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I understand the concept of the NULL hypothesis and how to reject it, however I would like to be able to explain why this is a better approach than trying to prove the alternative hypothesis true.

What would it entail do to "prove" the alternative hypothesis true? I put "prove" in quotations to exempliy that we cannot prove it with certainty, but perhaps we can assign some type of probability of it being true, similar to how we assign a probability of a result given that the NULL hypothesis is true. Can you create a completely fictitous example (for example where the hypothesis space is limited to 3 discrete values), which shows how you would try to prove the alternative hypothesis true? And why would this approach be worse than the NULL approach? Is it because we can only show something is true, by showing that every other possibility is false?

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    $\begingroup$ If you want to assign a probability to an alternative hypothesis being true, you need to use Bayes' Theorem. $\endgroup$
    – fblundun
    Commented Apr 2, 2022 at 9:59
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    $\begingroup$ A partial answer to the question: it is easier to control the size of the test than the power. $\endgroup$
    – Richard Hardy
    Commented Apr 3, 2022 at 12:52

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Frequentist statistics tell you the probability of obtaining a given data set for a given model. If the data are unlikely given a model, people interpret that as the model being incorrect. If the data are likely given a model, people interpret that as the model being satisfactory.

Bayesian statistics formalize this using a prior, allowing one to say how probable a model is given data.

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    $\begingroup$ You could say that more than formalizing it, Bayes allows it to happen. Without Bayesian thinking, quantities such as a p-value (P(data more impressive than mine if the null hypothesis is true)) are at most indirect measures of evidence. Frequentism deals with the probability of making an assertion and Bayes deals with the probability that a made assertion is true. hbiostat.org/bbr/md/alpha.html $\endgroup$
    – Frank Harrell
    Commented Apr 2, 2022 at 14:29

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