We begin with some quotes from a few writers, financiers and probability mathematicians. <\/p>\n\n\n\n
\u2022 Tagore: \u2018If you shut the door to all errors, the truth will be shut out (too).\u2019<\/p>\n\n\n\n
\u2022 Chekhov: \u2018Knowledge is of no value, unless you put it into practice.\u2019<\/p>\n\n\n\n
\u2022 Munger (4): \u2018You don\u2019t have to be brilliant, only a little bit wiser than the other guys on average, for a long time.\u2019<\/p>\n\n\n\n
\u2022 Markowitz (1987) (5): \u2018The rational investor is a Bayesian<\/em>.\u2019<\/p>\n\n\n\n \u2022 Jaynes (1976): \u2018..the record is clear; orthodox arguments against Bayes Theorem, and in favour of confidence intervals, have never considered such mundane things as demonstrable facts concerning performance.\u2019<\/p>\n\n\n\n \u2022 Feller (1968): \u201cIn the long run, Bayesians win.\u201d<\/p>\n\n\n\n Well, the mathematicians and financiers are clear about Bayesian probability, while the writers indirectly so: Tagore and Chekhov were astute. If no alternatives are given, then our approach and its probabilities will often be seriously affected, which fact Tagore appears to allude to. How often do we lament not using or remembering all we know – Chekhov seems aware of this. <\/p>\n\n\n\n We seem to be prepared in these times to exclude possibilities, `worlds\u2019, options. In so doing, when we look at Bayes famous theorem for a minute, we see our inevitable error. <\/p>\n\n\n\n Munger and Feller\u2019s quotations go hand in hand and lead us to the approach discussed here of logical or inductive, plausible inference. Our approach here acknowledges or is guided by the above motivating words. Let us try to spot places in life where we might think of some of these words.<\/p>\n\n\n\n How do you or your team make decisions<\/em>?<\/p>\n\n\n\n If you don\u2019t have a \u2018plan\u2019 for this, then my Art of Decision<\/em> book should give you something to work with. Many people choose to make their decisions intuitively and rapidly (2). This may well suit many of us. But can you justify it? Is there nothing that a bit more reflection and logic can bring, on average and over the long run, that might help improve the way we bash forwards in life, business, diplomacy and statecraft?<\/p>\n\n\n\n What is good and easy, and what is not good and hard?<\/p>\n\n\n\n If there was any \u2018give\u2019 at all from the reader in answering the first question above, then this second one is a nice followup. If you can assess what has been \u2018not good and hard\u2019 in terms of decisions, then read on: there may just be something here for you.<\/p>\n\n\n\n What is probability as extended logic<\/em>? (3) Logic is the mode of reasoning<\/em>. It is logic that is extended <\/em>from only applying to binary certainty <\/em>and impossibility <\/em>to. . .<\/p>\n\n\n\n . . . all states of degree of belief <\/em>(uncertainty) in between these two endpoints. <\/p>\n\n\n\n —-<\/p>\n\n\n\n 2 as described in the book \u2018Blink\u2019 by Malcolm Gladwell.<\/p>\n\n\n\n 3 This phrase was used, for example, by Jaynes (2003).<\/p>\n\n\n\n —-<\/p>\n\n\n\n What is decision theory<\/em>? most desirable <\/em>action available to the \u2018Agent<\/em>\u2019 (the person or organisation who acts).<\/p>\n\n\n\n These definitions are general and seem to allow wide application. As a bonus, the ideas that underpin decisionmaking, i.e. our topic, also relate to artificial intelligence and machine learning, and thus will be of interest to those trying to give themselves a good base for understanding those rapidly developing areas.<\/p>\n\n\n\n —-<\/p>\n\n\n\n (4) d. 2023 at age 99. With Warren Buffett, he built up a fund of almost $1 trillion over several decades. —-<\/p>\n\n\n\n We turn to the great Roman scholar de Finetti, who said in 1946: \u201cProbability does not exist\u2019<\/em>\u2019.<\/p>\n\n\n\n What did he mean by this? Here, we shall look at Bayesian subjective (6) probability as extended logic. We compare it with orthodox, frequentist ad-hoc statistics. We look at the pros and cons of probability, utility and Bayes\u2019 logic and ask why it is not used more often. In card game terms, like contract bridge or other games, there is partial, differing states of knowledge across the game players, and there is the different concept of \u2018missing a trick\u2019, which is to say, we made a mistake given what we knew; subjective probability and Bayes-backed decision logic is about being rational and avoiding missing costly \u2018tricks\u2019, especially after a certain time or in the long run, by virtue of \u2018playing\u2019 consistently optimally.<\/p>\n\n\n\n Above is a quote of de Finetti (7), who was known for his intellect and beautiful writing. He meant that your probability of an event is subjective rather than objective. That probability does not exist in the same way that the \u2018ether\u2019 scientists thought existed before the Michelson-Morley experiments demonstrated its likely non-existence in the early 20th century.<\/p>\n\n\n\n Probability is relative not objective. It is a function of your state of knowledge, the possible options you are aware of, and the observed data that you may have, and which you trust. When these have been used up we equivocate between the alternatives. We do this in the sense that we choose our probability distribution so as to use all the information we have, not throwing any away, and so as not to add any more \u2018information\u2019 that we do not have (8). As you find out more or get more data, you can update your probabilities. This up-to-date probability distribution is one of your key tools for prediction and making decisions. Many people don\u2019t write it down. Such information may be tacit and there will be some sort of \u2018mental model\u2019 in operation. If you try to work with probability, it is likely that you may not be using the above logic, i.e. probability theory. You may be making decisions by some other process. . .<\/p>\n\n\n\n In a 2022 lecture by the Nobel Laureate, Professor \u2018t Hooft (9)\u00a0expounded a theory in which everything that happens is determined in advance. (10)<\/p>\n\n\n\n Why do I bring this up? Let us go back to de Finetti. Since we can never know all the \u2018initial conditions\u2019 in their minute detail, then our world is subjective, based on our state of knowledge, and this leads to other theories, including that of probability logic,<\/p>\n\n\n\n which is my topic here. \u2019t Hooft\u2019s theory is all very well. (11) There may be \u2018groupthink\u2019. Alternatives may be absent from the calculations (we come back to this later). <\/p>\n\n\n\n —-<\/p>\n\n\n\n 6 Subjective because I am looking at the decision from my point of view, with my state of knowledge. —-<\/p>\n\n\n\n The famous ether experiment mentioned above is an example of the great majority of top scientists (physicists), in fairly modern times, believing in something, for a long time, that turned out later literally to be non-existent, like the Emperor\u2019s New Clothes.<\/p>\n\n\n\n In the \u2018polemic\u2019 section of his paper about different kinds of estimation intervals (1976), the late, eminent physicist, E T Jaynes, wrote \u2018. . . orthodox arguments against Laplace\u2019s use of Bayes\u2019 theorem and in favour of \u201cconfidence Intervals\u201d have never considered such mundane things as demonstrable facts concerning performance.\u2019<\/p>\n\n\n\n Jaynes went on to say that \u2018on such grounds (i.e. that we may not give probability statements in terms of anything but random variables (12)), we may not use (Bayesian) derivations, which in each case lead us more easily to a result that is either the same as the best orthodox frequentist result, or demonstrably superior to it\u2019.<\/p>\n\n\n\n Jaynes went on: \u2018We are told by frequentists that we must say \u2018the % number of times that the confidence interval covers the true value of the parameter\u2019 not \u2018the probability that the true value of the parameter lies in the credibility interval\u2019. And: \u2018The foundation stone of the orthodox school of thought is the dogmatic insistence that the word probability must be interpreted as frequency in some random experiment.\u2019 Often that \u2018experiment\u2019 involves made-up, randomised data in some imaginary and only descriptive, rather than a useful prescriptive (13), model. Often, we can\u2019t actually repeat the experiment directly or even do it once. Many organisations will want a prescription for their situation in the here-and-now, rather than a description of what may happen with a given frequency in some ad hoc and imaginary model that uses any amount of made-up data.<\/p>\n\n\n\n Liberally quoting again, Jaynes continues: \u2018The only valid criterion for choosing is which approach leads us to the more reasonable and useful results<\/em>?<\/p>\n\n\n\n \u2018In almost every case, the Bayesian result is easier to get at and more elegant. The main reason for this is that both the ad hoc step of choosing a statistic and the ensuing mathematical problem finding its sampling distribution are eliminated.<\/p>\n\n\n\n \u2018In virtually every real problem of real life the direct probabilities are not determined by any real random experiment; they are calculated from a theoretical model whose choice involves \u2018subjective\u2019 judgement. . . and then \u2018objective\u2019 calibration and maximum entropy equivocation between outcomes we don\u2019t know(14). Here, \u2018maximum entropy\u2019 simply means not putting in any more information once we\u2019ve used up all the information we believe we actually have.<\/p>\n\n\n\n \u2018Our job is not to follow blindly a rule which would prove correct 95% of the time in the long run; there are an infinite number of radically different rules, all with this property. <\/p>\n\n\n\n —-<\/p>\n\n\n\n 12 In his book, de Finetti avoids the term \u2018variable\u2019 as it suggests a number which \u2018varies\u2019, which he considers a strange concept related to the frequentist idea of multiple or many idealised identical trials where the parameter we want to describe is fixed, and the data is not fixed, which viewpoint probability logic reverses. He uses the phrase:\u00a0random quantity\u00a0<\/em>instead. —-<\/p>\n\n\n\n Things (mostly) never stay put for the long run. Our job is to draw the conclusions that are most likely to be right in the specific case at hand; indeed, the problems in which it is most important that we get this theory right or just the ones where we know from the start that the experiment can never be repeated.\u2019 <\/p>\n\n\n\n \u2018In the great majority of real applications, long run performance is of no concern to us, because it will never be realised.\u2019<\/p>\n\n\n\n And finally, Jaynes said that \u2018the information we receive is often not a direct proposition, but is an indirect claim that a proposition is true, from some \u201cnoisy\u201d source that is itself not wholly reliable\u2019. The great Hungarian logician and problem-solver P\u00f3lya deals with such situations in his 1954 works around plausible inference, and we cover the basics of this in this book.<\/p>\n\n\n\n Most people are happy to use logic when dealing with certainty and impossibility. This is the standard architecture for sextillions of electronic devices, for example. <\/p>\n\n\n\n Where there is uncertainty between these extremes of logic, let us use the theory of probability as extended logic.<\/p>\n\n\n\n The above is a draft adaptation of an early chapter section of the 2024 book The Art of Decision by Dr J D Hayward<\/p>\n","protected":false},"excerpt":{"rendered":" We begin with some quotes from a few writers, financiers and probability mathematicians. \u2022 Tagore: \u2018If you shut the door to all errors, the truth will be shut out (too).\u2019 \u2022 Chekhov: \u2018Knowledge is of no value, unless you put it into practice.\u2019 \u2022 Munger (4): \u2018You don\u2019t have to be brilliant, only a little bit wiser than … Continue reading Motivation for probability as degree of belief<\/span>
\u2022 Box and Tiao (1973): \u2018It follows that inferences that are unacceptable must come from inappropriate assumption\/s not from our inferential logic.\u2019<\/p>\n\n\n\n
Probability is our rational degree of belief <\/em>. . . in the truth of a proposition.<\/p>\n\n\n\n
The description of the process of how to use the information available to determine the<\/p>\n\n\n\n
(5) Mean-Variance Analysis, Oxford: Blackwell<\/p>\n\n\n\n
As human beings, we find this situation really tricky. There may be false intuition.<\/p>\n\n\n\n
7 See appendix on \u2018History & key figures. . .
8 This is how the \u2018Maximum Entropy Principle\u2019 works, and there is an explicit example of how this works mathematically in the first section of the Mathematical Miscellany later.
9 He won the prize in the late 1990s for his work with a colleague on making a theory of subatomic particle forces make sense.
10 This was called the \u2018N1\u00a0theory\u2019.
11 And to digress, we may wonder how (bad or good) it is for humanity to live life under such a hypothesis.<\/p>\n\n\n\n
13 What should we believe? What should we therefore do?\u00a0
14 See Objective Bayesianism by Williamson (2010)<\/p>\n\n\n\n
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