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The standard procedure for the hypothesis testing is that we state the null hypothesis and the alternative, and then we see if we can reject the null hypothesis.

Then, if we can reject the null hypothesis, we accept the alternative.

In my class, the professor explained the similarity between this argument and the proof by contradiction.

Now I got the question: in mathematics, we can prove theorems directly, not just by contradiction.

So, is there a way to “directly” test the validity of a hypothesis?

I am pretty sure this has been asked elsewhere, but I could not find the duplicate.

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  • $\begingroup$ In my opinion, Bayesian testing could do the job. You first define the priors over the hypotheses, then update them using the data, and finally, accept the hypothesis with the greater posterior. You should, however, be aware that the Bayesian probability is one's degree of belief, not a long-term frequency as in the standard hypothesis testing. $\endgroup$
    – Milos
    Dec 26, 2021 at 6:07
  • $\begingroup$ Likelihoods would allow you to assess the evidential support for all of the possible hypotheses in a statistical model. The ratio of likelihoods gives you the relative support, and it is possible to make inferences directly from the likelihoods. If you want probability-based inferences then you need to use Bayesian processing of the likelihoods. $\endgroup$ Dec 26, 2021 at 6:23

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As pointed out in the comments, Bayesian inference would be a way to "pile up" evidence in favour of a hypothesis. This, however, would always fall short of a proof.

I think an important point to bear in mind is that a single counterexample can establish that a claim is false, while no amount of evidence can establish that a claim is right; hence, using evidence alone we can reject hypothesis, not establish their validity.

Edit: On re-reading your question I have a minor comment: you say "...if we can reject the null hypothesis, we accept the alternative."

This is not necessarily the case. The evidence can be in strong disagreement with the null, and in still stronger disagreement with the alternative. This would force you to re-think your problem, and enlarge either the null or the alternative, as seemingly neither one provides an adequate model for your observations.

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