April 17, 2011
The is a follow up from Nathaniel’s post. One of the ways that the probabilities of probabilities can be used is in asking what experiments would be best for a scientist to do. We can do this because scientists would like to have a logically consistent system that describes the world but make measurements which are not completely certain – the interpretation of probability as uncertain logic is justified.
Lets make a probabilist model of scientific inquiry. To do this, the first component we need is a model of “what science knows”, or equally, “what the literature says”. For the purposes here, I will only consider what science knows about one statement: “The literature says X is true”. I’ll write this as and its negation as . This is a really minimal example.
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April 14, 2011
Most of us who’ve studied probability theory at University level will have learned that it is formalised using the Kolmogorov axioms. However, there is an interesting alternative way to approach the formalisation of probability theory, due to R. T. Cox. You can get a quick overview from this Wikipedia page, although it doesn’t really motivate it very well, so if you’re interested you’re much better off downloading the first couple of chapters of Probability Theory: The Logic of Science by Edwin Jaynes, which is an excellent book (although sadly an incomplete one, because Jaynes died before he could write the second volume) and should be read by all scientists, preferably while they’re still impressionable undergraduates.
For Cox, probability theory is nothing less than the extension of logic to deal with uncertainty. Probabilities, in Cox’s approach, apply not to “events” but to statements of propositional logic. to say p(A)=1 is the same as saying “A is true”, and saying p(A)=0.5 means “I really have no idea whether A is true or not”. A conditional probability p(A|B) can be thought of as the extent to which B implies A.
There are a couple of interesting differences between Cox’s probabilities and Kolmogorov’s. Cox’s is more general, but also less formal (people are still working on getting it properly axiomatised). One important difference is that in Cox’s approach a conditional probability p(A|B) can have a definite value even when p(B)=0 (this can’t happen in Kolmogorov’s formalisation because, for Kolmogorov, p(A|B) is defined as p(AB)/p(B)). This means that, unlike the logical statement , the probabilistic statement p(A|B)=1 doesn’t mean that A is true if B is false. So conditional probabilities are like logical implications only better, since they don’t suffer from that little weirdness.
Anyway, that’s cool but what I really wanted to write about was this: in Cox’s version of probability theory, it’s meaningful to talk about the probability of a probability. That is, you can write stuff like p(p(A|B)=1/2)=5/6 and have it make sense. I’ll get to an example of this in a bit.
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March 17, 2011
Yesterday, I went to a science communication conference thing at the University of Brighton. Here is what I learned…
So the first thing to say is that it was quite comforting to hear the science editor for The Observer saying that the medias reporting on genetic modification and the measles, mumps and rubella (MMR) vaccine were real failures of science reporting. He even described the reporting on MMR as “a deep burning shame”. Less favorable was the media representatives description of their roles. Both of them said that their job was not to educate, but rather to either entertain or to “hold our masters to account”. Education, if it occurs, is a side effect. I find this slightly worrying – but less so than other people I have talked to. I will not dwell as I have a different point to make.
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