The image below show mathematical expressions for the meanings of training and testing. I can understand the meaning of testing but the meaning of training is not so clear because of $M$ there. I took from a video lecture on youtube, the lecturer said $M$ means multiple bins which are equivalent of multiple hypotheses. When he said multiple bins, I thought of multiple samples, am I correct? I sort of understood what he meant by multiple bins but I don't think I have a solid picture of it. If somebody understands it, please give me an explanation about it.