Probability

The probability of a proposition is a quantification of all possible states of the world in which the proposition would be true, given one's present state of uncertainty.

[TODO yuck]

[Old definition, delete later: Probability is a quantitative method of exploiting the symmetry of a system to reach, with certainty, conclusions about aggregates of outcomes, even when each individual outcome is uncertain.]

The idea behind probability is that when faced with uncertainty, we quantify the outcomes which could be the case, based on our present knowledge. In some cases, this quantity is discrete, such as all the possible configurations of a deck of cards. In other cases, this quantity is continuous, such as all the possible places on the dart board where a dart could land.

The purpose of quantifying outcomes like this is that it makes them amenable to measurement, to the identification of quantitative relationships between uncertain situations.

Probability is related to symmetry, in the sense that there is a symmetry among any set of states of affairs which could be the case: they are by definition indistinguishable in the respect that we do not know which of them is the case.

The point of probability is to achieve certainty. [TODO elaborate]

Examples

The following are examples of things that occur with numerical probability:

- weather events

- dice rolls, coin flips, drawing a card from a shuffled deck, lotteries, etc

- pseudoRNGs. [Sometimes people ask: Are pseudoRNGs really random? The answer is yes, because a good pRNG is precisely one whose outcomes are not something we know how to predict. TODO move]

- the direction of cosmic rays

- equal energy microstates of a physical system

- brownian motion

- the measurement of a qubit in a generic basis

- the gender of a conceived embryo

But isn't everything possible?

A judgement that X is P is always made with some context in mind. In particular, this is also true of the judgement that some outcome is possible. In most everyday contexts, like that of flipping a coin a few times by hand (but unlike that of a scientist studying a nuclear chain reaction), an outcome which has a 1 in a billion chance of happening should be categorized as impossible.

Categorizing such rare outcomes this way is cognitively necessary, in the sense that if you operated on the principle that events with a 1-in-a-billion chance of happening are "possible," you couldn't function at all. Indeed, if something is categorized in your mind as "possible within the next minute," then that means you should be prepared for it; but there is no way to know about, let alone be thinking about, let alone be satisfactorily prepared for, all the 1-in-a-billion events that might happen to you in the next minute. It is not necessary to be prepared for such things, either.

Consider the following example.

If you are a seemingly healthy 30-year-old, who regularly gets checkups, there is still some sense in which you might randomly get a heart attack the next time you climb the stairs. Indeed, it is technically "possible"; things like that have actually happened to people before. However, even though it is technically "possible," you should not think of such things as possible in your every day life (this is not contradictory, because in the former usage of "possible" the context is that we are thinking of the tens of billions of people throughout history whose bodies have failed in various ways, and in the latter usage the context is that we're thinking about a specific individual).

Let's say that you don't take this advice, and instead think of the heart attack as "possible." It will wreak havoc in your mind. For example, it's also possible that this afternoon, your boss will want to talk to you about how your business report report is coming along. How could you concentrate on preparing for a potential meeting with your boss, when it's "possible" that you're going to get a heart attack today?? You couldn't. Or for example, by the same standard by which you accepted that it's possible you'll get a heart attack while climbing the stairs, you are also logically compelled to accept that it's possible you'll die in an elevator-related accident. How should you get to the 5th floor of your office building, when it's "possible" you'll die no matter which way you do it?? You better just stay on the 1st floor and not risk it. Etc.

And note that this prospect of a "possible" heart attack is just one of an endless list of "possible" ways that a healthy 30-year-old "could" die today. I chose the possibility of a heart attack rather arbitrarily, and there are many other things I could have added to the list.

[TODO I really don't like this example. Part of the reason not to be concerned about heart attacks is that you can only do so much to prevent them. Also, the heart has an identity, and it will definitely fail or not fail based on nature. It's not like the universe is a place where anything at all is possible. We might not know all of the reasons for heart failure (or we might). When we do, we can get rid of probability.]

The idea is that every context determines some small number $\epsilon >0$, such that any event which will only happen with probability less than $\epsilon$ is deemed impossible (IN THAT CONTEXT). [TODO really this should be a statement about propositions.] There's an objection that it's "possible" to flip a coin 100000x and get heads every time. This is mis-using the concept of "possible." We have concepts of false, unlikely, possible, probable, true, arbitrary, etc. because we need these categories cognitively. If you categorize a coin landing heads 100000x in a row as something that's "possible," then it destroys your cognitive categories and renders your mind impotent. More about this in section below.[TODO does this rly belong here?]

The traditional concept

Unlike what many people believe, probability is not something that comes in at the base of metaphysics. Probability is epistemological, not metaphysical.

One intellectual view asserting that probability is metaphysically central is Bayesianism. [TODO mention the principal principle]

Numerical probability (i.e. assigning actual numbers to uncertain outcomes) is a very high-level concept, and is only applicable in certain cases. An example where numerical probability is not applicable:

  This Monday, the president will announce his decision about whether or not to run for re-election.

Some Bayesians try to say like oh well it's a 30% chance he runs, 70% chance he doesn't. Prediction markets are based on this idea [TODO add a caveat about prediction markets; I think they could be useful]. But it's not a valid use of probability. Why is it not valid? [TODO think more about this]

  * you'd have to explain why it's a 30% chance he runs, which you can't. There is no symmetry present in the president's decisions, and they are not based on symmetric things like dice-rolling. [TODO is that the only basis for probability?]

Another view, sharply opposed to Bayesianism in many other respects, is held by David Deutsch and his followers. DD believes in the many worlds interpretation of quantum mechanics, and he thinks it is that metaphysical thing which gives rise to probabilities. He says things like

    Probability is not a measure of ignorance.

    Source: https://twitter.com/DavidDeutschOxf/status/1692513674798252452

[TODO elaborate more based on his actual writings] It's not completely clear that his view contradicts mine, because the many worlds interpretation also says that probability is epistemic; it's your uncertainty about which branch of the universal wave function you're living on.

Specification

Specification of probability. We should assign number $1/k$ to each of the $k$ possible states of system with a $k$-fold symmetry. This number is called the probability of the state.

Sometimes this specification is called a principle (see e.g. the "fundamental assumption of statistical mechanics"). The principle say something like "any state in a system with a symmetry is equally likely to occur." That approach assumes we have some prior concept of probability, and then tries to equate it with multiplicity.

Consider a two-sided coin. The coin has a twofold symmetry. It follows that two distinguishable coins have a fourfold symmetry, three distinguishable coins have an eightfold symmetry, and $n$ distinguishable coins have a $2^{n}$-fold symmetry. "Distinguishable coins" could mean the same coin at different periods in time (say after different flips). So we see that a history of $n$ coin flips belongs to a system with a $2^{n}$-fold symmetry. So by the specification of probability, it occurs with probability $2^{-n}$.

This principle generalizes of course.