The unexpected hanging paradox

I wrote about the Monty Hall problem a while back. An even more intriguing puzzle is the unexpected hanging problem, which was explored in a paper by Margarita Vázquez Knowledge, Information and Surprise in Triple C. This is the basic paradox:

A prisoner is told that he will be hanged on some day between Monday and Friday, but that he will not know on which day the hanging will occur before it happens. He cannot be hanged on Friday, because if he were still alive on Thursday, he would know that the hanging will occur on Friday, but he has been told he will not know the day of his hanging in advance. He cannot be hanged Thursday for the same reason, and the same argument shows that he cannot be hanged on any other day. Nevertheless, the executioner unexpectedly arrives on some day other than Friday, surprising the prisoner.

(Wording taken from the Wolfram Mathworld site)

As with the Monty Hall problem, it just makes no sense at all at first sight. But now, after talking it through at length with my son, the paradox has slowly vanished. (Disappointingly, since there's something satisfying in an impossible paradox.)

The key is that you have to include the possibility that he will not be hanged and/or it will not be unexpected. If you don't include that possibility he can't be hanged unexpectedly, which contradicts the fact that you have not included those possibilities. So you have to include them, and by including them, you allow for the hanging to happen and be unexpected.

(And actually I think that the telling of it from Mathworld above is slightly misleading, because it can be on Friday that he's hanged. If he gets to Friday he'll know that either the judge was lying about him being hanged or about it being unexpected. Since he doesn't know which applies, he still doesn't know whether he'll be hanged that day or not.)

Is income arithmetric or geometric?

Further thoughts inspired by that video posted yesterday...

The more slick the presentation, the more careful you have to be to check what's really behind it.

One thing that's bothering me, is that log scale for income. Note that the presenter did mention it, he wasn't trying to hide it, but he seemed very relaxed about it, and it needs thinking about. What it is saying, is that the differences in income are huge. This really does need thinking about, and Vedral makes reference to the geometric rather than arithmetric way that income distribution works in his book "decoding reality" which I'm reading at the moment. You can see how this might come about, your life is improved by percentage increases in income, not absolute amounts. If you earn £500k, a raise of £5k is not going to change your life, but if you are on the minimum wage, it will make a significant difference.

And yet, £5k buys you the same things (I'm talking about within the UK) whatever your income.

I think what this is saying, is that big income differences are really bad!

Data visualisation - debunking 'third world' myths

Thanks to Tony Hirst for this spotting this amazing video. It's about 20 minutes long, and well worth the time.

A video about information from YouTube

I came across this on YouTube:



It is nice, but:

"information has no form".

True, but in a sense, information is form. That's all it is. Then again, 'information is physical' as Vedral says.

More on consciousness

In the New Scientist, 9 January (or online 7th January) Ray Tallis writes

MOST neuroscientists, philosophers of the mind and science journalists feel the time is near when we will be able to explain the mystery of human consciousness in terms of the activity of the brain. [...]

[M]y argument [...] is about the deep philosophical confusion embedded in the assumption that if you can correlate neural activity with consciousness, then you have demonstrated they are one and the same thing, and that a physical science such as neurophysiology is able to show what consciousness truly is.

Hear hear! If I am understanding what Tallis is saying, it's that 'most neuroscientists, philosophers of the mind and science journalists' are still wedded to a very reductionist view ('reductionism gone mad'), which thinks that if you understand materials engineering you know what a chair is (to use my own favourite reductionist analogy). I'd better admit, here, though, that I've not yet followed Robin Faichney's advice to read some of Dennett's recent work, so I'm almost certainly being unfair. But this is a blog, so I don't have to be fair, right?

Later in the same piece:

We cannot explain "appearings" using an objective approach that has set aside appearings as unreal and which seeks a reality in mass/energy that neither appears in itself nor has the means to make other items appear.

Notice the 'mass/energy, not even mentioning information. It's like suggesting that you could get some understanding of what I'm writing here from looking at which parts of the computer memory are storing the text. Actually I do know that Dennett talks about information, so there's certainly something missing there.

Another part of the article I want to highlight:

This disposes of the famous claim by John Searle, Slusser Professor of Philosophy at the University of California, Berkeley: that neural activity and conscious experience stand in the same relationship as molecules of H₂O to water, with its properties of wetness, coldness, shininess and so on. The analogy fails as the level at which water can be seen as molecules, on the one hand, and as wet, shiny, cold stuff on the other, are intended to correspond to different "levels" at which we are conscious of it. But the existence of levels of experience or of description presupposes consciousness. Water does not intrinsically have these levels.

This relates to something I've been worrying about with the layers I've been talking about earlier (see the posts on trapeziums). I argued that that there is a hierarchy of meaning: voltage levels 'mean' binary 1s and 0s, patterns of 1s and 0s forming ascii codes 'mean' letters of the alphabet, letters 'mean' words, and so on. The trouble is, I'm not sure whether the meaning in there in, say the ascii codes, or whether the meaning only ever exists in my head. It is like as if the meaning only comes from the observer sitting alongside, looking sideways at the pile of trapeziums, one on top of the other.

Information content of colours

After writing that post yesterday, coincidentally that evening on TV (on 'The One Show', I think) there was an item about the use of colours to identify brands. They gave the example of the blue of Cadbury's Dairy Milk and the red of Coca Cola, I think, if I remember correctly.

I was thinking about my pink. What might it convey? Now I think about it, two things I might associate with a pink: 'Gay', and 'Breast Cancer awareness' ('pink ribbon'). I wasn't actually meaning either of those, so if it did convey that, it was false: the colour does not contain that information on this occasion.

The amount of information that a colour could convey is quite large, but depends on the viewer's colour resolution. This links back to my musings on the Damien Hirst's spot paintings a while back. Actually, I say it's quite large, but we're only talking about 24 bits for 'Truecolor', and as I was saying yesterday, that's tiny compared to what you can convey with a few words. I don't know, there's something not quite right with this argument.

The programme also talked about football strips. The information content of those is worth exploring. I think they convey three things:

1) the identity of the club
2) the identity of the sponsor of the strip
3) the season (clubs change their strip each season - in encourages fans to buy a new replica strip each year)

Sometimes some strips are very plain, sometimes they are intricate. I'd suggest that they don't convey any more information by being intricate. The added complexity is merely 'decoration', or information terms it is either redundancy or noise. In some cases thinking of it as noise would make sense, since I think it can make it more difficult to extract the identity of the club from it.

There's a lot more that could be explored - home and away strips, the experimental finding that teams wearing red gain a measurable advantage compared to clubs wearing blue. (Reported in the New Scientist a while back. Evidence includes the relative successes of Liverpool and Manchester United compared to Everton and Manchester City.)

Information content of personalisation

I've just upgraded to Firefox 3.6 and was given the option of choosing a 'persona'. This allows me to select images/colours to appear at the top and bottom of my screen. See http://www.getpersonas.com/en-US/gallery/All

I was taken aback to find that there's currently more than 76,000 to choose from, with more being added all the time (anyone can create one and upload it).

In Theories of the Information Society Frank Webster suggests that the choices available today in 'personalisation' are one example of there being more information around today than in the past. Specifically, in the context of clothing, he suggests that the medieval peasant’s smock contained a lot less information than the clothing of even the poor today.

I suppose by choosing one out of 76,000, you might argue that I am providing something like log₂(76,000) = 16 bits of information about myself.

The more popular the one chosen, the less information it represents in this calculation. I am currently using 'Olivencia Pale Pastel', and the website says there's '1 active daily user' of this scheme - I'm not sure whether that means me or someone else. Others have hundreds, thousands or tens of thousands of users, so mine is an obscure choice, and therefore probably represents rather more than 16 bits of information.

Except it doesn't. I certainly didn't go through all 76,000 and chose this one. A little bit of thought did go in to finding it, maybe 5 minutes, but a large part of the decision was random chance.

If I had time, I would create my own using my own image files. I guess that would then represent more information about me. But maybe 'if I have time' is the key. There's no shortcut to producing information, it takes time.

(And anyway, even if it did represent 16 bits, or 32 bits, say, that's still pretty small compared to a few words.)

Update:

Hehe, look what Olivencia Pale Pastel looks like with the colours in my blogger! Even I can tell that's terrible. It's so bad I've got to keep it now.

Information is shock resistance

..."arm yourself". So says Naomi Klein:

Info content of the FA cup draw

For the semi-final draw they dramatically select each of the four balls for the four clubs, but in fact there's only log₂(3) = 1.58 bits of information, and it could be done with a single draw. Since both matches play at Wembley, there's no 'home' or 'away', so there's only three possibilities:

A - B and C - D
A - C and B - D
A - D and B - C

During the draw:

- first team out of the box: gives no information
- second team out, select one from the three remaining, and gives all of the information
- third and fourth teams out, tell us nothing, because we already knew they were to play each other

Unless I suppose it makes a difference which match is played first, in which case there are six possibilities, adding one bit of information, 2.58 bits. (log₂(6) = 2.58)

A - B then C - D
A - C then B - D
A - D then B - C
C - D then A - B
B - D then A - C
B - C then A - D

- first team out. Tells us that team plays in the first match. That is a selection from two equal possibilities (that team could have played in the first or second match), so contributes 1 bit of information
- second team out. This is a selection of one from three (the three teams still in the hat) so is 1.58 bits of information. This is now all the information, because we know the other two teams play each other in the second match. Information adds, so total information is 2.58 bits.
- third and fourth teams out contribute no information.