*away*from 0, towards 50%, and then up towards 100%. New information might then push it back again. Information seems to be able to push probabilities around in any direction - so why do we say that its effect is predictably to push them towards the extremes? This post zooms in on this idea, since it's a very important one in analysis.

To start with, it's worth taking time to consider what it means to assign a probability to a hypothesis. To say that a statement - such as "it will rain in London tomorrow", "Russia will deploy more troops to Georgia in the next week" or "China's GDP will grow by more than 8% this year" - has (say) a 10% probability implies a number of things. If we consider the

*class*of statements to which 10% probabilities are assigned, what we know is that one in ten of them are

*true*statements, and nine in ten are

*false*statements. We don't know which is which though; indeed, if we had

*any*information that some were more likely to be true than some others, they

*couldn't*all have the same probability (10%) of being true. This is another way of saying that the probability of a statement

*encapsulates*, or summarises, all the information supporting or undermining it.

Now let's imagine taking those statements - the ones assessed to be 10% probable - and wind time forward to see what happens to their probabilities. As more information flows in, their probabilities will be buffeted around.

*Most*of the time, if the statement is true, the information that comes in will confirm the statement and the probability will rise.

*Most*of the time, if the statement is false, the information that comes in will tend to disconfirm it and the probability will fall. This is not an empirical observation - it's not 'what we tend to see happening' - but instead it follows from the fundamental concepts of inference and probability. It means that things that are already likely to be true are more likely to be

*confirmed*by new information, and things that are already likely to be false are more likely to be

*disproved*with more information.

This means that

*most*of the '10%' statements (the nine-out-of-ten false ones, in fact) will on average be disproved by new information, and the others (the one-in-ten true ones) will on average be confirmed with new information. By definition, this isn't a predictable process. It's always possible to get unlucky with a true statement, and receive lots of information suggesting it's false. It's just

*less likely*that that'll happen with a true statement than with a false one. And the more information you get, the probability that it's

*all*misleading becomes vanishingly small.

But we need to be careful here. When we say that

*most*statements assigned a 10% probability will be disconfirmed with new information, we're

*not*saying that, on average, the probability of '10% probable' statements will fall. Far from it: in fact, the average probability of all currently '10% probable' statements, from now until the end of time, will be 10%. Even if we acquire perfect information that absolutely confirms the true ones and disproves the false ones, we'd have one '100% statement' for every nine '0%' statements - an average probability of 10%. But as time (and, more pertinently, information) goes on, this will be an average of increasingly-extreme probabilities that approach 0% or 100%.

Perhaps surprisingly, we can be very explicit about how likely particular future probability time-paths are for uncertain statements. If we assume that information comes as a flow, rather than in lumps, the probability that a statement's probability will rise from p0 to p1, at some point, is rather-neatly given by p0/p1. For example, the probability that a statement that's 10% likely will (at some point) have a probability of 50% is (10% / 50%) = 20%. Why? Well, we know that only one in ten of the statements are true. We also know that for every two statements that 'get promoted' to 50%, exactly one will turn out to be true. So two out of every ten '10%' statements

*must*at some point get to 50% probable - an

*actually*-true statement, and a false fellow-traveller - before one of them (the true one) continues ascending to 100% probable and the other (the false one) gets disconfirmed again. (The equivalent formula for the probability that a statement will

*fall*from p0 to p1 is (1-p0) / (1-p1).)

*which*statements new information is going to push one way or the other - if we did, we'd have the information already and it would be incorporated into our probabilities. Assuming, of course, we were doing things properly.

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