How to assess the probability of pretty much everything

An ancient approach to effectively navigating a world of uncertainties

The Bayesian principle of updating one’s beliefs as a result of new evidence, from analyticsvidhya.com.

I’m pretty sure—almost certain, in fact—that right now I’m sitting at my desk, typing an essay on probability on my recently acquired MacBook Air M2 laptop. True, there could be a Cartesian demon messing around with my mind and creating the illusion that I’m typing on a keyboard, but I don’t think that’s very likely. It is only slightly more likely that I’m dreaming of typing on a keyboard. I’ve certainly had stranger dreams before. Then again, I can usually tell the difference between when I dream and when I’m awake, so I’ll stick with my initial assessment. I am, indeed, typing this essay on my laptop’s keyboard.

One thing I’m almost absolutely sure of is that the square root of nine is three. Unless I’ve suffered from a stroke that has impaired my basic reasoning functions. Which is unlikely, but certainly possible.

By contrast, I’m far less positive about what the weather will be next weekend in Brooklyn. That’s because I know that weather forecasts aren’t reliable if stretched over a period of more than 3-4 days. If pressed, I could guess based on Brooklyn’s typical weather this time of the year, though with climate change accelerating you never know for sure.

And so on. Our lives are made of constant assessments and re-assessments of probabilities. Rarely, if ever, can we seriously claim to have Knowledge of the Truth (notice the capitalized letters). But we also rarely, if ever, actually need Knowledge or Truth. A convincing evaluation is good enough to actually act in the world.

The word “convincing” is how the Greek term pithanon is often translated, and pithanon was the criterion for action invoked by Academic Skeptics like Carneades of Cyrene. Interestingly, the Roman philosopher Cicero translated that same word with the Latin probabilis, from which of course the English probability derives. When we say that something is probable, therefore, we mean that we have convincing evidence or reasons to provisionally accept it as true. And, more importantly, to act on it.

The level of pithanon related to the notion that I’m sitting and writing is so high that there is no point for me to question the idea. But I should be more hesitant to plan for an outing with my wife and friends next weekend, because the pithanon associated with there being good weather at that time is not high.

This general way of looking at things, that we should assess the relative likelihood of this or that proposition or situation and act accordingly is nowadays often framed in terms of Bayesian probabilities, after a fundamental theorem proposed in 1763 by Thomas Bayes. The basic idea of Bayes’ theorem is that we ought to begin the evaluation of a situation on the basis of what we already know and then update our assessment as new information comes in. As David Hume put it, at about the same time as (and independently of) Bayes, a reasonable person proportions their beliefs to the evidence.

For instance, take my concern with the weather in Brooklyn next weekend. Right now the forecast is morning showers on Saturday and partly cloudy on Sunday. The chances of said morning showers are given at a suspiciously precise—considering that we are talking ten days down the road—47%. What degree of trust should I put in this forecast? Based on what I know about long-term weather forecasting and on my own past experience, I’d say very low. Still, at the moment I may be cautious about scheduling a picnic with several friends on that Saturday, and perhaps lean more toward Sunday instead. But of course I’ll keep checking the forecast over the next few days, every time updating my bets for the weekend in accordance to how the forecast changes, by how much, and in what direction. At some point I’ll have to take a gamble and either schedule the picnic or not. After all, my friends need to plan ahead of time as well!

Bayes’ theorem actually provides us with precise, constantly updated estimates of the relevant probabilities. You plug in the relevant numbers and the equation gives you the best assessment of how strong your belief in X (where X is, say, “it will rain on Saturday”) should be. The problem is that for most of us it is simply not practical. Setting aside that many people are insufficiently familiar with (or have never even heard of) Bayesianism, rarely we have available the kind of quantitative information that can correctly be used in this context.

But there is an alternative framework, proposed by the Academic Skeptic Carneades of Cyrene during the second century BCE, which accomplishes nearly the same results, no math required!

To appreciate how this works, we need to recall what an “impression” is in ancient Greco-Roman psychology. The term, phantasia (pl. phantasiai) in Greek, refers to our first, usually instantaneous, assessment of a given situation. For instance, I walk by a gelato parlor and I get the impression that it would be good to walk in and get me a dark chocolate cone.

Before I act on such impulse (Gr., orgē), however, I need to step back for a moment and deliberate whether to give it (or not) assent (Gr., sunkatathesis). Assent is always the result of deliberation about something. In this particular example, I remember that I’m about to join my wife for dinner and that eating gelato now would spoil my appetite. Not to mention that we’ve just come back from a long vacation in Italy and I’ve put on a few pounds I’d like to shed. The gelato wouldn’t help there either. So I deny assent to the impression and decline to act on the basis of the corresponding impulse.

There is a nice modern interpretation of this framework, based on the work of Nobel prize winning psychologist Daniel Kahneman, and explained in his best selling book, Thinking Fast and Slow. The idea is that our brains work on two tracks: subconscious and instinctive (fast, System I) or conscious and deliberative (slow, System II). In some situations, like in cases of sudden and imminent danger, the fast brains takes over and acts quickly, if inaccurately. But when we have time to consider things a bit it pays to hit the brakes and allow the slow but more accurate brain to kick in. Essentially, acting on the basis of impressions alone is equivalent to engaging Kahneman’s fast thinking, while deliberating whether to assent to an impression or not calls into action Kahneman’s slow thinking.

Carneades thought that we can engage impressions with multiple levels of accuracy, depending on just how confident we wish (or need) to be about our assent. Consider the following diagram, from Anthony Long and David Sedley’s The Hellenistic Philosophers

Relative to the object (called the “impressor” on the left side of the diagram, because it is what generates the impression, like my gelato sighting) there can only be two situations: either the impression is true or false. Neither the Stoics nor the Skeptics were into denying the existence of objective truths about the world. The problem, of course, is that we don’t have direct, unmediated access to the external world, so we are stuck with the sequence on the right of the diagram, relative to the “percipient” of the impression, that is, us.

The first level is essentially a reflection of how the impression appears to us at first glance, as the result of the mediation by Kahneman’s System I: apparently true or apparently false. If there isn’t much at stake in a given situation—say, having to decide which wine to order with our dinner—then we can simply go with our unreflective impression and be done with it.

Should the decision, however, have more import than that we may want to pause and consider more carefully: is the impression “intensely apparent,” that is convincing, or not? For instance, if we are in charge of picking a good restaurant for a gathering with friends we should look closely at its characteristics before committing, or else we risk disappointing our friends and becoming frustrated ourselves.

Sometimes we require even more confidence than that, so we should consider escalating things to the third level. For that we investigate how well the impression under examination coheres with other, related impressions. If none of the other impressions we consider casts doubt on the focal one, then the latter is said to be “undiverted” and we can be reasonably confident about our assessment. For example, say that we are contemplating buying a house. Have we looked not just at the cost and general features, but also at the potential resale market, whether the neighborhood is getting better or worse, how far of a commute it will be from there to get to work, whether there are grocery stores and pharmacies nearby, and so on? Any of these additional considerations could divert our impression and convince us not to buy the house.

Finally, there will be cases where the consequences of our assessment are serious, the stakes high. In those instances we ratchet up our scrutiny even further and subject the pertinent impression to a thorough examination. Only if the impression survives such a heightened scrutiny we give assent to it. For instance,  we may be weighing the option of making an important career change, such as leaving our current job and seeking another one. We should do so only after we have had the time to investigate the available possibilities in some detail, because once taken the decision may be irreversible and highly consequential.

The Pyrrhonian Skeptic Sextus Empiricus summarized Carneades take in this way (italics mine):

“An impression never stands in isolation but one depends on another like links in a chain. … So whenever none of these impressions diverts us by appearing false, but all with one accord appear true, our belief is all the greater. For we believe that this is Socrates from his having all his usual features—color, size, shape, conversation, cloak, and his being in a place where there is no one indiscernible from him. … As in everyday life when we are investigating a small matter we question a single witness, but in the case of a larger one several, and in a still more crucial matter we cross-question each of the witnesses from the mutual corroboration provided by the others—so, say Carneades and his followers, in matters of no importance we make use of the merely convincing impression, but in weightier matters the undiverted impression as a criterion, and in matters which contribute to happiness the thoroughly explored impression. (Against the professors 7.176–84)

We can summarize Carneades’ handy scheme by attaching some subjective probabilities (“p” below) to each level of analysis of the impression, in the following manner [1]:

(i) Apparently true, it seems more probable than not (p>.5)

(ii) Convincing, it appears very likely to be true (p>.75)

(iii) Undiverted, it coheres with and is mutually corroborated by other factors (p>.9)

(iv) Thoroughly explored, it has been subjected to a meticulous cross-examination (p>.99)

So now you have a nice, compact, and easy to use guide to assessing situations, first suggested over two millennia ago and yet compatible with the best modern understanding of probability theory. When you have to evaluate an impression, begin by deciding how much confidence you reasonably need, then select the appropriate level of Carneades’s scheme and act accordingly. Happy decision making!

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[1] In Bayesian analysis probabilities are often understood as estimates of the strength of the agent’s subjective belief that something is, or is not, true. While in some cases it is possible to substitute objective probabilities in the equations, understanding belief in this fashion is both useful and of very general applicability.