Puzzled By Statistics

[Dancing David]

No one expects the expectation! Our chief weapon is...

no?

Well, it was worth a shot.

Our main weapons is the mean! And a standard deviation.

Our two main weapons...

Is that a comfy chair?

 

[technoextreme]

perhaps technoextreme meant the expectation (not the mean), the expectation need not correspond to any realisation of the process, and may be physically ridiculous. frightening then how often it is plotted in forecasting.

one need never expect the expectation.

The expectation is the mean.

You must specify a theory (or hypothesis, same thing). Then you can use the data to compare it to another theory (via Bayes). If your theory is that the coin is fair, all sequences are equally probable. If your theory is that the coin is weighted, some sequences are much more probable than others. Therefore, given some data, you can distinguish between the two theories.

And this is why I hate the question. It's so trivially simplistic that you don't need bayes to solve it at all.

 

[technoextreme]

But the math doesn't tell you which theories to consider - that's your job.

That's another reason why Dr. A's question is dumb because in this case the math tells you which theory to consider flat out with 100% no ******** involved certainty. Given an sufficient amount of independent events that are statistically uniform you will get a bell curve. I don't know why you would want to do it with sequences but that makes your life a heck of a lot harder.

We already have the data. It happened, however improbable we may think it was. Our job now is just to evaluate various competing hypotheses, based on it (and on our previous opinion of those hypotheses). Yes?

You can't make any opinions off of one test. As statistically improbable it was it's not statistically impossible.

The OP's question, at least in part, was why one should focus on the mean rather than some other statistic. There is no fundamental answer, other than that it's likely to be the sensible thing to do in the case of a series of coin flips.

Because it's the only statistic that actually gives you an average of what the hell is going on within the pdf. The second moment gives you the variance which is how much the probability density function is smooshed in and out. The third moment gives you skewness which is how much your probability density function is lopsided. The forth moment gives you kurtosis which is whether or not your probability density function is short and squat, or skinny or tall. It's not like there aren't other statistics but in this case it's the only one that is useful.

 

[sol invictus]

And this is why I hate the question. It's so trivially simplistic that you don't need bayes to solve it at all.

Not at all. In fact, at its root the question is so hard its undecidable in the sense of Godel, as I mentioned above.

That's another reason why Dr. A's question is dumb because in this case the math tells you which theory to consider flat out with 100% no ******** involved certainty.

No it doesn't, actually.

Given an sufficient amount of independent events that are statistically uniform you will get a bell curve. I don't know why you would want to do it with sequences but that makes your life a heck of a lot harder.

You seem to be thinking of a different problem.

You can't make any opinions off of one test. As statistically improbable it was it's not statistically impossible.

False, of course. If that were true we could never have any opinions at all, since the data accessible to us is obviously finite. In fact, you can stick any data you want into Bayes' theorem. In Dr. A's example of 990 heads out of 1000 flips you will get a very powerful constraint on the possible theories.

Because it's the only statistic that actually gives you an average of what the hell is going on within the pdf. The second moment gives you the variance which is how much the probability density function is smooshed in and out. The third moment gives you skewness which is how much your probability density function is lopsided. The forth moment gives you kurtosis which is whether or not your probability density function is short and squat, or skinny or tall. It's not like there aren't other statistics but in this case it's the only one that is useful.

That's true only if you assume the coin flips are independent. As has been discussed, a sequence of 500 heads followed by 500 tails indicates something very different than a more typical random sequence with the same totals.

 

[lenny]

No one expects the expectation! Our chief weapon is...

?the spanish inquisition?

 

[lenny]

model inadequacy

I was being somewhat informal. I figured that this would get the idea across better than just saying "use Bayes's theorem", which is the more formal version.

i agree.

I don't understand what you're trying to say here.

i was trying to open the discussion to include model inadequacy where the bayesian way, and indeed the probability calculus, are arguably uninformative/irrelevant.