frequentism

Bayesian Irrationality

I just came across a strange and revealing statement by Tyler Cowen:

I am frustrated by the lack of Bayesianism in most of the religious belief I observe. I’ve never met a believer who asserted: “I’m really not sure here. But I think Lutheranism is true with p = .018, and the next strongest contender comes in only at .014, so call me Lutheran.” The religious people I’ve known rebel against that manner of framing, even though during times of conversion they may act on such a basis.

I don’t expect all or even most religious believers to present their views this way, but hardly any of them do. That in turn inclines me to think they are using belief for psychological, self-support, and social functions.

I wouldn’t expect anyone to say something like “Lutheranism is true with p = .018”. Lutheranism is either true or false. Just as a person on trial is either guilty or innocent. One may have doubts about the truth of Lutheranism or the guilt of a defendant, but those doubts have nothing to do with probability. Neither does Bayesianism.

In defense of probability, I will borrow heavily from myself. According to Wikipedia (as of December 19, 2014):

Bayesian probability represents a level of certainty relating to a potential outcome or idea. This is in contrast to a frequentist probability that represents the frequency with which a particular outcome will occur over any number of trials.

An event with Bayesian probability of .6 (or 60%) should be interpreted as stating “With confidence 60%, this event contains the true outcome”, whereas a frequentist interpretation would view it as stating “Over 100 trials, we should observe event X approximately 60 times.”

Or consider this account:

The Bayesian approach to learning is based on the subjective interpretation of probability.   The value of the proportion p is unknown, and a person expresses his or her opinion about the uncertainty in the proportion by means of a probability distribution placed on a set of possible values of p….

“Level of certainty” and “subjective interpretation” mean “guess.” The guess may be “educated.” It’s well known, for example, that a balanced coin will come up heads about half the time, in the long run. But to say that “I’m 50-percent confident that the coin will come up heads” is to say nothing meaningful about the outcome of a single coin toss. There are as many probable outcomes of a coin toss as there are bystanders who are willing to make a statement like “I’m x-percent confident that the coin will come up heads.” Which means that a single toss doesn’t have a probability, though it can be the subject of many opinions as to the outcome.

Returning to reality, Richard von Mises eloquently explains frequentism in Probability, Statistics and Truth (second revised English edition, 1957). Here are some excerpts:

The rational concept of probability, which is the only basis of probability calculus, applies only to problems in which either the same event repeats itself again and again, or a great number of uniform elements are involved at the same time. Using the language of physics, we may say that in order to apply the theory of probability we must have a practically unlimited sequence of uniform observations. [P. 11]

*     *     *

In games of dice, the individual event is a single throw of the dice from the box and the attribute is the observation of the number of points shown by the dice. In the game of “heads or tails”, each toss of the coin is an individual event, and the side of the coin which is uppermost is the attribute. [P. 11]

*     *     *

We must now introduce a new term…. This term is “the collective”, and it denotes a sequence of uniform events or processes which differ by certain observable attributes…. All the throws of dice made in the course of a game [of many throws] from a collective wherein the attribute of the single event is the number of points thrown…. The definition of probability which we shall give is concerned with ‘the probability of encountering a single attribute in a given collective’. [Pp. 11-12]

*     *     *

[A] collective is a mass phenomenon or a repetitive event, or, simply, a long sequence of observations for which there are sufficient reasons to believe that the relative frequency of the observed attribute would tend to a fixed limit if the observations were indefinitely continued. The limit will be called the probability of the attribute considered within the collective. [P. 15, emphasis in the original]

*     *     *

The result of each calculation … is always … nothing else but a probability, or, using our general definition, the relative frequency of a certain event in a sufficiently long (theoretically, infinitely long) sequence of observations. The theory of probability can never lead to a definite statement concerning a single event. The only question that it can answer is: what is to be expected in the course of a very long sequence of observations? [P. 33, emphasis added]

Cowen has always struck me a intellectually askew — looking at things from odd angles just for the sake of doing so. In that respect he reminds me of a local news anchor whose suits, shirts, ties, and pocket handkerchiefs almost invariably clash in color and pattern. If there’s a method to his madness, other than attention-getting, it’s lost on me — as is Cowen’s skewed, attention-getting way of thinking.

More about Probability and Existence

In “A Digression about Probability and Existence” I address

the view that there is life as we know it — an outcome with a low, prior probability given the (theoretical) multitude of possible configurations of the universe — only because there are vastly many actual or possible universes with vastly many configurations.

I observe that

[i]n this view, life as we know it is an improbable phenomenon that we are able to witness only because we happen to exist in one of the multitude of possible or actual universes.

I should have pointed out that it is impossible to know whether life as we know it is a low-probability event. Such a conclusion rests on an unsupportable assumption: the existence of a universe which is “fine tuned” to enable life is a low-probability event. And yet, that assumption is the basis for assertions that the existence of our universe — with its life-supporting combination of matter, energy, and physical laws — “proves” that there must be other universes because ours is so unlikely. Such “logic” is an edifice of rank circularity constructed on a foundation of pure supposition.

Such “logic,” moreover, misapplies the concept “probability.” No object or event has a probability (knowable chance of happening) unless it meets the following conditions:

1. The object or event is a member of a collective of observable phenomena, where every member of the collective has common features.

2. The collective is a mass phenomenon or an unlimited sequence of observations, where (a) the relative frequencies of particular attributes within the collective tend to fixed limits and (b) these fixed limits remain the same for reasonably large subsets of the collective. (Adapted from “Summary of the Definition,” on pp. 28-9 in Chapter 1, “The Definition of Probability,” of Richard von Mises’s Probability, Statistics and Truth, 1957 Dover edition.)

Mises, obviously, was a  “frequentist,” and his view of probability is known as “frequentism.” Despite the criticisms of frequentism (follow the preceding link), it offers the only rigorous view of probability. Nor does it insist (as suggested at the link) that a probability is a precisely knowable or fixed value. But it is a quantifiable value, based on observations of actual objects or events.

Other approaches to probability are vague and subjective. There are, for example, degrees of belief (probabilistic logic), statements of propensity (probabilistic propensity), and “priors” (Bayesian probability). Unlike frequentism, these appeal to speculation, impressions, and preconceptions. Reliance on such notions of probability as evidence of the actual likelihood of an event is the quintessence of circularity.

In summary, there is no sound basis in logic or empirical science for the assertion that the universe we know is a highly improbable one and, therefore must be one of vastly many universes — if it was not the conscious creation of an exogenous force or being (i.e., God). The universe we know simply “is” — and that is all we know or probably can know, as a matter of science.

Related posts:
Atheism, Religion, and Science
The Limits of Science
Three Perspectives on Life: A Parable
Beware of Irrational Atheism
The Creation Model
The Thing about Science
Evolution and Religion
Words of Caution for Scientific Dogmatists
Science, Evolution, Religion, and Liberty
The Legality of Teaching Intelligent Design
Science, Logic, and God
Capitalism, Liberty, and Christianity
Is “Nothing” Possible?
A Dissonant Vision
Debunking “Scientific Objectivity”
Science’s Anti-Scientific Bent
Science, Axioms, and Economics
The Big Bang and Atheism
The Universe . . . Four Possibilities
Einstein, Science, and God
Atheism, Religion, and Science Redux
Pascal’s Wager, Morality, and the State
Evolution as God?
The Greatest Mystery
What Is Truth?
The Improbability of Us
A Digression about Probability and Existence