These theories are evidence about true distribution of data, so I construct a new theory based on them. I then could predict the next data point using my new theory, and if I have to play this game go back and choose one of the original theories that gives the same prediction, based only on prediction about this particular next data point, independently on whether selected theory as a whole is deemed better.
Having more data is strictly better. But I could expect that there is a good chance that a particular scientist will make an error (worse than me now, judging his result, since he himself could think about all of this and, say, construct a theory from first 11 data points and verify the absence of this systematic error using the rest, or use a reliable methodology). Success of the first theory gives evidence for it, which depending on my priors can significantly overweight expected improvement from more data points coming through imperfect procedure of converting into a theory.
These theories are evidence about true distribution of data, so I construct a new theory based on them. I then could predict the next data point using my new theory, and if I have to play this game go back and choose one of the original theories that gives the same prediction, based only on prediction about this particular next data point, independently on whether selected theory as a whole is deemed better.
Having more data is strictly better. But I could expect that there is a good chance that a particular scientist will make an error (worse than me now, judging his result, since he himself could think about all of this and, say, construct a theory from first 11 data points and verify the absence of this systematic error using the rest, or use a reliable methodology). Success of the first theory gives evidence for it, which depending on my priors can significantly overweight expected improvement from more data points coming through imperfect procedure of converting into a theory.