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Comment by Arnehuang | Hacker News Reader

I think of it like this:

Suppose I want to make a decision about whether to hedge for a market crash right now. Statistics can tell me the likelihood of a crash, and how bad. But if the market crashes, and very badly, how might that affect my life? To make a good decision I would need to think of all the things that come with a market crash (job loss, savings loss). This is not statistics.

I could again use statistics to say what is the chance I lose my job given a market crash (say 70%). But then I would need to estimate the impact on my life should I lose my job (Stress, etc). This is not statistics. But it should very well factor into my ability to do back of the napkin math on whether I should hedge or not.

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If your decision substantially involves or derives from making an estimate about a population based on a sample, it is statistics. "Making decisions under uncertainty" is well-studied in statistical literature, just like "quantifying uncertainty" is well-studied. It sounds like you think the latter is "actual statistics", but these things are both statistics.

In particular:

> But if the market crashes, and very badly, how might that affect my life? To make a good decision I would need to think of all the things that come with a market crash (job loss, savings loss). This is not statistics.

This is all statistics, not just the part where you're forecasting likelihood of the market crashing. The reason is because making decisions about the future under the constraints of uncertainty implicitly involves a forecast. When you decide how to diversify your personal investment portfolio, how much to allocate to your Roth versus traditional IRA or 401k, etc, you are making forecasts about which allocation will provide you with a more favorable outcome.

Stated more concisely: there is no rational reason to use statistics for forecasting market events but not for deciding what to do in the event specific market events occur.

This is exactly statistics. This is an expectation of a utility function with respect to some distribution.

> Statistics can tell me the likelihood of a crash

Statistics cannot tell you any such thing.

Do you mean to say that nothing can tell you such a thing?

What is a likelihood, but a statistic?

If there is any method to determine a statistic, it seems reasonable to me to say that that method involved statistics.

(Now, of course, except for possibly where quantum randomness is relevant, which might be quite often, I'm fairly confident that the only probabilities are subjective or relative to some set of assumptions, or something along those lines, because the future "already exists". But, given some fixed priors and some fixed evidence, there should in principle be a well defined probability of such a crash. So, insofar as peoples priors match up, there should, in principle, be a common well defined probability given "the information which is publicly available", or also, given whatever other set of evidence.)

Of course, that doesn't mean it is computationally tractable to compute such a probability.

> But, given some fixed priors and some fixed evidence, there should in principle be a well defined probability of such a crash.

:-)

How do you test this model?

It is easy to find things that fit one of the previous crashes.

Given that there is only one realization of history, the data we have is consistent with any model that puts a non-zero probability on a crash.

Well, what I gave isn't exactly a model of the market, so much as "a description of having a model of the world".

So, I'm not sure what you mean by "test this model".

You can refine your model-of/beliefs-about the world, by continuing to look at the world and make observations.

And obviously your beliefs should include a non-zero probability of a crash. That follows from non-dogmatism/Cromwell's rule.

And yeah, there is only one, (or, either that, or at least we can only observe one, which is practically the same thing) "realization of history". This doesn't produce any difficulty, because probability isn't defined by the proportion of trials in which the event occurred.

Probability is about degree of belief (or, belief and/or caring).

edit: I suppose you can also evaluate how calibrated your beliefs have been, which is kind of like testing a model.

> Probability is about degree of belief (or, belief and/or caring).

Not at all.

Probability is a countably additive, normalized measure over a sigma algebra of sets.

> This doesn't produce any difficulty, because probability isn't defined by the proportion of trials in which the event occurred.

You misunderstand the point.

Let's say you provide me a distribution of crash probabilities for every trading day for the next three months.

We all ought to know that P(event) = 0 does not mean event is impossible., Therefore, P(event) = 1 does not mean "not event" is not impossible.

What would allow one to state that your model is consistent/not consistent with the one observed history of events over the three months, regardless of whether there is a crash or not?

You have to come up with this criterion before observing the history.

Ok yes, that’s the definition of a probability measure. But I was talking about the concept of probability, in the world, contrasting with the “objectively defined via frequency in related trials”, which is something people sometimes claim. I misunderstood and thought that was the claim you were making.

Ok.

I would think that, if we have a continuous distribution, then the score should be the probability density of what is observed?

If you say beforehand “I think x will happen”, and I respond “I assign probability 1 that x will not happen”, and then x happens, then I’ve really messed up big time. I’ve messed up to a degree that should never happen.

(And, only countably many events can be described using finite descriptions, and a positive probability could, in principle, be assigned to each, while having the total probability still be 1, so that nothing that can possibly be specified happens while being assigned a probability of 0. Though this isn’t really computable..)

As a more practical thing, if I assign probability 0 to an event which you could describe in a few sentences in under 5 lines (regardless of whether you actually have described it), and it happens, then I’ve really messed up quite terribly, and this should never happen (outside of just, because I made an arithmetic error or something.)

> As a more practical thing, if I assign probability 0 to an event which you could describe in a few sentences in under 5 lines (regardless of whether you actually have described it), and it happens, then I’ve really messed up quite terribly, and this should never happen (outside of just, because I made an arithmetic error or something.)

I think this conversation has reached an impasse.

https://en.wikipedia.org/wiki/Cantor_set

I'm familiar with the cantor set, and I know it has 0 measure. Just because you can succinctly describe the cantor set, which has 0 measure, doesn't mean I've messed up. If I assign a uniform distribution over [0,1] to some number outcome in the world, and an element in the cantor set is the result, then I've messed up. But, when we measure numbers in the world, we don't measure specific real numbers, as all our measurements have some amount of error. So, that can't happen. We can measure that the result is in some interval, and that this interval contains some element of the cantor set, but the probability of what we observed, is not something that I assigned 0 probability to. Like, heck, every interval will have a rational number in it, and the rational numbers also have measure 0.

"the measured value is in the cantor set" isn't a thing that we can observe to have happened.

("the value, when rounded to the finite amount of precision that our measurement has, is in the cantor set" is something that would have positive probability, under the uniform distribution over the interval, and therefore something I shouldn't assign a probability of 0.)

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