The null hypothesis is the elaborate conspiracy theory that you happen to share with all your peers.
As I said above: you often cannot collect enough observations to rule out a possibility.
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Yes, the answer may not be yes/no. Your probability value would become your prior.
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If your null hypothesis is full independence, you might never collect enough evidence to leave it behind, if the factor is small enough, so your priors matter. If you model successively more interdependences, you may even miss the path into really fat optima.
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