Using Likelihood Odds Ratio

To be able to do inference, I need to provide the entire joint distribution, i.e. Prior + Likelihood.
What if instead of the likelihood, I have the likelihood odds ratio? i.e. \frac{P(D|H_1)}{P(D|H_2)}.
How can I do inference on this?

In a Bayesian setting, you’re talking Bayes factors or something similar. See here for a bunch of discussion and examples. If you’re looking to do the frequentist version (“prior-free”), then you’re talking standard likelihood ratios.