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What does it mean for A to be more likely than B? For example, suppose two people are throwing darts. The first person gets a bulls eye 6 out of 10 times. The second person misses every single time by a wide margin. Suppose now that A = first person is a professional dart player and B = second person is a professional dart player.

Is A more likely than B? If we look at priors, I suppose we can say that a professional dart player is more likely to get 6 out of 10 bulls eyes than a non professional dart player. However, the opposite implication doesn't necessarily hold. For example, a bad dart player could get lucky, or a good but non professional dart player could also get 6 out of 10 bulls eyes. Does this still mean that A is more likely than B?

Or should this question be answered if we imagine a bet. For example, if we knew that one of them is a professional dart player in advance, which one would you bet on? When phrased in this way, it definitely seems more rational to bet on A. However, we don't know in advance if one of them is a professional dart player. As such, in what sense is A more likely than B?

What about the scenario where there don't seem to be priors? For example, suppose person A and B tries to guess where ten coin tosses would land. Person A gets 3 coin tosses correct. Person B gets 9 coin tosses correct. Is it more likely for person B to be a psychic than person A? Well, if we go by priors, I'm not sure how there seems to be any. In the case of the previous example, we have data that shows that professional dart players atleast exist. In this case, we have no prior evidence that suggests psychics exist.

Again, we can transform this into a betting scenario. If we knew in advance that one of them was a psychic, what would you bet on? In this case, again, it probably makes more sense to bet on person B. But not only do we not know in advance that one of them is a psychic, we don't know in advance that psychic abilities are even possible.

So what does "more likely" mean in both of these cases? Or is this helplessly subjective?

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  • One doesn't need de Finetti's fair betting strategy or Dutch book argument to understand the colloquial "more likely" simply as "more plausible" in terms of one's degree of belief of some possibly necessarily idealized thing you roughly depicted above to be practically useful for anyone. Indeed in some pre-prior situation one's ignorance could be so complete that one's inner probability density about the probability of whether a certain outer state of affairs obtains or not could be totally flatly distributed between 0 and 1, but most probably wouldn't apply such ignorance to psychic ... Jan 28, 2023 at 6:12
  • Pick one of these, then it means $P(A)>P(B)$, where part of your question is choosing events $A,\,B$ carefully. I'm not saying your questions are purely mathematical, without philosophical components; I'm saying the latter are best analyzed after teasing out several sub-questions, then answering them in a preferred order.
    – J.G.
    Jan 28, 2023 at 22:11

5 Answers 5

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I believe you are asking the same question over and over again in all these posts: can probability theory help me with subjective decisions?

Cutting to the chase, probability theory is a mathematical tool that has proven rules to calculate and infer, but what you feed into it either has a basis in data (you have frequentist probabilities), or is subjective. If you don't have data, the input probabilities to the machinery are subjective, and that's that.

Once you are in subjective territory, making decisions will have to rely on personal preferences, or criteria external to probability theory, but you won't find in probability theory a definitive answer on how to make subjective decisions.

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    +1 Another theme is "What is the minimum information required to make an accurate prediction".
    – user64314
    Jan 28, 2023 at 18:08
  • @SteveSaban But this question is also subjective IMHO. Even in science, requiring "6 sigma" to claim a discovery is a somewhat arbitrary threshold. Probability theory, statistics, etc etc do not provide objective criteria for those thresholds. There aren't any hard and fast thresholds, only heuristics and established practices.
    – Frank
    Jan 28, 2023 at 22:29
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Your are supposing that 'more likely' has a well defined meaning, which is quite wrong. The way the phrase might be used by a physicist or statistician might be quite different to the way in which it might be used in a court of law or a betting shop.

There are certain circumstances in which the phrase can be applied entirely rationally. If you have the cliched urn containing one hundred black balls and one white one, you can say it is more likely that if you extract one without looking it will be black.

If you have two equally proficient darts players, you might reasonably suppose that the chance of one of them hitting six bull's eyes in a row is just as unlikely as the chance of the other, as you have no reason to suppose otherwise.

If you are at a race track and there are two horses in a race, one tall and muscular with a professional jockey, the other short and emaciated with an overweight beginner on its back, you might suppose it more likely that the tall horse will win.

What you might be starting to see as the common denominator in the examples I have cited so far is that something can be reasonably supposed to be more likely if there are one or more reasons to suppose so. Professional jockeys on fit thoroughbreds tend to win races more often than beginners on donkeys. Professional darts players tend to hit a bull's eye more often than occasional players. And so on.

In the case of the balls in the turn, the reasoning can be quite objective and readily quantifiable- almost everyone would agree that your are less likely to extra the one white ball from among the hundred black ones. In other cases there may be either scope to adopt a more-subjective view, or a risk of mistakenly quantifying the chances, because the factors involved are less clear cut.

In arriving at a view of the probability of a given event, you might apply a mix of mathematical reasoning, pure guesswork, extrapolation from experience, irrational subjectivity and so on. There is no absolutely right way to go about it in the general case.

I imagine, without having hard evidence to back up the view, that the assessment of probabilities is an inherent trait we possess as a result of evolution. After all, a primitive human who was the better judge of whether tigers might be in caves, or whether food might be found on a certain type of landscape, would be more likely to survive.

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    Most people are quite bad at estimating probabilities, or their consequences.
    – gnasher729
    Jan 28, 2023 at 18:19
  • @gnasher729 According to Daniel Kahneman, that includes professional statisticians.
    – J D
    Jan 28, 2023 at 19:38
  • I'm not supposing that hence why the question was asked
    – user62907
    Jan 29, 2023 at 6:38
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Excellent question. It is a tremendously broad question, as all interesting questions are because while all people use phrases that relay confidence in outcomes and events, very few people reflect on the meaning of the word. Briefly, it depends on the context and interpretation, and there are number of different ways to interpret.

While one could dissect the question from the perspective of philosophy of mind regarding propositional attitudes, I'm going to start you off with the philosophy of math. Where you should start is with the some basic probability and statistical knowledge, which I presume you have, and then introduce you to the fact that there are multiple interpretations of probability (SEP):

One regularly reads and hears probabilistic claims like these. But what do they mean? This may be understood as a metaphysical question about what kinds of things are probabilities, or more generally as a question about what makes probability statements true or false. At a first pass, various interpretations of probability answer this question, one way or another.

According to the article, there are a number of them:

3.1 Classical Probability
3.2 Logical/Evidential Probability
3.3 Subjective Probability
3.4 Frequency Interpretations
3.5 Propensity Interpretations
3.6 Best-System Interpretations

The TLDR is that depending on the context one can justify claims about probability in several ways. For instance, a frequentist interpretation is the probability you learned in school. For independent events, it's about possible outcomes. A six-sided die tells us that rolling a 1 is less probable than rolling a 1 or 2. The former P=1/6, the latter P=1/3. But can we apply that to scientific claims that are inductions from a qualitative theory? Probably not. In fact, to use frequentist fractions would be a species of fallacy called false precision. 'I'm 99% sure!' when self-appraising confidence is a claim that isn't generally subject to the language of statistical frequency.

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It's subjective because terms like "professional, non-professional, good, bad" have no objective meaning in your question. What is the statistical difference between a professional and an amateur? What's the statistical difference between good and bad? What is the statistical difference between a psychic and a regular person? .

Typically, applying objective values to subjective terms is done by a consensus of interested parties. Like these people did:

http://www.sejarchive.org/resource/IPCC_terminology.htm#:~:text=Likelihood%20of%20an%20outcome%20or,means%20greater%20than%2050%20percent.

The IPCC expresses its consensus conclusions using terminology that indicates the varying levels of certainty the authors have in them. (Reference: Working Group I Technical Summary, 18.64MB, pages 22-23.)

Likelihood of an outcome or result

"Virtually certain" means greater than a 99 percent probability of occurrence. "Extremely likely" means greater than 95 percent. "Very likely" means greater than 90 percent. "Likely" means greater than 66 percent. "More likely than not" means greater than 50 percent. "About as likely as not" means 33 to 66 percent. "Unlikely" means less than 33 percent. "Very unlikely" means less than 10 percent. "Extremely unlikely" means less than 5 percent. "Exceptionally unlikely" means less than 1 percent.

Relative degrees of confidence in a statement

"Very high confidence" means at least a 9 out of 10 chance of being correct. "High confidence" means about an 8 out of 10 chance. "Medium confidence" means about a 5 out of 10 chance. "Low confidence" means about a 2 out of 10 chance. "Very low confidence" means less than a 1 out of 10 chance.

This group decided on these values by consensus

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In short, "more likely" means a higher likelihood, which you can attempt to approximate using historical data, and which you can attempt to use in your favor when betting. Everything else below is offered to help provide a clearer way of thinking about these scenarios.

One source of confusion seems to be, "what if we don't know what the prior is" or "what if the prior is small". A can be more likely than B even if A and B are both very unlikely and even if I don't know how likely they are. For example, whats the chance that A: I win the lottery, vs B: the chance that I win the lottery AND am psychic? B is A with another condition so B < A no matter what. But both are unlikely, and I don't know what the numerical value of either is.

What does it mean for A to be more likely than B? For example, suppose two people are throwing darts. The first person gets a bulls eye 6 out of 10 times. The second person misses every single time by a wide margin. Suppose now that A = first person is a professional dart player and B = second person is a professional dart player.

Probability of First = Pro given that First scored 6/10 : P(Pro | 6/10)

vs

Probability of Second = Pro given that Second scored 0/10. : P(Pro | 0/10)

You can calculate from Bayes law:

P(Pro | Score) = P(Score | Pro) P(Pro) / P(Score).

Both have the same prior P(Pro) so you really compare P(6 | Pro) / P(6) and P(0 | Pro) / P(0). If being a Pro means you will get 6s more often than an average person and 0s less often than an average person then:

P(6 | Pro) / P(6) = More > Less = P(0 | Pro) / P(0).

Is A more likely than B? If we look at priors, I suppose we can say that a professional dart player is more likely to get 6 out of 10 bulls eyes than a non professional dart player. However, the opposite implication doesn't necessarily hold. For example, a bad dart player could get lucky, or a good but non professional dart player could also get 6 out of 10 bulls eyes. Does this still mean that A is more likely than B?

Yes, there may be unknown factors at play, but probability is a set of methods for dealing with our ignorance. So A is more likely than B on average, but some of the time even if you bet on the more likely outcome you'll be wrong. A coin might be weighted so that it lands heads 60% of the time. If forced to bet, you will bet heads. But 40% of the time, you will still be wrong.

If you imagine God's database of every time someone scored 6 and every time someone scored 0 and look at how often a 6-scorer was Pro and how often a 0-scorer was Pro, that takes into account all the possible scenarios and lucky days and unlucky days.

Or should this question be answered if we imagine a bet. For example, if we knew that one of them is a professional dart player in advance, which one would you bet on? When phrased in this way, it definitely seems more rational to bet on A. However, we don't know in advance if one of them is a professional dart player. As such, in what sense is A more likely than B?

If you know one is a Pro, now you're asking a slightly different question.

P(A is THE pro | A got 6/10 and B got 0/10)

P(B is THE pro | A got 6/10 and B got 0/10) = 1 - above

Now it is like you look into a database of every time a Pro and Non-pro competed, and look at the times that the scores were 6-0. The chance of A being Pro is the fraction of the times that the player with 6 was pro in this database.

What about the scenario where there don't seem to be priors? For example, suppose person A and B tries to guess where ten coin tosses would land. Person A gets 3 coin tosses correct. Person B gets 9 coin tosses correct. Is it more likely for person B to be a psychic than person A? Well, if we go by priors, I'm not sure how there seems to be any. In the case of the previous example, we have data that shows that professional dart players atleast exist. In this case, we have no prior evidence that suggests psychics exist.

Even without priors, since A and B have the same prior, when comparing which is more likely, you only compare the likelihood. Which is relative and based on how they scored.

Again, we can transform this into a betting scenario. If we knew in advance that one of them was a psychic, what would you bet on? In this case, again, it probably makes more sense to bet on person B. But not only do we not know in advance that one of them is a psychic, we don't know in advance that psychic abilities are even possible.

If you posit that you know one of them is a psychic that contradicts the part where you say we don't know if being psychic is possible.

So what does "more likely" mean in both of these cases? Or is this helplessly subjective?

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