Multivariate Bayes’ Theorem – the Probable Truth?
Francis Bacon, the philosopher who was a contemporary of Shakespeare, began one of his essays with the words: “What is truth? said jesting Pilate, and would not stay for an answer.”
Postulate the following joint event: we can be 100% certain that something did (or did not happen), but how do we know if it is the truth?
In terms of truth (T) being perceived as absolute, we have a Bernoulli Distribution (0, 1) where Probability(T) = 1 and Probability(NOT T) = 0.
This means that each real event (E) is governed by the subjective perception of truth (T). The joint probability of E and T is P(E*T) or P(E intersection T). Is this joint probability calculable?
It seems to me that P(T) is a prior subjective probability. Are we led to invoke Bayes' Theorem?
In the discrete Bernoulli Universe, the total probability of the distribution of (T) = 1.
But what if P(T) has a continuous distribution? What if there are degrees or shades of T? T can then be considered as a subjective measurement on a continuum.
I contend that it would be an oversimplification to assume that P(T) follows a regular pattern. I contend that its graph may follow either a totally chaotic pattern, or else a quasi-chaotic pattern. The truth may be that we have a third probability to contend with, namely the degree of Chaos (C) in the graph of P(T) or P(C).
We may then have a trivariate distribution of (E) AND (T) AND (C). Is this calculable?
Are we heading into the realm of some multivariate form of Bayes’ Theorem? What form would a multivariate Bayes’ Theorem take? I find it baffling enough to cope with the simplest version of Bayes’ theorem.
This idea needs to be studied. Here are two real life situations.
1. In psychoanalysis, a patient may be in absolute denial (the discrete Bernoulli situation). As treatment progresses, the multivariate Bayes’ Theorem may become progressively applicable. The continuous distribution of (T) may oscillate.
2. Prior to a legal case being taken to a court of law, a witness might be subjected to a polygraph lie detector test. While a witness, at any given time (H), may be subjectively totally convinced that an event did or did not happen = P(T), the reality of it happening may have been quite different. In such a situation, the polygraph test will return a result of ‘TRUE’ when it is ‘FALSE’ and ‘FALSE’ when it is ‘TRUE’. Are we in the realm of Type I and Type II errors?
Where several witnesses (W) differ in their evidences of matters of fact, we need some measurement of the accuracy of their accounts. Is there a most likely account? We may then have a quinvatiate distribution of (W) AND (H) AND (E) AND (T) AND (C).
In such a situation, can a judge make a decision beyond any reasonable doubt? Individual judges (J) can, and do, differ in their conclusions. Legal cases are frequently overturned or quashed on appeal. A panel of judges may agree to abide by a majority verdict. Have we a further distribution of (J) ? We may then have a hexavariate distribution of (J) AND (W) AND (H) AND (E) AND (T) AND (C).
Lastly, we must allow for Random Error (R) due to chance. We may then have a heptavariate distribution of (R) AND (J) AND (W) AND (H) AND (E) AND (T) AND (C).
Is the probability of the outcome of this joint event calculable? Are its component events independent? Or is there some quantifiable interdependence between some of them or between unknown numbers of their various combinatorial groups? Can we test what interactions we should include and what orders of these interactions are significant?
All this may seem to be flying in the face of the principle of Occam’s Razor. Conversely, would we be happy to condemn an innocent person? How far do we have to go to convict beyond any reasonable doubt? How humanly reasonable does that doubt have to be? Is there always an element of doubt?
What if the priors are not independent? What if they are partially correlated? What if their interactions are partially or completely confounded? Will the place orders of the priors in the run of the joint probabilities be significantly important in affecting the result of the calculation? Do we have to make some rigorous assumptions? Or would a multivariate Bayes’ Theorem be statistically robust?
Does the conclusion seem to be that we need to measure the degree of Truth? Is this a matter more proper to pure philosophy, or perhaps to jurisprudence, than to statistical science? The Truth may be absolute, but our perception of it may be faulty. Perhaps probability theory can be a true measure of that perception.
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