You might die today. You might suffer a stroke, for example. Or, if you venture to the streets, you might be hit by a car. Or if you happen to be outside, you might be hit by a lightning. Or a meteor might strike near enough to you. Or World War III might start with a megaton warhead exploding nearby.
You might, also, if you are single, find your true love today. Or you might be crowned the Supreme Ruler of the Known Universe. Or you might find an important person’s wallet and be able to collect a huge finder’s fee. Or you could find yourself the single winner of a jackpot in the national lottery.
One purpose of the long childhood and adolescence of humans is to allow time for one to be taught and otherwise acquire the necessary skills in the great gamble that we call life. One learns to pay attention to the important things: take care to look both ways before crossing a road, for example. One learns to avoid the really dangerous things, such as touching a hot stove burner with an unprotected hand, or poking inside a live electric socket with an iron nail. Most importantly, one learns to learn, to adapt to new situations.
One learns to emphasize taking into account dangers and opportunities that one regards as likely: for example, when crossing a road, one looks left and right, since those are the directions where one is likely to see approaching vehicles that pose a collision danger. One learns to ignore extremely unlikely dangers: when crossing a road, one does not look up to see if a helicopter is about to land on the road. One also learns to adjust these likelihood estimates based on observations: if there’s a loud noise above, looking up is warranted; there actually might be a helicopter approaching to land.
Most people would not describe making the decision to cross a road as rolling dice, but that’s effectively what it is. Even if one is extremely careful, even if one has looked both left and right, and up for good measure, when one steps on the road and starts walking across it, one exposes oneself to risk. The risk is first that something out of the ordinary happens: a speeding motorcyclist traveling 200 km/h will not be noticed by a pedestrian in a routine safe-to-cross check, and the motorcyclist will not have time to take action to avoid a collision. The risk is also that one’s vigilance might be below par: a pedestrian who always crosses this road at this time of the day, not having encountered any conflicting traffic in the years and years they have taken this road, might not look as carefully as one should. There is also the risk of the extremely unlikely case of an airplane with a total engine failure making a silent forced landing on that road just as one is crossing it (bad news for both the plane and the pedestrian).
The gamble aspect is clearer (but sometimes misunderstood) when a doctor offers a patient the choice between an elective surgery and continuing with noninvasive treatment. Any surgical operation has the risk of death on the operating table; for most elective operations, where the surgeon can screen out high-risk patients, the risk is low, but patients do die in elective surgery all the time. An operation also has the risk of other adverse events, ranging from later death due to, for example, a massive pulmonary embolism, to less drastic ones. Any patient realizes this, and weighs whether the rewards justify the risks. What people sometimes miss is to take account that one is not adding risk but trading risk, for the noninvasive option also carries risk (in some cases even the risk of sudden death at about the time of the surgery would have taken place). However one looks at it, the key point is that one is rolling dice.
There is a philosophical theory that looks at life in this way. The practical knowledge each person carries in their head is modeled as a large table containing an entry for every eventuality that the person can conceive, listing for each the probability of it occurring at this instant, as that person reckons the probability. Each person’s table is different, reflecting differences in life experiences and personality. Each time one makes a new observation about the current situation or takes a decision that changes the situation, the table is instantly updated to match their personal probability for each eventuality in light of the new information or change of circumstances.
The theory requires that the probabilities in the table follow the laws of the formal theory of probability. The theory also says that a person is rational if their table of probabilities never leads them to placing a set of bets that is a sure loser; for example, betting for the equation 2+2=5 (assuming conventional meanings for the symbols in the equation) is a sure loser, while betting for Elvis being alive is not (it’s just very very unlikely). In technical terminology, a set of bets that will surely lose is a Dutch book (I do not know the etymology of that phrase); the theory thus states that rationality means not being vulnerable to a Dutch book.
This theory is, for historical reasons, called Bayesianism (though that term also encompasses other closely related theories); some authors use the more descriptive name of subjective probability. There are three key ideas: first, a probability is always defined in relation to some agent (a person, a computer program, etc), whose history shapes the probability; second, an agent learns by adjusting its probability estimates based on new data; and third, an agent’s actions can be viewed as bets.
Consider any particular moment when an agent receives a new piece of information. The probability the agent has assigned to a particular eventuality just before it receives that information is called its prior probability or just its prior for that eventuality. Conversely, the probability the agent assigns to an eventuality in response to the new information is called its posterior probability or its posterior for that eventuality. There are a number of proposed rules for deriving the posterior from the prior and the nature of the new information; the oldest and best established is based on Thomas Bayes’s Theorem of formal probability theory, which explains the name Bayesianism.
A number of philosophers and scientists of the early 20th Century found the inherent subjectivity of Bayesianism repugnant, and developed several alternative theories, common to all being the idea of the probability of an eventuality being an objective trait of the eventuality, not tied to any particular agent like Bayesianism decrees. Standard statistical inference, as it is taught in most courses of statistics and employed in most statistical studies over the 20th Century and even in this century, is the best known of these alternative theories. All approaches share the same formal theory of probability, but the way they apply it to real-world situations is different. While standard statistics is defensible on its own right, it has been mistaught and misapplied by scientists for nearly a century… but that’s a whole another blog post.
Nevertheless, I find the informal version of Bayesianism that I have described to be a very good rule-of-thumb model of many things. Consider, for example, the question of truth I discussed in my two previous posts (1, 2). Two witnesses may have a totally different view on whether the defendant handled a knife at the scene; this can be viewed as the result of ambiguous information (what each witness actually saw) combining with their personal priors to yield their respective posteriors, which they then offer to the court (no pun intended!). It also neatly explains why people come to different conclusions on such things as the existence of God, the efficacy of homeopathy, and the danger from global climate change; different people have different priors (which is another way of saying that they have different prejudices), and the evidence they have seen is, for many people, not sufficiently impressive to make an appreciable difference to their posteriors. As the saying goes, extraordinary claims require extraordinary evidence… but each person has their own idea of what is extraordinary.
There is, however, some hope. A theorem has been proved saying, in effect, that no matter how divergent the viewpoints of people on some point, sufficient evidence can be imagined to make them all agree. Whether that imagined evidence can, in practice, be found, is another matter.
There is another point to consider, as well. There are eventualities that are fatal. If one’s prior for even one of those eventualities is way too low, one will eventually be killed. On the Internet, we call that earning the Darwin Award; it is the Nature’s way to manage our collective priors.