Continues from previous post (Fallacies of Frequentism and Subjectivism); this is Section 4: Forcing People to Believe, of my paper on “Subjective Probability Does Not Exist“
We now have the conceptual framework require to reveal the amazing secrets of one of the most spectacular magic tricks ever performed in human history, which continues to deceive millions. We will show how we can force a probability belief upon an unwilling Agent. Agent Ansa claims not to know the probability that it will rain in Tokyo one week from today. We will not only prove her wrong, we will actually produce the probability belief that she has, to within 1% accuracy.
In order to force a belief upon her, we must start by undermining her self-confidence. We ask her if she knows anything about any probability. She claims to know the probabilities of coin flips, dice, and cards. We ask if she has personal experience with flipping coins for a long time. When she acknowledges her ignorance, we can browbeat her by citing Clark and Westerburg (2009) who show that coin flips can be manipulated to produce bias towards heads. She should be flattened upon learning that experimental evidence shows that the coin will land on its edge about 1 in 6000 times. As the opening shot of De-Finetti’s book (“Probability does not exist”) shows, the first step of magic happens when Ansa relinquishes her intuitive conceptions of probability and cedes to our authority to define this for her. This step is made much easier by an empiricist mindset which creates doubts about existence of the unobservable and unmeasurable, invoking the widely believed epistemic fallacy.
At the second step, we cast around for suitable alternatives to intuitive probability. We ask Ansa if she remembers how probability was defined in the textbook which told her that coin flips lead to 50% probability of heads. She vaguely recalls the limiting frequency definition, which she memorized to reproduce on the exams, even though it didn’t make much sense to her. We re-assure her that her doubts about the legitimacy of this maneuver are justified. There is no way to observe a limiting frequency in the real world, and no way to make the definition applicable to the probability of a single coin flip. One of the leading authorities on probability, William Feller explained the problem as follows: “There is no place in our system for speculations concerning the probability that the sun will rise tomorrow. Before speaking of it we should have to agree on an (idealized) model which would presumably run along the lines “out of infinitely many worlds one is selected at random…” Little imagination is required to construct such a model, but it appears both uninteresting and meaningless.” Feller does not seem to realize that sunrise is not special in this respect. For any real-world event, infinite replications can only take place in an imaginary world. Once probability is defined as a limiting frequency, it is easily seen that this definition has no implications – none whatsoever – for any finite sequence of trials of any real-world event.
It requires only a simple sleight of hand to convert this rejection of frequency theory to a rejection of ontic probability. To Ansa, we can just say that all efforts by leading experts to define probability as a characteristic of real-world events have failed. For example, in the eminently practical context of weather forecasting, de Elía, Ramón, and René Laprise (2005) show that there is no agreement on how to define the probability of rain tomorrow. It is worth noting that the sleight-of-hand consists of assuming that if frequency theory is wrong, then probability cannot be defined; the possibility of other coherent definitions is ignored.
It is only after we have knocked out the intuitive and the external objective conceptions of probability that room is created to implant the notion of subjective probability. For this third step, we have to encourage Ansa to emulate the audacity of Kant, who dared to think that space and time are characteristics of our minds which we project onto external reality. We encourage Ansa to think that probability is really a projection of our minds onto external reality. When a coin is flipped, we do not know the initial conditions and forces acting on the coin. Thus, a statement about probability is really a statement about our internal states of (lack of) knowledge about the real world. Once Ansa accepts this radical (but reasonable) idea, we are well on our way towards the goal of implanting a belief into her mind.
Let G be event of rainfall in Tokyo one week from today. We now propose to calculate the personal probability she attaches to the event: PA(G). This probability is hidden within the mind of Ansa, unknown even to her. The easiest way to extract it is via comparison with some benchmark probabilities. But we have just destroyed probabilities, and so we need to reproduce them in disguise, as outcomes of rational behavior on part of Ansa. We can accomplish this as follows.
We ask Ansa to contemplate an Urn and Ball setup. 100 Balls will be placed into an Urn. N of the balls will be black and 100-N will be white. We will shuffle the balls vigorously to create an even and uniformly distributed mixture of black and white balls within the urn. Next, we will blindfold a picker, and ask him to reach deep into the urn and pick out one of the balls. We will say the event U(N) has occurred if a black ball is drawn, while U*(N) denotes the complementary event of drawing a white ball. Whether on not ontic single-case probabilities are well defined is subject to controversy and discussion. However, the knowledge of N, the number of black balls, does have a logical implication for our actions and decisions. In particular, define LU(N) to be a lottery which pays $5 if event U(N) occurs, and a black ball is drawn. It seems a matter of pure logic that any process which treats all balls in the same way will lead to increasing occurrences of U(N) as N increases. We must persuade Ansa that rational agents would have preferences over these lotteries which are monotonic in N:
Monotonic Preferences over Benchmark Lotteries: All rational agents prefer LU(N) to LU(M) if N is greater than M.
For our arguments, we do not need consensus over rational agents; it is enough that Ansa has monotonic preferences over these lotteries. Having set up our benchmarks, we are now in position to measure the subjective probability that Ansa assigns to G. G is the event of rainfall in Tokyo one week from today, to be ascertained by the official pronouncements of the Japanese Meteorological Agency. The lottery LG pays $5 if it rains in Tokyo, and $0 if it does not. We will ask Ansa make choices which compare LG with the benchmark lotteries, in order to calculate the personal probability that Ansa assigns to the event G.
Seven Steps to Revealed Probability: At the first step, ask Ansa to choose between LU(50) and LG. Her choice “reveals” one of two possibilities: (i) or (ii) . Thereafter, at each step, ask her to choose between the midpoint of the range of revealed probabilities and LG. Each step halves the interval of possible beliefs. Since 27=128, in seven steps we can determine Ansa’s belief to with ±1%.
Suppose that at the end of the seven steps, we find that Ansa chose L(G) over LU(85) and she chose LU(87) over L(G). We triumphantly pronounce that Ansa has revealed a personal probability of about 86% for rainfall in Tokyo. Just like the magician pulls a rabbit out of a hat, we reached into the mind of Ansa, and pulled out a probability that nobody knew was there! Suppose that Ansa resists this conclusion. She complains that her choices were arbitrary. She does not have any beliefs or knowledge about rainfall in Tokyo. We have a powerful counterargument to this claim. We claim that rationality commits her to use this “revealed” 86% probability for all subsequent decisions over the lotteries LG and LU(N). This can easily be proven.
Coherence, Commitment, and Rationality: Any choice of Ansa which conflicts with a prior choice is irrational.
Explanation: Suppose that Ansa chose LG over LU(85), revealing that . Next we offer her a choice between LU(60) and LG. She could now choose LU(60), revealing that , in conflict with her previous ‘revelation’. This would show that her choices were arbitrary, not aligned with any prior belief. However, inconsistency (or incoherence) is irrational. Consider the effect of the inconsistent choice. Ansa ends up with LG from the first choice and LU(60) from the second choice. Suppose she reverses both choices to make them consistent. Then she would have LU(85) from the first choice and LG from the second. The pair of lotteries (LG,LU(85)) dominates the pair (LG,LU(60)), because LU(85) is better than LU(60). Rationality commits Ansa to make choices consistent with the probabilities she revealed in the seven steps.
We have already re-defined the meaning of probability for Ansa. To clinch the argument, we also need to re-define the meaning of “knowledge”. We inform her that internal mental states are in constant flux, and she should ignore her “feeling” that she does not “know” about the probability of rainfall. The correct understanding of knowledge is that it is a guide to action. It is obvious from her choices that guides her choices over lotteries. On this basis, we may conclude that this number represents her knowledge about the probabilities of rainfall in Tokyo. Reluctantly, Ansa concedes to having “knowledge” about Tokyo weather, without her own awareness of this knowledge. The seven choices have served to reveal this knowledge to both external observers and to herself.
Applause – Bows – Curtains
This concludes Section 4: Forcing People to Believe, of my paper on “Subjective Probability Does Not Exist“. Next Post on Section 5: Differentiating Between Choice and Preference. Previous post was Fallacies of Frequentism and Subjectivism.
WEA Pedagogy Blog 
The Tokyo weather bureau when announcing the probability of rain at a future date did not consult Ansa so her preferences among bets about the outlook were no part of their assignment of probability.. It is perfectly possible to define something called subjective probability and to measure it by revealed preferences about which bets to take. (It is possible of course for this approach to fail to give rise to tightly defined subjective probabilities. People may just express indifference among a range of the possible bets, indicating their prior opinions are diffuse). Still it is unlikely the bureau made its forecast by organising a betting market among people in the office. Let’s follow Wittgenstein’s dictum: don’t think, look. What did the bureau do? They fed weather observations into a stochastic probit model which produced the 40 per cent result. In effect it told them that whenever such weather data had been observed in the past it had rained 40 per cent of the time. This is not an ontic probability because no such thing exists. The model reflects what the bureau knows about the causes of Tokyo weather and contains “random” elements representing all the things they don’t know. The forecast depends on a retained hypothesis about the causes they understand and a supplementary hypothesis that the things they don’t understand are random in the sense that they will cause deviations in the outcome but these deviations will be have no discernible pattern or bias. Given the hypothesis of randomness of unknown elements a probability can be defined that depends on past observations not on subjective probabilities. You can regard the notion of randomness as metaphysical if you wish but no empiricist would object.