[ The following discusses my Web page, Black, White, and Gray. Do computers already use certainty factors under the name of Bayesian analysis? And if certainty factors are different from Bayesian analysis, which is better for everyday human use?
[ I tried writing to my correspondent, but my message to him bounced with an error that said `MX domain ... has target domain name ... which is an invalid CNAME.' I have written to his DNS host asking for a fix. ]
Context:
In 1944, Louis Guttman noted that all forms of measurement belong to one of four types of scale: categorical, ordinal, interval, and ratio. Guttman was thinking of measurement, but his scales to apply to human social structures and, more generally, to basic mathematics. Guttman was talking about the different ways people can make judgements about what they see, with or without the help of a ruler or other gauge.
In the 1980s, David McAllister invented `certainty factors', which can be used to express how accurate, truthful, or reliable you judge perceptions, and the arithmetical procedures to go with them.
In Black, White, and Gray, I foresaw more use of certainty factors in computers.
A correspondent wrote
Computers already use certainty factors under the name of Bayesian analysis, ...
I responded by saying that I understood, but wondered whether certainty factors and Bayesian values are identical, in practice and in theory? I do not think so.
My correspondent said
Most people view probability as fraction ("1/6 of past rolls for this die were one") ...
This is the `insurance company' or `probability reflects history' view, versus
... but the Bayesian view is one of certainty ("There is a 1/6 chance the next die roll will be one, based fully upon the history of the die").
which is the `judgement about the future' view.
It is indeed true that certainty factors provide a way to take a `judgement about the future' view, and in this sense they are similar to Bayesian values.
However, the Bayesian view must involve a judgement across time. That is what probabilities are about, regardless of whether you think of them as produced by actuaries or otherwise.
On the other hand, certainty factors
express how accurate, truthful, or reliable
you judge the observation. Certainty factors have nothing to do with time intrinsically. They are a way of combining judgements.
You may, if you wish, use certainty factors to take a view across time. You can use the conclusions to make a prediction, but that is different from their intrinsic nature.
So your comment that a certainty factor is not a probability is incorrect; the modern view is that they are equivalent. The CFCombine rules presented are really just the Bayesian rules for combining independent and dependent probabilities in disguise. ...
As far as I know, the combination rules for certainty factors are different from the Bayesian combination rules. Perhaps my understanding of mathematics fails me; but as far as I can see, both the reasoning leading to and the expressions themselves are different:
Bayesian combination rules:
If a and b are the probabilities associated with two independent
pieces of evidence, then combined they indicate a probability of:
ab
-------------------
ab + (1 - a)(1 - b)
For example
(.60)(.72)
-------------------------------
(.60)(.72) + (1 - .60)(1 - .72)
or in Emacs Lisp
(/ (* 0.6 0.72) (+ (* 0.6 0.72) (* (- 1 0.6) (- 1 0.72))))
which is 0.79
[ Bayesian combination rules from Paul Graham's site. ]
Certainty factor combination rules:
If a and b are the certainty factors associated with two
independent pieces of evidence, then combined they indicate:
a + b(1 - a)
For example
(.60) + (.72)(1 - .60)
or in Emacs Lisp
(+ 0.6 (* 0.72 (- 1.0 .6)))
which is 0.88
[ Certainty factor combination rules ]
My correspondent went to say,
Any modern form of predictive analysis uses probabilities, ...
and I responded, `Yes, you are right.'
... and the systems produced are so complex that they defy analysis.
Yes, this is also true.My correspondent continued by writing,
In one sense this is the true tragedy: for legal reasons, a predictive system must be able to explain HOW its decisions are made; .... People who design systems for medical and safety are required to use less effective but more understandable approaches (such as decision trees) for legal rather than efficiency reasons.
And in one way it is a tragedy. But in another, it shows an interest in humans keeping control of their own understanding.
I do not think that predictive systems based on probabilities, the `modern form', are necessarily the best form.
Whoever or whatever makes a judgement should be able to explain how it reached that judgement. Otherwise, there is no way you can check that judgement. It is like paperless electronic voting systems. No matter how often the systems do right on tests, in an actual election, if you worry about bugs or fraud, and you have no form of accountability, no `paper trail', you are sunk.
... There is a social barrier to fuzzy systems: people tend to demand absolute certainty in an uncertain world and expect more than educated guesses from their machines, even if the educated guesses would be far better than the result of an absolute but erroneous deductive process.
My correspondent is right: there is social barrier. No doubt about that. But humans do make judgements all the time. I can see people accepting computer made judgements after following through with and coming to understand a few. They will figure they can always, if they want, go through the whole procedure to check. Of course, they could also do this with a Bayesian calculation, too. But people understand Bayesian analysis less than they understand the combination of two or three (or more) `suggestive' hints.