Likelihood ratio vs. Bayes factor

The Likelihood ratio uses the Maximum Likelihood Estimator (MLE) and subs it in for to find the max probabilities of each model. So it uses a point estimate and doesn’t take into account possible unknowns about the parameter

The Bayes Factor integrates across the entire $\Theta$ space so it takes the (weighted) average of different probabilities of $\theta$‘s.

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References

Criticisms of MLE - similar idea here. The likelihood ratio doesn’t mean that the data is specifically the most likely realization @lancaster2004 - Chapter 2