# M-open, M-complete, M-closed

May 30, 2016 — July 23, 2023

Placeholder.

Yuling Yao, The likelihood principle in model check and model evaluation

We are (only) interested in estimating an unknown parameter \(\theta\), and there are two data generating experiments both involving \(\theta\) with observable outcomes \(y_1\) and \(y_2\) and likelihoods \(p_1\left(y_1 \mid \theta\right)\) and \(p_2\left(y_2 \mid \theta\right)\). If the outcome-experiment pair satisfies \(p_1\left(y_1 \mid \theta\right) \propto p_2\left(y_2 \mid \theta\right)\), (viewed as a function of \(\theta\) ) then these two experiments and two observations will provide the same amount of information about \(\theta\).”

This idea seems to be useful in thinking about M-open, M-complete, M-closed problems.

## 1 References

*Lecture Notes-Monograph Series*.

*Bayesian Theory*.

*The Journal of Machine Learning Research*.

*Bayesian Theory and Applications*.

*arXiv:2202.04744 [Cs, Stat]*.

*Journal of Machine Learning Research*.

*Bayesian Analysis*.

*Proceedings of the 32nd International Conference on Neural Information Processing Systems*. NIPS’18.

*Proceedings of the 34th International Conference on Neural Information Processing Systems*. NIPS’20.

*Journal of the Royal Statistical Society Series B: Statistical Methodology*.

*arXiv:2104.03889 [Stat]*.

*arXiv:2011.08644 [Stat]*.

*Bayesian Analysis*.