Sound and Fury

I’ve been thinking about the Dutch Book Argument (DBA) recently. I think the constraints on rational betting preference that underpin the force of the argument are unreasonable. At least they are not always reasonable. I think a better way to think of the argument is as a conditional argument: “Given these rational betting preference conditions, this is how rational degrees of belief should be constrained.” Then you can have a whole series of different DBAs with different betting preference conditions leading to different constraints on credence. It would be interesting to see how one would have to constrain betting preferences in order to have you beliefs behave like upper and lower probabilities or Dempster-Shafer belief functions…

I think this isn’t to undermine the force of the DBA, but to reinforce it. With this wider framework we can understand why people often fail to reason probabilistically. We can understand what aspects of rational betting preference are “non-probabilistic”.

That’s not to say that the DBA isn’t without its flaws. Some elements that concern me are:

• Using betting behaviour as a proxy for belief
• Existence of exactly specific numerical credence (and utility)
• Reasoning as calculating expected utilities
• The idealisations involved in discussing “ideal rational agents”: utility maximising, purely self interested, perfect calculating agents…

The title is a quote from Henry Adams.

I despair for modern politics when David Cameron tries to ally himself with “the common man” while at the same time engaging in petty point-scoring about Titian’s age. Thanks Dave. Really constructive politics we’re doing here. And it is faintly worrying in a 1984 kind of sense when someone then tries to edit wikipedia to make almighty Dave look better. And worrying in a kind of Mr. Bean sense when they get it wrong. And worrying in a more serious sense that time was wasted on this exchange when the country is in dire financial straits. Not to mention issues of climate change that only grow more urgent.

That’s not to say I would never vote Conservative (although instinctively I’m probably closer to something like what Labour used to stand for.) I’d never vote for the current Labour government either. I’ve nothing against Mr. Brown. I think he’s serious and possibly even boring, but experienced and probably fairly good at politicky stuff. And I’m in no position to make any stronger judgements of his ability etc. What I object to is Jacqui Smith. (a) she spells her name in a really stupid way. (b) her voice grates on me whenever I hear her interviewed. She looks permanently put-upon and harrassed and sounds it too. (c) she seems to be forging ahead with all sorts of surveillance type policies despite widespread disapproval. This point is particularly galling for me because in theory I would be open to some sort of national ID card scheme if it could be made useful and worthwhile. The current (unpopular) scheme is hamstrung not only by the anti-ID card lobby, but also by worries over security of the data and the cost involved. And the fact that the card as it is wouldn’t be useful. My suggestion is make the card such that people will find it a convenience to have it, rather than force them to carry it. (d) she has twice ignored the advice of groups set up to explore the reclassification of drugs. Why this is annoying is because it speaks of a basic disregard for the scientific facts which, I feel, should be at the basis of policy decisions. The argument given for ignoring the advice is that it “would send the wrong message”. But if the decision was effectively made before the advice was given, why spend money on having these people produce the advice in the first place? Commissioning a report seems to carry with it an implicit duty to pay attention to the recommendations put forward and to make any decision at least partly on the basis of the report.

None of these are particularly well thought out arguments, nor are they based on any careful collating of all the relevant information. But that’s exactly the problem. I would in general be predisposed to go out and vote, but the impression I get of the current crop of candidates is fairly negative. Politicians should be trying to convince me they deserve to be in power. And I am going to be convinced only by a cogent set of principled policies. And I am not going to go out and read party political manifestoes. It is the duty of the politicians to get their message across to me. My impression is that manifesto promises are often reneged. And politics seems to be all about criticising the other guy. (That could be because in terms of ideology or policies, there is no real difference between the two main parties any more.)

This is all a bit of rant really. I’ve probably done nothing more than show how ill informed I am. Never mind, eh?

Prize for the best title for a conference ever: Tickle your catastrophe.

The number of partitions on an N-membered set is called its “Bell number“. The number of strict total orders, I’m not sure about that… There is some relevant information here.

Well it’s good to see those problems solved, and I’m glad they weren’t trivial… Here is a more general issue. How many relations are there on a set? How many transitive? And so on.

So, a relation is simply a set of ordered pairs. So there are 2^X possible relations if there are X ordered pairs. For a set with N elements, there are N^2 ordered pairs. Thus 2^(N^2) relations on N elements. To give you an idea of scale, there are 65,536 relations on the 4 element set. But there are only 15 partitions on the 4 element set (not 18 as I said before; I counted each 2+2 partition twice.) So the properties of being transitive, reflexive and symmetrical must seriously cut down the number of relations! The only thing I’ve concluded in answer to this question is that there is an upper bound on the number of possible total relations on N elements: 2^(N^2) – 2^([n^2]/2). This is because the property of being total means that the size of the set of ordered pairs has to be big enough to contain at least one of (a,b) or (b,a) for every a,b. So small sets of pairs are ruled out.

I have a question: for a set of N elements, how many different ways are there of partitioning it? In other words, how many distinct equivalence relations are there on a set? Similarly, how many distinct strict total orderings are there? How many weak partial orderings?

All of these things (equivalence relations, orderings…) are determined by sets of ordered pairs on the set. So is it easier to calculate numerical formulae for each type of structure on the set, or is it easier to demonstrate that, for example there are more weak partial orderings than there are partitions… It’s trivial, for example, to show that there are at least as many weak partial orderings as there are strict total ones, since each strict total order is equivalent to a weak partial order.

There could be quite a lot of partitions. A partition is a subset of the power set. Or an element of the power set of the power set. So the absolute upper bound on partitions for a 4 element set is 2^(2^4) = 65,536. Obviously this is a ridiculously high figure, but I can’t see an easier way to get closer to the actual number. I am no good at combinatorics. But for the 4 element set you could probably just list all the possible partitions. First list all the ways the sets could break down:

• 1,1,1,1
• 2,1,1
• 2,2
• 3,1
• 4

Then list the number of ways of splitting up the 4 elements, A,B,C,D into those sets.

• Only one {1,1,1,1} partition
• Six {2,1,1}
• Six {2,2}
• Four {3,1}
• One {4}

For a total of 18 partitions. So there are more partitions than there are subsets. But for the 2 element set, the power set has 4 elements, whereas there are only two possible partitions. My guess is for bigger sets there will be more partitions than there are subsets.

As for strict total orderings, I think there will be N! of them. Because there are N choices for the maximal element, N-1 choices for the next “biggest” and so on. So the 4 element set has 24 strict total orderings on it. N! > 2^N for N>3. As for weak partial ordering, there’s probably even more of them…

What other kinds of structure can one impose on a set? If one thinks of graph theory in terms of set theoretic structure, what then? In undirected graphs, the edges of a graph are pairs of elements of the set of vertices. I just did a little googling and learned that there is something called “graph ennumeration” which is where one studies the number of graphs with a certain number of vertices that have certain properties. Much like the classification of finite groups, I suppose. So for example there are no asymmetric graphs with less than 5 vertices. (Or possibly 6?)

Anyway, that’s just what I was thinking about this morning, when I should probably have been trying to understand the game theory lecture…

This New Scientist article made me wonder about time. The latest super-clocks are sensitive to differences in height (actually differences in gravitational field) small enough that their accuracy would be affected if they were placed on a slightly lower table, for example. This is madness. Maybe we’ve reached a point where time doesn’t really mean all that much any more. On those kind of scales, at that sort of precision, maybe time just stops being a useful concept. Like the length of a coastline stops being a useful concept when you measure it too accurately. (Because it’s a fractal). Or temperature; if you keep zooming in, you reach a point where the “temperature” of the volume becomes meaningless – if you zoom in enough that the volume contains maybe one atom or even no atoms, then surely temperature is unhelpful. Perhaps the same thing happens to time when we start trying to subdivide it into 10^-18ths of a second or whatever it is they are doing…

Here’s another article on time where some people seem to be reaching the same conclusions… I’m not sure these articles will be available – I’m on the campus network and they might have some sort of subscription to the magazine…

I just read about the Titanoboa. This is a snake that lived soon after the Dinosaurs died out. It was huge. I don’t think that quite does justice to it. It has been estimated that these things weighed upwards of 1,100kg and were 13 metres long. That’s a snake taller than your house weighing as much as your car! This thing was a metre wide! HUGE! The craziest part was that these things probably ate crododiles. Let me say that again; it ATE CROCODILES! You could not make up something like that up.

I think the reaction to the recent snowy weather has been more than a little hysterical. Sure, transport was disrupted, sure schools closed and so on. But it’s simply not weather that we are prepared to deal with. So what?

I’d like to illustrate what I mean with a story that may have some basis in fact, and is partly reconstituted from half-memories I had of visiting Derby. So the story is that they built a flood barrage that was supposed to resist a “once in a hundred years” flood. A few years after this, Derby was hit by a “once in 200 years” flood. The barrier was overcome and much damage was done. Was this anyone’s fault? Well, no. Not really.

I don’t know if this story is true, but I think it illustrates a good point – sometimes, bad things happen and there is no one to blame. Deal with it.