Let me clarify that I'm not defending the notion of statistical significance in data analysis -- I'm merely saying that the advice to publish likelihood ratios is not a complete answer for avoiding debate over priors.
I analyzed some data using two versions of a model that had ~6000 interest parameters and ~6000 nuisance parameters. One of my goals was to determine which version was more appropriate for the problem. The strict Bayesian answer is to compare different possible models using the Bayes factor, which marginalizes over every parameter in each version of the model with respect to each version's prior. Likelihood ratios are no help here.
It turned out to be a lot easier and more convincing to do a simple residual plot for each version of the model. For one version the residual plot matched the model's assumptions about the error distribution; for the other, it didn't. This is a kind of self-consistency check: passing it doesn't mean that the model is adequate, but failing it definitely means the model is not adequate.
(BTW, the usual jargon goes, "statistically significant at the 0.05 / 0.01 / 0.001 level.")
Your ability to distinguish them that way means that there was a large likelihood ratio from the evidence.
I still can't see the relevance of Bayesian Statistics over Frequentist Statistics, and I take Less Wrong as evidence that this is a cause for clarification.
I'm looking for a historical narrative of the development of mathematics that tells me what mistake lead to frequentism over Bayesianism, which is supposedly the correct view. Alternatively, you can just say "Read PT:TLOS!" if it's that silly of a question.