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I am writing a paper that uses a Bayesian method. I have a parameter $\theta$, I have null and alternative hypothesis $H_a(\theta)$, $H_o(\theta)$, and a value $P(H_o(\theta)\mid D)/P(H_a(\theta)\mid D) = k$. I conclude in favor of the alternative hypothesis if $k < \eta$ where $\eta$ is a threshold where $P(H_o(\theta)\mid D) = \alpha$. Does all the same old frequentist terminology hold? Is $\alpha$ still described as an upper bound on the significance level? Do I still say "$\theta$ is statistically significant if $k > \eta$?", or do Bayesians use a different lingo?

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Not really. Bayesian testing is very liberating for statisticians and non-statisticians. A distinction, and one which frequentists should consider, is the point of being rigorous in apriori specifying decision rules, or alternative hypotheses of interest. It allows one to report the results of a trial or study in terms of "We have $Pr(H_a | X)$ belief that $H_a$ is true" if a suitable, possibly informative prior has been chosen for $Pr(H_a)$.

If one insists that a declaration of "statistical significance" is really, truly, honestly necessary (which it usually never is), a decision theoretic approach can be used to make this rigorous. This is analagous to the Neyman-Pearson method of testing. If one defines a loss function, for choosing $H_a$ over $H_0$, or vice versa, one can report which decision has a minimal loss, which is informed by the data as well as some decisions about the cost of type I and type II errors.

An analogous approach to Fisherian testing is using Bayes factors. Bayes factors are useful to report for statistical tests, and more closely resemble classic hypothesis testing via $p$-value without comparing it to any arbitrary $\alpha$ threshold. Again this benefits from an appropriately stated alternative hypothesis and prior for the decision. A Bayes factor again, is a reflection of belief, whereas for frequentists probability is considered a frequency.

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This issue doesn't come up often because when people use fully Bayesian methods, they don't usually create a null hypothesis and then make a choice on the basis of a hard threshold of relative probability, as you have. After all, a lot of the motivation for using Bayesian methods is to avoid such things.

I'd recommend avoiding the various forms of the word "significant" to avoid any confusion, although the analogy with classical null-hypothesis significance testing is clear, and is worth pointing out in your paper. Just say something like "Using the decision rule specified by Equation 2, I accepted $H_a$."

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