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What is the semantics of assigning probabilities to future events?

by Michal1 min read1st Apr 20216 comments

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Probability & StatisticsRationalityWorld Modeling
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My question concerns the semantics of making future predictions such as 'the probability of X winning the election is 70%'. There are programs, e.g., Good Judgment, IARPA ACE, aimed at excelling at this kind of predictions.

In the classic mathematical interpretation of probability we can say 'the probability that one gets 8 heads in a row when flipping an unbiased coin is 1/256'. This statement can be derived mathematically from the assumption that the coin is unbiased and it can be verified empirically by performing the experiment iteratively and counting the number of successes. In the Bayesian statistics the things get slightly more fuzzy but still we can make a reasoning as follows. We model our knowledge as 'the value of the radius of Saturn in km is a random variable with normal distribution with mean 60.000 and variance 0.1'. Then we make several independent measurements, possibly burdened with some inaccuracy, and refine our prior distribution to have mean = 59.300 and variance = 0.01. This does not make sense in the previous interpretation but we can still attach some clear semantics to the sentence above by treating this random variable as a result of a measurement which is a repetitive random event. If one does not agree with such a statement, they must either question the choice of the prior distribution or the mathematical derivation.

Now suppose that two forecasters were asked in 2018 the questions below.

a) Will Donald Trump win the 2020 election?

b) Will USD/EUR exchange rate drop below 0.8 in 2020?

c) Will Sumatran orangutan become extinct by 2020?

d) Will humans land on Mars by 2040?

The first forecaster provided the following probability scores for these events: 43%, 90%, 45%, 42%. The other one gave numbers: 54%, 50%, 52%, 43%. We already know that the events (a,b,c) did not occur. First, how can we settle who has been a better forecaster so far? Secondly, their forecasts for the event (d) differ slightly. What kind of argumentation can the first forecaster make to convince the other one that 42% is the 'correct' answer? And what does this numerical value actually mean, as landing on Mars is not a repetitive random event nor it is a quantity which we can try measuring like the radius of Saturn? If one believes the 42% is a better estimation than 43%, how can it help making any choices in the future?

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4 Answers

Probability is in the mind. It's relative to the information you have.

In practical terms, you typically don't have good enough resolution to get individual percentage point precision, unless it's in a quantitative field with well understood processes.

First, how can we settle who has been a better forecaster so far?

 

The first forecaster thought it was less likely that 2 out of 3 things that didn't occur -  wouldn't. The second forecaster thought it was more likely that 2 out of 3 things that didn't occur - would. So I think that the first forecaster has got a pretty easy case on this one.

I think the rest of your questions seem to be thinking that the percentages are measuring something in the real world.  They are a measure of the predictor's confidence. A way to tell the world how seriously they think you should take their prediction.

What kind of argumentation can the first forecaster make to convince the other one that 42% is the 'correct' answer?

I don't think he can. He is technically a little less sure that humans that will land on the Mars than second forecaster. (or, if you prefer, a little more sure that they won't) And a 1% difference is functionally 0 difference in this situation.

If they had vastly different levels of confidence, they could discuss the gaps in the optimism/pessimism,  but at 1% difference....that's just personal preference 

And what does this numerical value actually mean, as landing on Mars is not a repetitive random event nor it is a quantity which we can try measuring like the radius of Saturn?

To repeat self, They are a measure of the predictor's confidence. A way to tell the world how seriously they think you should take their prediction. 

If one believes the 42% is a better estimation than 43%, how can it help making any choices in the future?

Even if you had predictors with so many predictions that you could actually take a 1% difference seriously....I still don't know when that 1% would matter much.

I find this question really interesting. I think the core of the issue is the first part:

First, how can we settle who has been a better forecaster so far? 

I think a good approach would be betting related. I believe different reasonable betting schemes are possible, which in some cases will give conflicting answers when ranking forecasters. Here's one reasonable setup:

  • Let A = probability the first forecaster, Alice, predicts for some event.
  • Let B = probability the second forecaster, Bob, assigns (suppose B > A wlog).
  • Define what's called an option: basically a promissory note to pay 1 point if the event happens, and nothing otherwise. 
  • Alice will write and sell N such options to Bob for price P each, with N and P to be determined.
  • Alice's EV is positive if P > A (she expects pay out A points/option on average).
  • Bob's EV is positive if P < B (he expects to be paid B points/option on average). 

A specific scheme can then stipulate the way to determine N and P. After that comparing forecasters, after a number of events, would just translate to comparing points.

As a simple illustration (without claiming it's great), here's one possible scheme for P and N:

  • Alice and Bob split the difference and set P = 1/2 (A + B).
  • N = 1.

One drawback of that scheme is that it doesn't punish too much a forecaster who erroneously assigns a probability of 0% or 100% to an event.

A different structure of the whole setup would involve not two forecasters betting against each other, but each forecaster betting against some "cosmic bookie". I have some ideas how to make that work too.

And what does this numerical value actually mean, as landing on Mars is not a repetitive random event nor it is a quantity which we can try measuring like the radius of Saturn?

I don't see how we could assign some canonical meaning to this numerical value. For every forecaster there can always be a better one in principle, who takes into account more information, does more precise calculations, and happens to have better priors (until we reach the level of Laplace's demon, at which point probabilities might just degenerate into 0 or 1). 

If that's true then such a numerical value would seem to just be a subjective property specific to a given forecaster, it's whatever that forecaster assigns to the event and uses to estimate how many points (or whatever other metrics she cares about) she will have in the future.

I like the idea of defining a betting game 'forecasters vs cosmic bookie'. Then saying 'the probability that people will land on Mars by 2040 is 42%' translates into semantics 'I am willing to buy an option for Y<42 cents that would be worth $1 if we land on Mars by 2040 or $0 otherwise'.

To compare several forecasters we can consider a game in which each player is offered to buy some options of this kind. Suppose that for each x in {1, \dots, 99} each player is allowed to buy one option for x cents. If one believes that the probability of an event is 30... (read more)

1Dmitriy Vasilyuk4dThat's perfect, I was thinking along the same lines, with a range of options available for sale, but didn't do the math and so didn't realize the necessity of dual options. And you are right of course, there's still quite a bit of arbitrariness left. In addition to varying the distribution of options there is, for example, freedom to choose what metric the forecasters are supposed to optimize. It doesn't have to be EV, in fact in real life it rarely should be EV, because that ignores risk aversion. Instead we could optimize some utility function that becomes flatter for larger gains, for example we could use Kelly betting.

This is a topic I have found myself thinking a lot lately as well. I have found it useful to decompose a non-repeatable event (will X will the elections?) in two parts: one consisting of a combination of repeatable events and a "specific residual". 

Let's start with a coin toss. It is, in a very Heraclitean sense, a one time event which we can decompose as a throw of an ideal coin plus a tiny, negligible "specific residual". 

Let's go back to the problems in question. You could decompose them into combinations of events for which we have historical frequencies (how many times an incumbent politician...? how many times an election during an economic crises...? how do the probabilities of wining an election relate to the polls three months before...=), plus conceivably larger "specific residual" given the particularities of the question.

This approach is more useful than vague considerations on "probability is in your head" or "it just relates to information". It is actually how predictors work: decomposing the question into subquestions on which frequency considerations are easier to elicit, recombining them, and adding an extra layer on uncertainty on top.