Hi, Thanks for all the helpful comments. Some quick replies:

Graham: firstly, you're right that the most important point is that $P(M=m)$ isn't needed for the MCMC acceptance ratio. I'm interested in why you think the definition of the rule with two cases (ratio>1, otherwise) is better? Ah, perhaps the second sentence ought to be run into the definition of the rule, the point just to make explicit the implicit poitn that the case where the ratio is greater than one then every random probability will be less than the ratio, and hence be accepted. (We've phrased it as is partly because we're admitting that we don't have the space to actually justify some of the stuff, in which case that distinction used in the proof doesn't get used. But if people think the other presentation is better we perhaps ought to change it.)

Jacob: I don't think we mean multiset; what we mean is that an individual measurement comes from a finite set of possibilities, with the "time series" for measurements forming a sequence or possibly, making an outrageous conflation of mathematical structures, a vector where the index corresponds to a timestep.

Everyone: thanks for pointing out the other typos and cases of phrasing that could be improved.

Graham: firstly, you're right that the most important point is that $P(M=m)$ isn't needed for the MCMC acceptance ratio. I'm interested in why you think the definition of the rule with two cases (ratio>1, otherwise) is better? Ah, perhaps the second sentence ought to be run into the definition of the rule, the point just to make explicit the implicit poitn that the case where the ratio is greater than one then every random probability will be less than the ratio, and hence be accepted. (We've phrased it as is partly because we're admitting that we don't have the space to actually justify some of the stuff, in which case that distinction used in the proof doesn't get used. But if people think the other presentation is better we perhaps ought to change it.)

Jacob: I don't think we mean multiset; what we mean is that an individual measurement comes from a finite set of possibilities, with the "time series" for measurements forming a sequence or possibly, making an outrageous conflation of mathematical structures, a vector where the index corresponds to a timestep.

Everyone: thanks for pointing out the other typos and cases of phrasing that could be improved.