There's two ways to train algorithms 'multiple times' on the same data. The bad one is data duplication, but cross-validation is the good one. Data duplication is the sort of thing that Jaynes would have been worried about, because it means you're counting evidence from the same piece of data twice, thus your model has illusory precision. But what does cross-validation do? There's an issue called "overfitting," where any statistical procedure performed on a training set will fit both the noise and the signal in the training set, but while the signal on a test set will presumably be the same, the noise will be different and thus the model will do worse. Single validation is when you split your data into two parts, the training set and the test set, so that you can see how well your model trained on the training set does on the test set. When there's a tunable parameter in the training method, people will sometimes optimize the tunable parameter given data in the test set.* But to do one split and leave it at that is wasteful. Cross-validation is when you partition the data many times, and fit many different models, and can thus talk about how the population of models behaves. In particular, consider the case of 'leave-one-out' cross-validation, where in a dataset of n points, we train n different models, each time using n-1 datapoints to fit the model parameters, test them on the 1 datapoint left out. This gives each individual model as much training data as possible while still leaving us a test dataset to determine how resilient to overfitting our model-generation procedure is. * The principled way to do this is to split the data three times, into a training set (which the algorithm always has access to), a validation set (which the algorithm only has access to when setting the tunable parameters), and then a test set (which the algorithm never has access to, but is used to assess how well the model does after the tunable parameter has been optimized).

^ Above post is the illustration of the danger of LW's style Bayes. Below is a non-crazy discussion (e.g. one where people don't scream): http://andrewgelman.com/2013/12/10/cross-validation-bayesian-estimation-tuning-parameters/