projpred

https://avehtari.github.io/modelselection/bodyfat.html
https://link.springer.com/article/10.1007%2Fs11222-016-9649-y
https://arxiv.org/abs/1810.02406
https://avehtari.github.io/modelselection
https://www.youtube.com/watch?v=Re-2yVd0Mqk&t=87m6s

From Aki Vehtari after I mentioned my use of 'model approximation in 2001 RMS':
The discussion in your Section 5.4.2 is similar as by Lindley (1968). When noise is unknown the situation is a bit different. The term projection comes from Leamer (1979). Some additional historical Bayesian references can be found in Section 5.4 of 
https://projecteuclid.org/euclid.ssu/1356628931