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I am paraphrasing the two quotes:

  You can say there is a 25% increase in chance of dying
  if you smoke.  You cannot say there is a 25% chance that
  the nurse murdered those people - she either did or did 
  not.  (what we lack is the knowledge of the past event)

second para:

  On the preponderance of evidence, the court believes 
  beyond a reasonable doubt that the nurse *did* kill those
  people.  That event A *did* happen - we have used evidence
  to fill in the gaps of our knowledge.
  We are not assigning a probability to it, but we recognise
  that life is all probabilities, but thats what reasonable 
  doubt it all about.

I think that those two quotes (modified as I understand them) are consistent. They are also not entirely Bayesian but I would prefer to be in a court of law than a court of probability.

Hmmm, I guess I agree. But I think a Bayesian would argue that this is an argument over wording not meaning. If we formulate beliefs and act on beliefs, then how we want to word it doesn't make a difference.

Its post facto versus a priori. There is a 25% chance of something happening, but after the fact, it either did or did not happen. Averaged over many cases we can say that after the fact there is a 25% chance it happened, but criminal trials look at one specific instance, and its accurate to say that in an instance something either did or did not happen.

You should still use probability to quantify that belief. Whether it happened in the past or is going to happen in the future doesn't matter. If you throw a coin, and the coin is flying in the air, what is the probability that it lands heads? By your reasoning, the motion is already in place, and it's either going to land heads or it isn't. With a high speed camera and some physics we might even be able to predict which side it will fall. Even if we go back to a little before you threw the coin, the electrons in your brain are already in place and it's already determined which way you'll throw the coin. Would it be invalid to say that there is a 50% chance that it will land heads? So even when we are talking about the future, we use probability to quantify our belief. Or perhaps more accurately, we use it to quantify our lack of knowledge: in this case our lack of knowledge about how exactly the coin will move. Since we are already using probability not just for "fundamental randomness" but for quantifying lack of knowledge about the future, what's so special about the future vs the past? It makes total sense to apply the same reasoning to the past, even if we are talking about a specific instance.

You're shifting the argument. Sure you should, in some cases,[1] use probabilities to determine your belief about whether something happened or not. That's precisely what the court says in the second paragraph. Your beef was ostensibly with the wording of the first paragraph, but as I pointed out that when talking about things that have happened in the past, the wording is not incorrect.

[1] Of course you also shouldn't sometimes use probabilities. If we have statistics saying that witnesses lie 15% of the time, is it helpful to present expert testimony to the effect of "there is a 15% probability the witness is lying?" Such evidence is not useful and also impinges on the fundamental right of the jury to judge witness credibility.

My only point is that whether something is in the past or in the future has nothing to do with whether you can use probability. If you're consistent then either you don't use probability at all to model lack of knowledge, or you use it for both past and future. For example if you find it valid to say "when he tosses that coin, it has 50% probability to land on heads and a 50% probability to land on tails", then you should also find it valid to say "he tossed that coin, and it now has a 50% probability to be on heads and 50% probability to be on tails" if you did not yet see the coin. That's because in both cases, the outcome was already determined (by the laws of physics, disregarding quantum mechanics). The thing is that you just don't know which way it will land or has landed because you lack perfect knowledge.

I don't have a problem with not using probability to model lack of knowledge, but you have to apply that principle consistently (and whether it is ethically desirable to use probability is an entirely different and even more difficult question than whether or not it leads to accurate beliefs). That said, I do prefer to use probability to model lack of knowledge, because what is the alternative? Ad hoc gut feeling based reasoning?

I actually read it as not contradictory. This was about two events: one which was a priori more probable than the other, based purely on the previous experiences of the judge. That is, he considered "X probably happened because Y has a history of happening less than X". I take that to be that the judge committed a classic fallacy: he took a frequentist, limit-in-the-long-run subjective judgement, and tried to apply it to a situation where it's really not applicable. He took his priors from a mistaken frequentist consideration, that is what I interpret the first quote to say.

What he should have done, the second quote says, is to "arrive[s] at its conclusion by considering on an overall assessment of the evidence". Instead of considering what happened before, he should have taken his priors from the evidence. So, in the end, it's a sentence that supports a Bayesian methodology, in this case. Or maybe the Court of Appeal just was confused at the time, who knows.