Why does probability work?

Discuss the language of the universe.

Moderators: Calilasseia, ADParker

Why does probability work?

#1  Postby BWE » Nov 18, 2018 7:03 am

There is no reason it should.

Eta: is this a well known issue in statistics or philosophy of mathematics?
User avatar
BWE
THREAD STARTER
 
Posts: 2498

Print view this post

Ads by Google


Re: Why does probability work?

#2  Postby Thommo » Nov 18, 2018 7:45 am

It's not clear what your question means.

If it means "why does a dice land with any given face upwards one sixth of the time?" or something akin to that, then the answer is that it might, or it might not. Probably it doesn't.

Statistical modelling, like any mathematical modelling, scientific modelling or logical modelling depends on making assumptions, and fitting those assumptions to real physical problems. The better the assumptions fit, the better the model fits and the more reliable its predictions are. The benefit of assumptions is that they allow you to use sophisticated derived results.

Perhaps on a broader level we could say that the answer to "why does probability work?" is essentially the same as "why does Euclidean geometry work?" or "why does Newtonian gravitation work?". If the assumptions are good enough for the situation, then it works, if they aren't, then it doesn't, which in those latter examples would be the case for sufficiently large scales.
User avatar
Thommo
 
Posts: 27174

Print view this post

Re: Why does probability work?

#3  Postby Evolving » Nov 18, 2018 8:34 am

I was going to say something similar -

- why do Newton's laws of motion operate? and the laws of thermodynamics: what's that about? -

but Thommo has given a more complete answer.

Maybe other, alternative laws might make sense too (and there used to be other ideas about the laws governing motion, for instance); but the fact is (or seems to be) that these laws, including the laws of probability, actually operate in the reality that we are observing.
How extremely stupid not to have thought of that - T.H. Huxley
User avatar
Evolving
 
Name: Serafina Pekkala
Posts: 11993
Female

Country: Luxembourg
Luxembourg (lu)
Print view this post

Re: Why does probability work?

#4  Postby Thommo » Nov 18, 2018 8:48 am

If you want a philosophical treatment of interpretations of probability (which doesn't seem to really be the question, but might be tangentially relevant or illuminating), you could try something like this:
http://www.siue.edu/~evailat/interp-prob.htm
The main interpretations of probability are best divided into into two groups:

  • Epistemological interpretations, according to which probability is primarily related to human knowledge or belief.
  • Objective interpretations, according to which probability is about a feature of reality independent of human knowledge or belief. Sometimes reality is taken to be the physical world; at times it is taken to include a sort of Platonic realm of mathematical and logical entities.


Or this:
https://plato.stanford.edu/entries/dutch-book/
The conclusion of the DBA is that the degrees of belief, or credences, that an agent attaches to the members of a set X of sentences, statements, or propositions, should satisfy the axioms of probability.

(be warned, there's an economist's "should" in that sentence)

Or this:
https://plato.stanford.edu/entries/prob ... interpret/
‘Interpreting probability’ is a commonly used but misleading characterization of a worthy enterprise. The so-called ‘interpretations of probability’ would be better called ‘analyses of various concepts of probability’, and ‘interpreting probability’ is the task of providing such analyses. Or perhaps better still, if our goal is to transform inexact concepts of probability familiar to ordinary folk into exact ones suitable for philosophical and scientific theorizing, then the task may be one of ‘explication’ in the sense of Carnap (1950). Normally, we speak of interpreting a formal system, that is, attaching familiar meanings to the primitive terms in its axioms and theorems, usually with an eye to turning them into true statements about some subject of interest. However, there is no single formal system that is ‘probability’, but rather a host of such systems. To be sure, Kolmogorov's axiomatization, which we will present shortly, has achieved the status of orthodoxy, and it is typically what philosophers have in mind when they think of ‘probability theory’. Nevertheless, several of the leading ‘interpretations of probability’ fail to satisfy all of Kolmogorov's axioms, yet they have not lost their title for that.


I can't say that (aside from some aspects of the Dutch book arguments) they do a lot for me, but maybe that's what you want.
User avatar
Thommo
 
Posts: 27174

Print view this post

Re: Why does probability work?

#5  Postby newolder » Nov 18, 2018 8:52 am

I had some confidence that this question would be asked before the year ended but there was no surety and I didn't place a bet.
I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops. - Stephen J. Gould
User avatar
newolder
 
Name: Albert Ross
Posts: 7310
Age: 1
Male

Country: Feudal Estate number 9
Print view this post

Re: Why does probability work?

#6  Postby surreptitious57 » Nov 18, 2018 9:06 am

Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other
A MIND IS LIKE A PARACHUTE : IT DOES NOT WORK UNLESS IT IS OPEN
surreptitious57
 
Posts: 10195

Print view this post

Re: Why does probability work?

#7  Postby Thommo » Nov 18, 2018 9:13 am

Just to be clear, probability theory does not say a fair coin tossed 100 times will land 50/50 heads and tails. Consequently if a coin lands 51/49 it does not entail the conclusion that one side is marginally heavier than the other, that would be an example of The Gambler's Fallacy.
User avatar
Thommo
 
Posts: 27174

Print view this post

Ads by Google


Re: Why does probability work?

#8  Postby Evolving » Nov 18, 2018 10:44 am

newolder wrote:I had some confidence that this question would be asked before the year ended but there was no surety and I didn't place a bet.


Can you name a pointless answer?
How extremely stupid not to have thought of that - T.H. Huxley
User avatar
Evolving
 
Name: Serafina Pekkala
Posts: 11993
Female

Country: Luxembourg
Luxembourg (lu)
Print view this post

Re: Why does probability work?

#9  Postby newolder » Nov 18, 2018 10:56 am

Image
I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops. - Stephen J. Gould
User avatar
newolder
 
Name: Albert Ross
Posts: 7310
Age: 1
Male

Country: Feudal Estate number 9
Print view this post

Re: Why does probability work?

#10  Postby BWE » Nov 18, 2018 6:39 pm

surreptitious57 wrote:Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

But these are all tautological answers. My question is why does the coin flip vary around a bell curve? Why wouldn't it be always heads for no reason at all?
User avatar
BWE
THREAD STARTER
 
Posts: 2498

Print view this post

Re: Why does probability work?

#11  Postby BWE » Nov 18, 2018 6:42 pm

surreptitious57 wrote:Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

The first part is tautological, probability works because it works. But the second part gets at my question kind of. If it was 51-49, that wouldn't be statistically significant but 80-20 would definitely indicate that there was a missing variable. Why?
User avatar
BWE
THREAD STARTER
 
Posts: 2498

Print view this post

Re: Why does probability work?

#12  Postby Evolving » Nov 18, 2018 6:45 pm

Don't know if this helps, but just to narrow down the enquiry: the question and its answer or answers relate to science, not to mathematics. Mathematics just defines the functions that describe the observed behaviour.
How extremely stupid not to have thought of that - T.H. Huxley
User avatar
Evolving
 
Name: Serafina Pekkala
Posts: 11993
Female

Country: Luxembourg
Luxembourg (lu)
Print view this post

Re: Why does probability work?

#13  Postby Evolving » Nov 18, 2018 6:49 pm

So it's not a mathematical law that says that probability works; it's a scientific observation. It seems to make sense (Why a bell curve? Because when a coin flips, there is no reason why it should come up heads rather than tails, so we can expect it to do both about equally often), but that's an intuitive, emotional, and - as you say - tautological argument equivalent to saying that Newton's laws of motion apply because they're common sense.
How extremely stupid not to have thought of that - T.H. Huxley
User avatar
Evolving
 
Name: Serafina Pekkala
Posts: 11993
Female

Country: Luxembourg
Luxembourg (lu)
Print view this post

Re: Why does probability work?

#14  Postby laklak » Nov 18, 2018 7:04 pm

It only works a percentage of the time.
A man who carries a cat by the tail learns something he can learn in no other way. - Mark Twain
The sky is falling! The sky is falling! - Chicken Little
I never go without my dinner. No one ever does, except vegetarians and people like that - Oscar Wilde
User avatar
laklak
RS Donator
 
Name: Florida Man
Posts: 20878
Age: 66
Male

Country: The Great Satan
Swaziland (sz)
Print view this post

Re: Why does probability work?

#15  Postby BWE » Nov 18, 2018 7:10 pm

Heh. The jokes almost write themselves. But yeah. it's a related issue to the unreasonable effectiveness of mathematics in the natural sciences.
User avatar
BWE
THREAD STARTER
 
Posts: 2498

Print view this post

Ads by Google


Re: Why does probability work?

#16  Postby scott1328 » Nov 18, 2018 7:17 pm

If probability theory didn’t work, we wouldn’t use it.

The OP’s question seems to be Hume’s problem of Induction rephrased.
User avatar
scott1328
 
Name: Some call me... Tim
Posts: 8695
Male

United States (us)
Print view this post

Re: Why does probability work?

#17  Postby Thommo » Nov 18, 2018 7:24 pm

BWE wrote:
surreptitious57 wrote:Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

But these are all tautological answers. My question is why does the coin flip vary around a bell curve? Why wouldn't it be always heads for no reason at all?


Coin flips are not distributed by a bell curve (AKA Normal distribution or Gaussian distribution), they are binomial distributions.

There's an important theorem in mathematics that governs the distribution of sample means from a wide range of arbitrary distributions, known as the https://en.wikipedia.org/wiki/Central_limit_theorem, which says that for large sample sizes the distribution of sample means is approximately normal regardless of the shape of the original distributions.

In terms of the question "Why wouldn't it be always heads for no reason at all?" it's poorly phrased. A series of coin flips of a fair coin of arbitrary length might be all heads, probability theory does not forbid this. And similarly a coin might be biased (such as by having two heads) such that it only ever flips heads. However in this latter situation the assumption that the coin is fair does not hold, and a model is only as good as its assumptions.
Last edited by Thommo on Nov 18, 2018 7:26 pm, edited 1 time in total.
User avatar
Thommo
 
Posts: 27174

Print view this post

Re: Why does probability work?

#18  Postby BWE » Nov 18, 2018 7:26 pm

scott1328 wrote:If probability theory didn’t work, we wouldn’t use it.

The OP’s question seems to be Hume’s problem of Induction rephrased.

Maybe. But it does seem like there's a deeper issue going on. There's the brute fact aspect involving observation but there's some kind of modeling mystery in there too. At least for me.
User avatar
BWE
THREAD STARTER
 
Posts: 2498

Print view this post

Re: Why does probability work?

#19  Postby Evolving » Nov 18, 2018 7:27 pm

On the bell curve, I interpreted the statement as referring to a large number of experiments, each consisting of a certain number of coin flips: the results of the experiments would cluster around 50-50 in the shape of a normal distribution.
How extremely stupid not to have thought of that - T.H. Huxley
User avatar
Evolving
 
Name: Serafina Pekkala
Posts: 11993
Female

Country: Luxembourg
Luxembourg (lu)
Print view this post

Re: Why does probability work?

#20  Postby BWE » Nov 18, 2018 7:28 pm

Thommo wrote:
BWE wrote:
surreptitious57 wrote:Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

But these are all tautological answers. My question is why does the coin flip vary around a bell curve? Why wouldn't it be always heads for no reason at all?


Coin flips are not distributed by a bell curve (AKA Normal distribution or Gaussian distribution), they are binomial distributions.

There's an important theorem in mathematics that governs the distribution of sample means from a wide range of arbitrary distributions, known as the https://en.wikipedia.org/wiki/Central_limit_theorem, which says that for large sample sizes the distribution of sample means is approximately normal regardless of the shape of the original distributions.


I think you understood my point though.
User avatar
BWE
THREAD STARTER
 
Posts: 2498

Print view this post

Next

Return to Mathematics

Who is online

Users viewing this topic: No registered users and 1 guest

cron