How to know that a posterior probability is significant in bayesian inference First of all I would like to apologize if I am making any misconceptions or not using the right vocabulary since I am just getting started in Bayesian inference.
That said, the problem I am facing is the classification of a series of devices in different classes using the measurements that they are reporting. In order to do that, I am using the new evidences (data reported by the devices) to obtain the likelihood probability for each class with which I then obtain the posterior probability for each class. As we know, the posterior probabilities of each class obtained by the Bayes inference will sum 1. In this sense, to finally classify to which class each device belongs, I am taking the hypothesis for which the posterior probability is higher.
The problem is that, sometimes, the posterior probabilities are too close to each other so the probability of the hypothesis with the maximum posterior probability is not significant compared with the others.
Example: imagine that we have three hypotheses with the posterior probabilities being 0.4, 0.3 and 0.3. Currently I am classifying the device to the first class based on these probabilities but, as can be seen, the three probabilities are too close each other, not being any of them significant to the others.
In this sense, is there any test that I can perform to obtain if the maximum posterior probability is significant enough to the rest of them?
Thanks in advance and sorry again if I am making any mistake with the vocabulary or any misconception.
 A: I think you could approach this in two different ways.  The first way you would select a certain hypothesis as the default choice and only if its posterior probability is below a certain threshold would you choose against it.  This would mimic frequentist null hypothesis testing and is similar to Go/No-Go decisions in phase 2 clinical development.  A second way you could proceed is to always select the hypothesis with the largest posterior probability, regardless of how close it is in size to the posterior probability of competing hypotheses.  This is similar to early phase clinical dose finding studies.  I don't think there is a right or wrong approach.
A: The criteria for 'significant'/'not significant' in a Neyman–Pearsonian hypothesis test are based on the error rate characteristics of the test, determined in advance of seeing the data. (See this for an explanation of hypothesis tests and p-values: https://link.springer.com/chapter/10.1007/164_2019_286) The dichotomy is not as universal or as useful as many beginner statisticians think.
With a Bayesian posterior you usually do not have access to the error rate considerations and so the 'significant'/'not significant' dichotomy is usually not applicable.
You might choose to set up criteria for inclusion into a set of classes for your devices on the basis of probability ratios for a specified hypothesis (i.e. a parameter in the statistical model that makes up the x-axis of your likelihood functions) compared to the maximally probable hypothesis but you must not call it 'significant'/'not significant'! However, the characteristics of the test that you invent would be generally unknown and so the utility and desirability of such a procedure would need to be explored.
Perhaps you should be asking a different question. Explain your data and inferential objectives and ask for suggestions as to how you might proceed. It might be that a Bayesian posterior is appropriate, but that is not clear to me.
(You failed to say anything about the prior used to convert the likelihoods into posterior probabilities. That is bad practice.)
