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Bayesian inference field: given a dataset, if I assume a normal a priori distribution on the average parameter with zero mean and a given variance, those hypothesis tests can be carried out on the posterior distribution to assume (with a certain confidence level) that is the average really zero? Does it make sense to apply the t-test statistic on a posterior distributions (t.test in R) or is the t-test an exclusive hypothesis test of the sample media?

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It does not make sense to apply a t-test on a posterior distribution - a t-test is applied to do data (although you could represent a prior through pseudo observations and then do a frequentist analysis such as a t-test).

One would more typically do one of the following:

  • look at how much posterior mass there is on the parameter being exactly zero (with a normal prior that would be zero, but you could create a point mass at zero in your prior - but see Lindley's paradox - or you could look at the posterior mass in an interval around zero that includes values that are in practical terms more or less zero)

  • Bayesian model averaging between models with the parameter set to zero vs. not set to zero

  • look at posterior credible intervals for the parameter (do they include zero and exclude values that are meaningfully different from zero?)

  • look at the Bayes factor between a model with the parameter to be estimated versus the parameter fixed at zero

Which of these makes the most sense may depend on what you are trying to do.

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  • $\begingroup$ Thank you so much for the answer. I am considering daily financial returns. In practice, I start from the assumption (assumed by now considered in the literature) that the parameter of the average financial returns is zero (the a priori is a normal trocatated with average zero and variance 1000 in such a way as not to be informative). I would like to state that this assumption is valid by inferring this parameter, so that it can be set to zero, for later estimates, within the model that describes the daily returns. $\endgroup$ – Mike9 Jan 3 at 19:13
  • $\begingroup$ As you suggested @Björn, I had thought about a statistical test that counts how many extractions from the rear are greater than zero and how many less than zero. If I have about 50% then I assume that the average parameter is effectively zero. The problem that "about 50%" statistically is not a correct form. I would need a confidence interval $\endgroup$ – Mike9 Jan 3 at 19:16
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    $\begingroup$ 50% less 50% more is not a good criterion. With your prior took have that "nicely" without any data (!) and you could even have a posterior that has no mass near zero, but is symmetrical (e.g. most mass around 1 and -1). A credible interval may be closer to what you want. $\endgroup$ – Björn Jan 3 at 20:29
  • $\begingroup$ But in what sense is a credible interval?. For example, I have 1000 estimates of the average parameter. Again I would say: I calculate the sample mean of the averages and I do a hypothesis test with the t-statistic that this is equal to zero with a certain level of confidence (!?) $\endgroup$ – Mike9 Jan 4 at 9:07
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    $\begingroup$ Forget about doing t- test I posterior samples, that really is not appropriate. One way of getting a credible interval is to look at the MCMC samples and the 2.5th to 97.5th percentile is a 95% credible interval. Most software like Stan will automatically provide it. $\endgroup$ – Björn Jan 4 at 9:36

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