and other sources was not as definitive as the Senator desired. Witnesses insisted upon saying, "On one hand, the evidence is so, but on the other hand...." Thus, the call for one-armed scientists.

This incident illustrates a fundamental dilemma of the scientist or engineer in communicating with his patron, the lay person. Laymen conceive of scientific fact as an absolute shades of gray and uncertainty are not acceptable. Scientific investigations are to produce unequivocal answers, according to the popular notion. On the other hand, scientists know that there are very few absolutes that will stand up for long. Those few that do are enshrined as "laws of nature."
The modest influence of science in affairs today rests largely on its reputation for objectivity. To the degree that we abandon that virtue, we lose influence and are considered merely another self-serving, politically biased, ax-grinding constituency. We hear already the sinister thoughts from politicians that a reputable scientist can be found to support any side of any controversy, that scientists have used their disciplines to reinforce their political convictions, and that scientists are interested primarily in feathering their own nests.

There are obvious reasons why we as scientists should not abet this tendency by lending our credibility to any side without stating all the caveats that go with a responsible scientific report to our own community. It is indeed difficult to qualify properly theoretical results and their speculative implications, tentative conclusions from limited data, and social impacts without creating more uncertainty than previously existed and thereby weakening the basis for action. Yet that is the responsible path. We should be encouraging greater respect for the mystique of the scientific process and its role in uncovering reality. We should be emphasizing the complexity of important matters, their unknowability, and yet their promise for the future.

A probability tree is not inductive logic or inductive reasoning.  But, it will serve as a useful inductive scoreboard for those theories that are competing for use.  Furthermore, when theories are favored in order to reinforce political convictions or feather nests, objectification of the competition will even the playing field.  Yeah, it happens; quite often in fact.  When research budgets start getting scrutinized in a declining economy, political justification can easily trump scientific justification when the peer reviewers foxes are guarding the entrance.  Keeping score can prevent that.  Judging theories is a little like judging gymnastics or figure skating; beauty is in the eye of the beholder.  But still, if a theory keeps winning even after it keeps falling flat, an open competition will show that the judges have been bought.

The probability trees employed throughout this blogthing are concerned with food risk.  The larger current issue that cries out for a tree treatment is global climate change.  It is abundantly clear by now that there is no reliable climate change theory, but there are multiple plausible climate change theories – and they can’t all be right (Morgan, 2015).   If you want to separate the science from the policy, grow a tree.

Tree Planting

Although not widely used in public health, probability trees are a recognized tool for formal decision analysis.    In fact, you can buy an add-in for Excel.  But, to integrate the probability of causes into a larger model that has statistical uncertainties as well, it is better to roll your own.  It is not hard.  Here are two ways to do it with Excel Visual Basic; the same logic can be used in just about any programming language:

Excel Probability Tree Examples

Note: The two node if-then-else tree is shown using both a worksheet function and with VBA.  A five node tree using the select case statement is VBA only.

References

David, EE (1975).  One Armed Scientists?  Science 176: 679

Morgan, MG (2015).  Commentary: Our Knowledge of the World is Often Not Simple: Policymakers Should Not Duck That Fact, But Should Deal with It.  Risk Anal 35:19-20.

Rescher N (1993).  27 Probability Logic: A Non-truth-functional System.  In: Many-valued Logic, Gregg Revivals, Aldershot pp. 184-188.

Official Post Soundtrack

Fleetwood Mac (1972).  Bare Trees.  In: Bare Trees, Track 5.

Post Notes

Thesis Post #20.  This is a third branch off of Dictionary of Probability, arguing for probability trees as an alternative to Bayesian approaches to theoretical uncertainty.  A discussion of using WoE to assign probabilties is an anticipated follow up. Yes, more quotes than original prose.  This is also my first attempt to link to some sample code; if it doesn't work, send me a note.  I expect to do more of that later.