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>P=0.05 is best interpreted as "There is a 95% chance that this data isn't pure noise".

No, that's not what P=0.05 means. This misconception is exactly why p values are so problematic. The question we want to answer is unanswerable, so instead we subtly substitute a different question in its place and don't even realize we've swapped concepts.

When I say that "There is a 95% chance that this data isn't pure noise" I'm being slightly indirect. More directly it would be something like "There is 5% chance that pure noise generator you would give you this result" but since I want to reason about the actual world that the experiment is trying to study, I think it is legitimate to do the Bayesian flip and infer the more useful first statement that the world is likely too noisy to detect an effect using this experiment.

I think I understand what you're trying to get at, but I don't think describing a p-value in terms of "pure noise" is quite right. The null hypothesis may well carry with it all sorts of specific assumptions: independence, finite variance, maybe even specific probability distributions (and not necessarily the normal distribution). Typically somewhere in there we define a θ parameter and then include something like θ = 0 that provides grounds for possible rejection. We think of "θ = 0" as the null hypothesis, but all the other assumptions are part of the null hypothesis as well; the statement "θ = 0" does not make sense without them.

If you see clouds how likely is rain? If you see rain how likely is it there are clouds? Are the answers to these at all the same? No.

There is absolutely nothing valid about your "flip" (called transposing the conditional) and the damage done to the human species by people doing the "flip" is too huge for most to comprehend. Sorry, but I really cannot overstate the problems with this "flip".

Perhaps people who get paid by grants should get three chances regarding this flip. If they strike out they are banned from ever being funded by NIH, etc for life. The result (either 99+% of current researchers would be culled, or p-values would become appropriately rare) would do more to advance science than any other idea I have seen.

If you want your science to be about the real world and not about some hypothetical scenarios that don't exist you have no choice but to do the transposition. Experiments are useless otherwise. There is a reason humans naturally and instinctively do it when they talk about experiments. This is how you describe things in real world human language.

Sure there are debates about priors. These are unavoidable. The hidden assumption in my example is a 50/50 prior on too noisy, not too noisy hypotheses. IMO a reasonable uniform (maximum entropy) prior to start a discussion from. Of course, one should be more precise when doing this type of analysis in a paper. The reason a good prior is hard to define here is not because you shouldn't do the flip, it is because the question that the t-test is trying to answer doesn't make much sense.

>"If you want your science to be about the real world and not about some hypothetical scenarios that don't exist you have no choice but to do the transposition."

How can saying false things make your science "about the the real world"? The only thing I can think of is that by "real world" you are referring to using the publication of massive amounts of incorrect/questionable information as a metric for success. Yes, that is indeed the sham going on.

>"There is a reason humans naturally and instinctively do it when they talk about experiments. This is how you describe things in real world human language."

I have no problem talking about data without transposing the conditional, if you do that is some issue with your training.

>"Sure there are debates about priors."

The error you advocate (of transposing the conditional) has nothing to do with priors.

There is no point in talking about the data if it doesn't tell you useful things about the population or type of thing you are studying, if it's not for building a predictive model of the subjects. This is the entire reason for research.

Also P(A|B) = P(B|A) if the prior probabilities are equal, often a reasonable prior.

>"There is no point in talking about the data if it doesn't tell you useful things about the population or type of thing you are studying"

Ok, but I fail to see the relevance of this to anything.

>"Also P(A|B) = P(B|A) if the prior probabilities are equal, often a reasonable prior."

Here, A = Hypothesis and B = data (or vice versa). You are claiming that the "probability of seeing your data whether or not the hypothesis is correct", is often near the "probability the hypothesis is correct independent of your data"?

Based on what? What principle leads you to think these two probabilities should be near each other? BTW, if you actually have one this is like the holy grail of statistics.

> You are claiming that the "probability of seeing your data whether or not the hypothesis is correct", is often near the "probability the hypothesis is correct independent of your data"?

Shouldn't that be "the probability of seeing the data, _given_ that the hypothesis is correct" and "the probability of the hypothesis being correct, _given_ that the data has been observed"?

No, the other poster is referring to Bayes' rule P(A|B) = P(A)*P(B|A)/P(B).

The claim is that it is reasonable to assume P(A) ~ P(B), so that P(A|B) ~ P(B|A). I am referring to P(A) and P(B) in the equation.

Ah, OK, got it.

Its hard to tell over the internet, but if you are being sarcastic please let me know. I have no problem with clarifying further.

No sarcasm! I believe I understand what you were saying now.

vaguely P(observations|pure noise generator) ~ P(pure noise generator | observations)

I agree there are debates surrounding priors, especially when you get into higher order, meta or purely hypothetical territory (there is no actual noise generator here) like this, but I think these are unavoidable.

>"vaguely P(observations|pure noise generator) ~ P(pure noise generator | observations)"

For this to be true, P(observations) ~ P(pure noise generator) as you said earlier. Anyway, you can look at it from either perspective, doesn't really matter.

On what basis do you claim this is reasonable to expect? I suspect you have none whatsoever... in fact simple thought experiments will lead us to the opposite conclusion, and there is plenty of data telling us the same.

> P=0.05 is best interpreted as "There is a 95% chance that this data isn't pure noise".

This is incorrect. A p-value is the probability of observing a value of the test statistic that is at least as extreme as the reported value, assuming the null hypothesis is true.

Which sounds like a fancy way of saying that "there is a 5% chance of the data being noise (while the null hypothesis is true)".

"there is a 5% chance of the data being noise" is a statement on the probability of whether the phenomenon is not real given the data: p(it's just noise | data). A p value is a statement on the probability we'd see this data given that the phenomenon isn't real: p(data | it's just noise).

Now, the exact values of a given data set are vanishingly unlikely, so we actually ask about the probability we'd say this data or even more extreme data given that we're just looking at noise. That's a typical p value.

That's what I meant: P(data | H0). So there is a 5% probability for the data to arise by chance (being noise) given H0 is true.

The whole terminology with values being "extreme" is just unintuitive and unwieldy, IMO.

"There is a 95% chance that this data isn't pure noise". That's not what P=0.05 actually means.

It's much closer to, this result rises above the noise floor. Suggesting but not implying a possible link between X and Y, fell free to investigate.