It's actually P(data|null-hypothesis).
There are both Bayesian and Frequentist approaches in statistics!
They represent very different methods to statistics, but they are also quite similar. My apologies, I couldn't find one link that gave a good description of Bayesian vs. Frequentist. Here are a couple links to get started:
http://jakevdp.github.io/blog/2014/03/11/frequentism-and-bay...
If anyone comes across a link that describes Bayesian and Frequentist clearly, please share it if you don't mind.
Either way I disagree. Both approaches have pros and cons depending on the type of analysis.
At the very least, just the presence of competing approaches in the field has pushed statisticians to have more rigour and do way more double checking than they might have, out of fear the other side actively looking to poke holes. It's easy to lie with statistics and the only people who can call statisticians out on their BS is other statisticians...Need more of this.
Nope.
> Both approaches have pros and cons
What are the pros of the frequentist approach?
Now. You might read this as me saying that Bayesian statistics is all made up, and that's not what I'm saying. I'm saying that if you go Bayesian all the way, and your result is even remotely controversial, someone could easily challenge you by saying "this result would have been different with different priors, and why won't they just come out and say p < 0.01? What are they hiding?"
When you do frequentist statistics, the equivalent of priors were, instead, made up for you by long scientific tradition. They're not very good priors, as the relevant xkcd illustrates. But at least it's not you making them up.
In a more complex model, there's also the thing where you can't calculate the result exactly, you have to approximate it with Markov Chain Monte Carlo or something, leading to another way to doubt your results (did the MCMC converge correctly?)
So instead you do frequentist statistics. You use your favorite stats package and it spits out a nice comforting p-value that will satisfy the reviewers. You have tons of guidance about how to do things. It's not a great thing, but it is a definite advantage of using frequentist statistics.
How about this: find me a published scientific paper with only Bayesian results in it, no frequentist statistics at all. Argue for its superiority all you want, but I don't think anyone does it. It would be seen as a stunt.
https://www.amazon.com/Proving-History-Bayess-Theorem-Histor...
Trivial¹, surely?
> [B]ut I don't think anyone does it. It would be seen as a stunt.
What did you mean by "scientific"?
You'd have to hunt for a narrow reading s.t. the above holds — extant counterexamples aren't limited to any one branch.
Check it out² for yourself.
Further: cogsci's bayesian adoption is rapidly accelerating.
The replication crisis is brutalizing huge swathes of psych:
• loss of confidence in NHST is becoming near-total for many
• journals are purging in turn — e.g. BASP's p-value ban³⁴
• others are overhauling stats-in-psych entirely
J Math Psych alone has two recent special issues⁵⁶ on this.
I mean:
> no frequentist stats at all
isn't even a strawman lately, let alone an absurdity.
In some fields, it's a battle-cry.
All that being said… re:
> Argue for its superiority all you want
I wouldn't even go that far. Don't give em that.
Probability interpretation fundies
• are all wrong,
• narrow minds and waste lifespans with the cultism, and
• should at least learn of the other interpretations.
Pitching probability as a 1v1 isn't merely wrong-prime⁷ — it's doublepluswrong″⁸.
It's pseudofundamentalism: fundies uphold foundations.
Flamewars predicated on ignorance of the same gotta go, no matter how fashionable they may be.
____________________________________________________________
[1] http://enwp.org/bayesian_game
[2] http://google.com/scholar?q=fully+bayesian
[3] http://doi.org/4z8 | BASP 37
[4] http://doi.org/34p | Nature re: BASP
[5] http://doi.org/btqx | J Math Psych 72
[6] http://doi.org/btqz | J Math Psych 74
Here is a simple example: lets say you flip a coin 10 times and get 8 heads, what is the probability that the coin is not fair? In frequentist statistics you only have to calculate the likelihood of the results for a null hypothesis, and then you use a p-value. While this approach is flawed, at least you can quickly do the calculation and get an approximate answer. In Bayesian statistics you have to specify the prior distribution, and calculate the likelihood of your results under every possible hypothesis. Correctly specifying this prior distribution and calculating the results is quite challenging - especially if you want to use a realistic prior (not just uniform). This is a pretty simple example, you can imagine how much more challenging this becomes in real-world problems. On the other hand, it is true that the frequentist approach doesn't really answer the question asked, so it is misleading (especially if you choose a p-value that isn't specific to the problem). If you choose p-values based on prior knowledge, than the differences between frequentist and bayesian are less extreme.
This is often a feature, forcing you to actually look at the complexity head on before you sweep it under a rug.
There is no Zuul.
Statistics can tell you whether a model is consistent with the data. But you need to deduce the null hypothesis from your model rather than use the default "no difference" (of course, sometimes no difference is deduced from a real model, but not often, in that case: great!).
In fact, that is the proper use of statistics. I would guess >99.99% of current usage is incorrect (ie pseudoscience) and amounts to a waste of time at best. The usual usage turns scientific reasoning on its head, and has lead to a (literally for most people) unbelievable amount of trouble.
This was pointed out most aptly by Paul Meehl long, long ago: http://www.fisme.science.uu.nl/staff/christianb/downloads/me...
Yes, that's true, but it badly misses the point. The power of statistics is to tell you when a model (the null hypothesis) is (most likely) inconsistent with the data so that you can confidently rule it out. Any finite data set is consistent with an infinite number of models, so knowing that a model and the data are consistent tells you absolutely nothing about whether or not that model has any relationship with reality (which, at the risk of stating the obvious, is what science actually cares about). This is the reason that rejecting the null hypothesis is considered a positive result.
In practical terms, we're not interested in true models, but useful ones, so the description of a model's consistency with observed data is often the more useful metric in practice than rejecting nulls :/ especially in applications where you can't set up repeated experiments.
OK, I realise its more nuanced than that too, but given how many papers and practitioners seem incapable of understanding that evidence against the null it's not explicit evidence for an arbitrary alternative, practically and consequentially I don't think that's how we should be working...
No, that's not true, because experiments are not done in a (figurative) vacuum. They are done in the context of an explanatory theory that has already gone through a vigorous filter and shown to be consistent with the all prior experimental data and has better explanatory power than all of its competitors. It is only when more than one theory survives this filter that an experiment is done, and the experiment is designed specifically to distinguish between the surviving theories.
So while it is true that an experiment allows you to eliminate an infinite number of theories, it's irrelevant, because by the time the experiment is done nearly all of those theories have already been eliminated anyway.
If you know whether your model is consistent with the data, you know whether it is inconsistent... I think you are talking about some other issue than I am.
The point about deducing the null hypothesis from your explanatory model is that the null hypothesis is precise. In that case you will get a strong test of the model, and it will get stronger as more data gets collected. Using a default null and and vague alternative is the exact opposite. (check the Meehl 1967 paper I linked earlier in this thread).
No, that's not true. And in fact I got it wrong earlier when I agreed with you that statistics can tell you when data is consistent with a model. They can't. At best they can tell you whether the data are not inconsistent. That sounds like the same thing, but it isn't. It's like the distinction between "not guilty" and "actually innocent." At best (or at worst depending on how you look at it) a statistical test can tell you, "This theory cannot be confidently ruled out on the basis of that data under the following assumptions..."
> I think you are talking about some other issue than I am.
That is quite possible.
[UPDATE:] BTW, I read the Meehl paper, and I completely agree with what he says. So you and I may be in "violent agreement" here.
The null hypothesis never has explanatory power. The null hypothesis is always a statement of the form, "The explanatory hypothesis under test is wrong for some unknown reason." This is why rejecting the null hypothesis, i.e. showing that the data are (with high probability) inconsistent with the null hypothesis, is considered a positive result.