Possibility Versus Probability & Decision

A Logical Probabilist (note the large forehead) explains that the interocitor has three states.

Philip Pilkington, author of the buyable The Reformation in Economics: A Deconstruction and Reconstruction of Economic Theory, asks a follow-up question to the material on the precautionary principle.

If probability is not decision then, I hope you’ll grant, that decision is not probability. (Where’s my Nobel Prize!?). But it seems to me that many probability enthusiasts/fetishists do indeed think that decisions should be made in line with probability theory.

If Px = 0.49, then “naw”. If Px = 0.51, then “yeah, boi”. If Px = 0.5, then crushing indifference, the absurdity of all life and meaning, and so on.

I think what Taleb is getting at (in characteristically cack-handed fashion) is that, as the far more interesting GLS Shackle would say, probability and possibility are two entirely different fields. Probability can only be applied to situations with a known, limited number of outcomes – dice rolls, n = 6. While possibility is relevant when the number of situations is unbounded and subjected to the potential infinities of our imaginations – n = 1… ?. (We mere mortals are only able to conceive of approaching the ? rather than realising the ? in our imaginations – the latter being the province of the Other Being).

Decisions are almost always made in a sphere of possibilities rather than probabilities. Probabilitists often do not realise this. And, in not doing so, they assumed that their models act like dice rather than being the products of their mind’s possibility-generating function which, naturally, is constrained by n = 1… ?.

There is another distinction in British and American English between the words possibility and probability. In Brit, possibility means of small chance, while probability means not unlikely (and not a plain likely). In Yank, possibility means it isn’t impossible, while probability means some kind of quantification or statement of a chance, regardless of size, large or small.

But I do not quite agree with this distinction in Pilkington’s question. Probability is the study of propositions in relation to evidence, and nothing more. Thus C = “The god Zeus had at least three wives” given P = “The god Zeus had at least four wives” has probability 1, and this is so even though if the god Zeus existed or not, i.e. whether we speak of the myth or the superman.

In Uncertainty, I liked to use the example of interocitors, fictional devices from the planet Metaluna, featured in This Island Earth. If we accept P = “This interocitor can and must be in states 1, 2, or 3 only”, then the probability C = “This interocitor is not in state 1″ is 2/3. But if we modify P to “This interocitor will be in some state”, then the probability of C is the uncapped unit interval. There is no information in P to say anything about what the states are except that some states exist. Tacit in that is the possibility of state 1; state 1 is certainly not impossible give P. So there is no fixed number.

What is C = “This interocitor is not in state 1 or 2″? We’re asking more of P. But P has nothing more to give. The probability doesn’t change. We don’t have certainty of C’s falsity or truth. We only have its possibility.

So there is a distinction. Possible is not impossible. In interocitors were real, we would say C is contingent.

Is it possible the number of states of the interocitor are infinite? There are no real-life infinities, which we could say is tacit in P, but the interocitor is not real life, and, anyway, in mathematics there are plenty of infinities (to say the least). If we’re to move to infinities, as folks do when used boxed models, nearly all of which rely on the infinity of the continuum, then we need to be especially careful about the infinities we invoke. People are usually not so careful.

If in some real-life situation we do not know the possible number of states (or situations), we have to assume some boundaries, else probability measures won’t be forthcoming, nor decisions based on them.

The bad but true pun is that infinity is a large subject, which I cover in detail in Uncertainty. There is something else here that I think of greater potential interest, and is also a subject I cover in Uncertainty, but not well enough, I think. And that is weight of evidence, related to comments above about probabilities of 0.49, 0.50, and 0.51. More to come…

(I only just realized I typed all the quotation marks as if I were in LaTeX. Occupational hazard.)

10 Comments

  1. My psych professor always said he did not care what was possible, since anything is actually possible (even physics laws may not be true—that is “possible”). It’s only what is probable that we need to look at and determine the likelihood of.

    He would never have endorse the “precautionary principle”, except to label it as a neurotic fear reaction to reality.

  2. There are many Decision Analysis tactics that rely not on probability but one’s reaction to favorable or unfavorable outcomes. Google “The Peal of Great Price + Pascal’s Wager” to see one such, the mini-max regret. (I can’t put the URL down because the Overseer says it’s spam.)

  3. Who is the audience for such parsing of concepts as “probability vs possibility”??

    Drunks around a casino craps or baccarat table understand that just as they understand the probability distinctions of 0.49, 0.50, to 0.51 are, for them while visiting the casino, meaningless, while for the casino operating on large-scale economies a very slight advantage over the long run yields a very significant return with near-certainty. The common folk understand “probability” vs “possibility” and can and do apply the distinctions. Who out there fails to grasp this??

    RE: “The Peal of Great Price + Pascal’s Wager”

    Here’s an excerpt from one such discussion:

    “Blaise Pascal (1623-1662) offers a pragmatic reason for believing in God: even under the assumption that God’s existence is unlikely, the potential benefits of believing are so vast as to make betting on theism rational.”

    “There are two kinds of argument for theism. Traditional, epistemic arguments hold that God exists; …. Modern, pragmatic arguments hold that, regardless of whether God exists, believing in God is good for us, or is the right thing to do; …”

    That is such a common argument (and itself a variant of the Precautionary Principle) — choose to believe in something.

    Nobody can “choose to believe” — making such a choice, then making it real as a true belief is always a delusion. No exceptions.

    Either one believes or one doesn’t. Evidence might lead one to consider a possibility that a situation may be one way versus another (innocent), or not (guilty); and a juror might be willing to act and believe accordingly by rendering the given verdict. At the same time, unless the evidence is conclusive or particularly compelling, a juror will seldom be certain or even pretend to be. ‘Certain enough’ suffices.

    Faith is like that. Except, unlike the juror’s degree of uncertainty after the verdict, believers will pretend to believe, and typically come round to believe they are not pretending at all — while concurrently holding the self-serving motive as the foundational reason for their “belief.” Confusing a self-serving profession of faith for actual faith even while consciously acknowledging the self-serving motive that makes the “belief” a choice rather than intrinsic.

    Such mental gymnastics are hypocrisy at its finest as it enables one to suppress the associated cognitive dissonance. Not only does such mental manipulation lead one to delude oneself that they “believe,” it also enables them to ignore their very un-Christian-like behaviors toward others while also letting them believe that they are “good” “morally upright” and so forth [often even as they abuse someone else]. All one need do is attend Church and observe the faithful … and then how they act out in the wild.

    The same basic mechanism occurs elsewhere. Lovallo and Kahneman describe this in their [in]famous paper, “Delusions of Success: How Optimism Undermines Executive’s Decisions.” https://hbr.org/2003/07/delusions-of-success-how-optimism-undermines-executives-decisions

  4. Of course you–if you are a mature adult–can choose to believe. You “choose to believe” every time you listen to the nightly news and swallow what is presented–no matter how far from Orwell’s “neutral truth” it may be. If you take vitamins, you choose to believe that you will gain some benefit from it. If you exercise, you choose to believe that your heart is greatly grateful for the opportunity to squeeze blood through its ventricles. If you cross the street on a green light, you choose to believe that you will not be mowed down by a car. Nearly every single waking moment is dominated by a choice to believe something.

  5. Ken and Anon, I think what Ken was trying to say is that you can’t choose to believe in something for which you do not have any evidence–and that evidence may be observations, testimony, or intuition. But that evidence–faith–is given to some of us, not to others, and here I say is where Grace enters the picture. There are very intelligent and rational people who believe in God and very intelligent and rational people who don’t believe in God. They’re confronted with the same evidence in terms of observations and testimony; what’s the difference? Grace imparted to some and the acceptance of that gift.

  6. Anon,

    Your examples only demonstrate what you believe through your actions on a belief and in no way indicate how you came to that belief. If it was by choice then you need to show how that is true.

  7. There are … people who believe in God and … people who don’t believe in God. They’re confronted with the same evidence in terms of observations and testimony; what’s the difference? Grace imparted to some and the acceptance of that gift.

    Some follow up on “Grace::

    How does it affect beliefs?
    Is it necessary for a “correct” belief?
    Do I need to know that I have it before I can accept it?
    If so, how do I know I have it?

  8. It seems that it is probable that you are talking about Theism.
    Looking at evidence and coming to conclusions. However it appears that Jewish Christian and to an extent Moslem belief in the One God is reliant on Revelation of Himself to Abraham by this One God. Why Abraham and why his descendants ? Do we know? He also said to Moses ” I AM THAT I AM: and he said, Thus shalt thou say unto the children of Israel, I AM hath sent me unto you.”
    God took the initiative.
    So speculations about gods and their existence are perhaps the stuff of philosophy and are of interest. But do not refer to these groups Judaism ,the Christian Church
    and Islam. They are based on direct verbal contact at the least.

    (http://www.tertullian.org/fathers/irenaeus_02_proof.htm) for an early view of the Jewish and Christian beliefs. Islam came later.

  9. Ken is correct about ‘chosen faith.’ It’s where I disagree with Peter Hitchens.
    that he choses to believe in God and I find this hard to fathom except that I do believe he is being truthful. It is also true to say that everybody is a hypocrite. Christians most of all because nobody is perfect. If you set a standard which is unachievable in an ordinary non cloistered life and even then, to be cloistered is to lack courage and does not rule out sin completely. This is separate from the issue of faith really. It is easy to spot someone’s sins and say this must mean they don’t have faith.

    It is not a choice.

    I say it is a feature of a part of the mind which is not entirely accessible to naming a source or an origin. Strictly speaking it could be said to be irrational.
    The same could be said of atheism if rational has some special extra meaning where some mathematical type self evident, frank proof is required.

    If you consider as I have that it is an emotional response to the unknown just as atheist may be seen in the same way, it is to miss rather more complex and hard to reach elements of thinking or notion which would not be characterised as pure emotion or pure logical thought.

    So when DaV asks what is Grace he asks the right question and when it is not answered properly this is because it cannot be answered. Which sounds like a kind of cop out. If it were easy to explain everybody would be the same in their thinking and nobody would deny the truth of it.

    Grace in the sense which is not the everyday sense of outward movement, act or appearances is an effect not the thing itself. It can also be an affect.

    When putting one foot in front of the other one has faith that all reflexes and normal movement will prevail and the limb won’t give way. Actually ignorance is bliss, if you know how the thing works you would wonder that it ever does.

    Nobody choses to believe in this kind of function. They simply accept it if they give it any thought. Only those who have had cause to question such a thing must be retrained to believe, to trust, or forget, in fact, that the leg won’t give way. Once bitten twice shy.
    Older people in particular learn this kind of fear through experience. Some people go off their legs completely as a result.

    To describe faith as a delusion is really not correct though an understandable thing to say. I don’t think people should be offended by this really. If a person does something unordinary and on the basis of a faith which they hold and another does not, it will look delusional. So what?

    One could say the same ting of someone who does not have faith and acts accordingly which is where the argument against atheism is strongest, around moral behaviour. Ethics and morals are not the same. It is one reason why the clash between the two world views. Atheists and Theists insulting each other over their morals. Each claiming superiority.

    It is also true that intellect alone has nothing do do with the question.
    In all faith there is an element of blindness which polite pole call trust, by definition it involves the unknown.
    If evidence or argument can defeat faith in God then of course it is a delusion because it isn’t true given some evidence. If God is eternal and creator of all things he is outside of our understanding of time and space. All we know is what is revealed to us. Not everybody is equal in that department for obvious reasons.

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