How to update P(x this week), upon hearing P(x next month) = 99.5%?

It seems like you need a lot more context. If Assad were to say he was going to resign in one year, I'd expect him to be out of power far sooner. If David Cameron said the same thing, I'd expect his statement to be fairly accurate.

I'm trying to decide how much of this context is expressed within the value of the prior probability. I have a very low prior on Assad voluntarily resigning. Much higher (relatively speaking) on Cameron.

Hypothetically: I assign a 40% prior to Smith resigning within the year. I assign a 75% prior to Jones resigning in that same period. Previously, both Smith and Jones denied any intent to resign (incorporated into the prior).

Now, Smith and Jones announce they will step down in 18 months. My intuition is that the reasons why Smith-prior was lower mean that I should adjust less than the Jones prior. Yet if the Smith prior was 10%, I would probably adjust the prior more than I adjust the Jones prior.

That makes my math-error sense tickle, so I'm probably be doing something wrong.

In most cases I think the probability would go up, but it depends on what other information you have, especially information about the stability of the PM's government and how transitions of power take place in that country.

The main reason for the probability to go up is that the announcement provides evidence that the PM has a weak hold on power. If he's announcing now that he'll be resigning soon, then there's a decent chance that the opposition is capable of driving him out of office at any time.

However, it's possible that you already knew how weak a hold the PM had on his job. And it may also be the case that transitions of power tend to happen in an orderly way in this country, and that the PM's announcement initiates the orderly procedure. So now that an election has been announced, they're very likely follow through and hold the election as planned. In that case, you might adjust the probability downward.

I'd start with one of the simplest models possible: I'd put December and March into a simple hope function with a flat distribution over time periods http://www.gwern.net/docs/1994-falk The relevant short-run equation being:

To put it into the box terms: December is 1 box, January/February/March are 3 boxes, and there's some probability l0 that his resignation is in any of these 4 boxes so n=4 (and l0-1 that there won't be a resignation in March or before). We want to know our belief that the resignation will be in the first box, box 1, or i=1. I think most promises to resign are kept or accelerated in first world countries, so I'd give l0 around 90% that he will resign before or during March. With all that, my expectation that the resignation will be in the first of 4 boxes given a 90% belief that the resignation is in the boxes at all is 29%

(It'd be more different from 1/4 if we had opened more boxes, or if there were more boxes to begin with.)

Incidentally, was this inspired by Monti of Italy?

I will note that this seems as though it ought to be a problem that we can gather data on. We don't have to theorize if we can look find a good sampling of cases in which a minister said they would resign, and then look at when they actually resigned.

Additionally, this post is mostly about a particular question involving anticipating political change, but the post title sounds like a more abstract issue in probability theory (how we should react if we learn that we will believe something at some later point).

Well, if p(x next month) = .995 then p(x next week) < .005, so your odds possibly went down.

If, on the other hand, p(x before the end of next month) = .995, then p(x next week) may be > .005

The short answer is that there isn't enough information in the problem description. Your priors are incorrect, incidentally; they should match historic data. If p(x) is always 50%, this implies that p(x next week) and p(x within the next year) are identical, which cannot be the case, because the latter probability includes the former. p(x within the next year) > p(x next week).

Your best strategy is to examine historical data, and see what the odds of a prime minister, once specifying a time frame for retirement, actually meeting that time frame are.

I'd say that if he plans on holding the vote in March, he'll probably resign some time around April. If he resigned in December, that would leave several months with no Prime Minister.

I'd guess that March would be chosen because doing it any faster would cause too many administrative problems. His opponents wouldn't push for doing it faster. His problems aren't going to help them, and he's going to be resistant to it, so why bother? That wouldn't make the probability go down that much if you already understood how long it would tend to take, since you'd have known he'd have to announce it beforehand, but I guess if you didn't know that, the probability might go down because you just found out it takes longer than a month.

In your model of the decision you only have two actors, the prime minister and his opponents. In the real world you have more actors. In the real world there are more than two different actors. Within the party of the prime minister there's somebody else who's going to take power of the party when the prime minister stepped down.

That other person might have made a deal with the prime minister that he succeeds the prime minister if he in return steps down within three months.

Different opposition groups might compete with each other over being the central opposition group. There are foreign countries that have interests.

Holding relections takes some time and if a prime minister stays 3 months in office till you have relections, what's the problem? You don't want months without a prime minister.

The question is not hypothetical: I was faced with precisely this problem in December, and got it wrong.

You can't decide based on one data point that you got it wrong.