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Let me illustrate this question with an example. Imagine you were to compare your credence or your belief of you winning the lottery twice with your belief in the devil’s existence.

Some argue that me winning a lottery two times has a defined probability whereas we don’t know if a devil exists. Hence, we can’t assign any probability to the devil’s existence and thus we can’t say that one should have a higher credence in my double lottery win than the devil.

However, let’s change the devil to the event where I toss a fair coin and have it land on heads 5,000,000 times. Now, it seems obvious that I should place higher credence in winning the lottery twice than tossing a fair coin and having it land on heads that many times. Even though the first is improbable, the latter is much moreso.

But here is where I see a problem that my mind is having a hard time wrapping my head around. The event of me landing a coin on heads is extremely improbable, yet it is possible. On the other hand, we can’t say that the devil is possible since the devil may not exist. In a sense, we have evidence for the coin event to be possible but not the devil.

But then this creates a scenario where I’m ultimately putting higher credence in an event with no evidence (I.e. the devil) than an event with evidence (I.e. the coin event) when comparing it to the lottery event. Even if I claim that my credence in the devil should be unknown, I am de facto giving it higher credence than the coin event since when compared to the lottery event, I don’t say that my credence in the devil is lower than it. This doesn’t seem right.

What’s going on here and how should I navigate through this?

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  • Instead of asking a question with a very weird title linked to a very specific example, why not ask a more general version of the question that captures as many similar examples as possible at once? For example, something along the lines of "Are epistemic probability and empirical/statistical probability comparable?".
    – Mark
    Mar 28 at 12:58
  • @Mark Good point, made the edit! Mar 28 at 13:00
  • I didn't mean that my title was the ideal title, it was just a suggestion (feel free to improve upon it), but thanks.
    – Mark
    Mar 28 at 13:01
  • An interesting example to ponder: what's more probable, the devil or abiogenesis? We have never observed abiogenesis take place empirically, so the statistics associated are essentially 0. However, many people have reported encounters with the devil.
    – Mark
    Mar 28 at 13:06
  • @Mark What’s tough with these examples is that there’s no correct probability: in the real world, devils either exist or don’t. Abiogenesis is either true or not. It seems as if one must just make a bet? Mar 28 at 14:30

3 Answers 3

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Let's call the two orderings at play, the 'arithmetic/empirical probabilistic' one and the 'epistemic/credence' one, (A, <') and (C, <'), respectively, so that the question

Are epistemic probability and empirical probability comparable?

becomes one about existence of monotone (non-decreasing) functions in either direction

Now, it seems reasonable that < is a linear order in A, [we may well take A to be the unit interval with the usual ordering, or at least the rational points] while <' may be neither antisymmetric nor linear in C, so even if we (somewhat artificially) passed to a quotient to obtain antisymmetry, the existence of a monotone f: C -> A would exactly mean linearity of <'; besides that, one of your points is that there's no way to meaningfully asign an(y) aritmetic probability to a certain d in C, which here just means that "there is no total function f: C -> A whatsoever, much less a monotone one", so in a sense 'comparability in one of the directions' is really doomed to fail

On the other direction, we may construct an f: A -> C (canonical in a sense) by taking a belief (?) like "throwing a dice with n faces will result in a face with value at most m < n" to be f(m/n), so that it makes sense to speak of winning lotteries ('l') and tossing coins ('c') in both contexts. Notice also that f cannot be surjective, by the previous paragraph, but it doesn't forbid one from having more credence in s = "the Sun will rise tomorrow" than in l, so that l <' s makes sense, even if there's no f-¹(s) to assert l < f-¹(s)

So where's the devil in these details? In

But then this creates a scenario where I’m ultimately putting higher credence in an event with no evidence (I.e. the devil) than an event with evidence (I.e. the coin event) when comparing it to the lottery event.

you worried that the previous points, not(d <' l) and (c < l) - and hence also (c <'l) - , somehow would imply that (c <' d), but it should be clear now that such points are not sufficient to conclude so

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When determining a probability:

  1. An event must be defined and triggerable
  2. The number of possible unique states must be determined.

For a coin toss:

  1. There are two unique states: Heads and Tails
  2. The triggerable event is the physical toss/flip of the coin.

The probability can then be calculated by triggering multiple events and recording the outcome: 50/50 for a fair coin as the number of events approaches infinity.

For the Devil's existence:

  1. There are two states: Exist or Doesn't Exist
  2. There is no defined triggerable event

The probilities are incalculable so they cant be empirical. Assigning an epistemic probability to the Devil's existence is meaningless speculation without a defined triggerable event.

This is one case where epistemic and empirical probabilities are not comparable.

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    Can you give a reference on the need for a "triggerable event" in the definition of a probability? If you change the question to "what is the probability of experiencing a visitation from a devil?" then that does have a "triggerable event", the visitation. Obviously the probability of devils existing must be greater than that. Apr 2 at 8:30
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The key to this is understanding the difference between epistemic and empirical probabilities. As it's name suggests, epistemic probabilities are statements about your state of knowledge. Some might prefer to call this belief rather than knowledge as there is no direct link between your state of knowledge and the true state of reality. Empirical probabilities are probabilities that have been measured or observed. I suspect that the question is actually talking about frequentist probabilities, which are defined by long run frequencies, but don't necessarily have to be measured. For example, you could have a statistical model of an coin, for which the long run frequency of flipping the coin and seeing heads is 0.5. This isn't an empirical probability because it is an imaginary coin created for a model of reality (in practice a real coin is not going to be exactly unbiased). To summarise:

  • empirical probability: an observed frequency of a particular outcome;
  • frequentist probability: a probability defined by a long run frequency (so cannot be applied to e.g. the truth of a particular proposition or the outcome of a particular random event);
  • epistemic probability (aka Bayesian probability): A numeric representation of the relative plausibility of a proposition or of observing a particular outcome of a random event (but you can also reason about long run frequencies using epistemic probabilities).

An epistemic probability will only correspond to reality if your prior state of knowledge is consistent with reality and your likelihood function correctly represents the conditional probability of the variable of interest given the hypothesis. It is a bit like logic, if your premises are false or your chain or reasoning is fault, then your conclusion will not correspond to reality.

In the comments, Baby_philosopher writes: "What’s tough with these examples is that there’s no correct probability: in the real world, devils either exist or don’t."

In exactly the same way, there is no correct probability that the next coin flip I make will come down a head - it either will or it won't. It is a single event, without a long-run frequency,so it doesn't have an empirical probability or a frequentist probability (other than the trivial 0 or 1). You can however have a numeric degree of plausibility that it will be a head, based on your prior understanding of the mechanics of coin-flipping (and the evidence provided by observing earlier coin flips).

If you ask a frequentist statistician for the probability the next coin flip I make being a head, they will give an indirect answer (normally without telling you they are not giving you a direct answer) by assuming that it is a random sample from a large population of coin flips. However, what they are doing there is forming a Bayesian (epistemic probability) based on a long run frequentist probability. Usually these kinds of thing are benign and don't cause a problem, but sometimes they do. Try explaining why the probability of the true value lying in a 95% confidence interval is not 95%.

"Some argue that me winning a lottery two times has a defined probability whereas we don’t know if a devil exists. Hence, we can’t assign any probability to the devil’s existence "

This is a lack of understanding of epistemic probability. We don't need to know if a devil exists to assign a probability to it's existence. Indeed it is a non-sensical statement because if we knew devils exist, then an epistemic probability of 1 would be fully rational. If we knew they didn't exist, then an epistemic probability of 0 would be fully rational.

An epistemic probability is an expression of your state of knowledge regarding the proposition. It doesn't necessarily correspond to a probability in reality, because some things have no (non-trivial) empirical probabilities. The benefit of empirical probabilities is that they allow you to explicitly state your beliefs numerically in a way that allows them to be rationally updated as you acquire evidence and to explore the expected consequences of your beliefs (i.e. make testable predictions with a well documented chain of reasoning).

Now another problem with the question is that it assumes that you are restricted to point values for empirical probabilities, but that isn't true. Competent Bayesians tend to prefer distributions that capture the uncertainty in their knowledge. For examples if you had no knowledge at all about devils, you might adopt a uniform prior on the interval [0, 1], indicated a lack of preference for any particular value for the probability that devils exist. It would capture your complete ignorance on the question - you would not be putting a "higher credence" on anything.

So as soon as you understand what epistemic probabilities are for (the hint is in the name) the difficulty goes away.

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  • @downvoter and indication of what you disagree with in my answer would be appreciated. AFAICS it is fairly standard. Mar 30 at 18:20

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