Further very important points about the philosophy of time.
OK, yesterday's stopped clock post was a bit of a gash job, and was at least partly trolling the mathematicians. Do you want the proper Bayesian analysis? Well here it is anyway. It's clear that what I should actually have said is that
(1) for any X, your prior probability P(T=X) will be a number (0,1), usually nonzero as you are aware that your watch might not be right.
(2) P(T=X|a stopped clock says T=X) = P(a stopped clock says T=X|T=X)*P(T=X) divided by P(a stopped clock says T=X).
(3) but for any Y, P (a stopped clock says T=X|T=Y) = P (a stopped clock says T=X). Otherwise it isn't stopped.
(4) Substitute Y=X into 2, and cancel P(a stopped clock says T=X|T=X)/P(a stopped clock says T=X) = 1
(5) P(T=X|a stopped clock says T=X) = P (T=X)
ie, for any X, seeing a stopped clock does not cause you to change your estimate of P(T=X). QED.
So far so good, but that's basically because we've been talking about P(T=X), considered as a function of X. If someone asks you the time, then you can't answer in the form of a function over the real numbers, it would take too long. You have to give a point estimate. What's your point estimate?
Well, remember, you have a watch, which tells you that the time is W=f(T) (ie, unlike a stopped clock, your watch tells you a different time depending on what time it is). It's perfectly consistent to say that for any X, P(T=X) = 0, because time is continuous (physicists - spare me), but that W is a reasonable estimator of T; you can even calculate the expected error.
And note, for any X, there may be cases such that W(T)=X, ie it is possible that there exists a stopped clock that is showing you the same time as your watch. But it's not possible that W=X, because W is a function of T and X isn't.
So:
1) stopped clocks are
2) stopped clocks provide no information about the time because for all X, P(T=X|stopped clock says T=X)=P(T=X) (note that the term after the equals sign refers to a probability distribution)
3) watches provide information about the time because W=f(T) is an estimator of T - ie P(T=X|W=X)≠ P(T=X)
4) however, if you read off W(T)=X as a number, P(T=X)=0, ie, in so far as they show a point estimate of the time, watches are also wrong.
5) therefore, the correct answer to "What time is it?" is "Time you got a watch".
The key thing here is that time isn't like space - a watch doesn't tell you that "the time is 10 am" in the same way in which a diagram tells you that a particular point is on a particular line, precisely because the watch changes with the time. Now this one can safely be handed over to the philosophers.
Cruisin'. After all that, I think we've earned ourselves a nice refreshing glass of jazz-rock fusion. I think the lesson of this week's episode is to use lowercase for specific values of x and uppercase for general X, or some such sensible convention.
Update: I can't believe I forgot about this important generalisation of the stopped clock theorem.
If W is a useful estimate of the correct time then the stopped clock provides a useful estimate of the correct time for a period of positive measure. It surely depends on f and f is also a function of D, where D is displacement from the observer, dD/dT and d^2D/dT^2.
ReplyDeletePossibly time for a drink and a return to Granger causality
then the stopped clock provides a useful estimate of the correct time for a period of positive measure
ReplyDeleteBluebottle What time is it Eccles?
Eccles Err, just a minute. I, I've got it written down 'ere on a piece of paper. A nice man wrote the time down for me this morning.
Bluebottle Ooooh, then why do you carry it around with you Eccles?
Eccles Well, umm, if a anybody asks me the ti-ime, I ca-can show it to dem.
Bluebottle Wait a minute Eccles, my good man...
Eccles What is it fellow?
Bluebottle It's writted on this bit of paper, what is eight o'clock, is writted.
Eccles I know that my good fellow. That's right, um, when I asked the fella to write it down, it was eight o'clock.
Bluebottle Well then. Supposing when somebody asks you the time, it isn't eight o'clock?
Eccles Ah, den I don't show it to dem.
Bluebottle Ooohhh...
Eccles [Smacks lips] Yeah.
Bluebottle Well how do you know when it's eight o'clock?
Eccles I've got it written down on a piece of paper!
Bluebottle Oh, I wish I could afford a piece of paper with the time written on.
Eccles Oohhhh.
Bluebottle 'Ere Eccles?
Eccles Yah.
Bluebottle Let me hold that piece of paper to my ear would you? - 'Ere. This piece of paper ain't goin'.
Eccles What? I've been sold a forgery!
Bluebottle No wonder it stopped at eight o'clock.
Eccles Oh dear.
Bluebottle You should get one of them tings my grandad's got.
Eccles Oooohhh?
Bluebottle His firm give it to him when he retired.
Eccles Oooohhh.
Bluebottle It's one of dem tings what it is that wakes you up at eight o'clock, boils the kettil, and pours a cuppa tea.
Eccles Ohhh yeah! What's it called? Um.
Bluebottle My granma.
Eccles Ohh... Ohh, ah wait a minute. How does she know when it's eight o'clock?
Bluebottle She's got it written down on a piece of paper!
Granger Causation in the world of post-punk
ReplyDeleteI still think Ajay's application of Shannon information was pretty conclusive. The only unexpected information you receive from the stopped clock is the fact that its hands don't move.
ReplyDeleteyou have a watch, which tells you that the time is W=f(T) (ie, unlike a stopped clock, your watch tells you a different time depending on what time it is) ... watches provide information about the time because W=f(T) is an estimator of T
ReplyDeleteI think you need to say more about why your estimator applies in some cases but not in others.
The 'stopped clock' might just be extraordinarily mal-adjusted and slow running; that is, on close inspection, it might turn out to be related to the machinery of the universe in a way you deem appropriate for a timepiece, and then you'd have to count it together with your watch.
This is an idealised, abstract mathematical entity kind of "stopped clock". It's the stopped clock equivalent of a friction-free slope, a rigid body or homo economicus.
ReplyDeleteHere's the echt Goons experience, no need for Flash:
ReplyDeletehttp://en.wikipedia.org/wiki/File:What_time_is_it_Eccles.ogg
Here's a proper piece of paper:
http://www.whattimeisiteccles.com/
But if these are all ideal clocks, then this:
ReplyDeleteIf someone asks you the time, then you can't answer in the form of a function over the real numbers, it would take too long
isn't a problem.
Oh dear.
ReplyDeletestopped clocks are always wrong because a stopped clock says T=X for some particular X and for any particular X, P(T=X) is probability zero
No, stopped clocks are almost always wrong on this basis. If you'd said that at the beginning I doubt anyone would have disagreed...
More pedantry time...
ReplyDeleteNow that we're in estimator world I think you're in genuine trouble, because your claim has now become "a stopped clock, unlike a non-stopped clock, is a poor estimator of the time." But now you are claiming nothing interesting about the saying, since the original saying on this translation is "even poor estimators sometimes generate decent estimates."
You seem to want to claim that it's not an estimator of the time, but of course it is, since there's nothing wrong with the constant function. There are many phenomena that are well-estimated by constants; my "will the sun rise tomorrow" estimator is extremely reliable, and of course myopic forecasts more generally count here too. You can work yourself down to "a stopped clock is a poor enough estimator that its estimates should be discounted even when they randomly happen to be close to reality" which is of course true but being as this is essentially the argument against listening to people who were right about the Iraq War I'm amused to read it on your blog.
I think Larry is right about "almost always". I don't agree with Dennis though; I didn't assert that a stopped clock wasn't an estimator (unless I did so accidentally through poor notation). But unless there's something wrong with my Bayesian proof (and I don't think there is), I've shown that a stopped clock is an estimator with zero information value - your expectation conditional on this "estimator" is the same as your unconditional expectation.
ReplyDelete"Will the sun rise tomorrow" is only a constant function relative to a particular filtration of the time series of sun-observations and thus I don't think is analogous, and a myopic forecast clearly isn't subject to the same Bayesian argument and so I don't think is analogous either.
In general I think the stopped-clock argument against listening to people who were right about the Iraq War is a valid schema - the problem with it is that it is typically applied with a false initial premis in the shape of an assertion about who was and wasn't a stopped clock. George Galloway, for example, was right about Iraq, but I am indeed not interested in his opinions about it.
You're most welcome. Having made that amendment, I'm not sure exactly what it is you're trying to prove any more.
ReplyDeleteBut on your Bayesian argument, if you look at your equation 2), it looks very much to me as if it reads "0=0", since P(T=X|a stopped clock says T=X) is 0 on the left and P(T=X) is 0 on the right.
I hope P(a stopped clock says T=X) is not also zero otherwise the whole thing really is buggered beyond repair, but I suspect it might be.
Either way, if you think you're extracting anything useful out of that, I would humbly suggest that you're mistaken...
I hope P(a stopped clock says T=X) is not also zero otherwise the whole thing really is buggered beyond repair, but I suspect it might be.
ReplyDeleteReplace all the probability masses (inappropriate for a continuous variable) with probability densities (appropriate for a continuous variable) and the argument is fine.
Yes, P(T=X) in that line isn't zero, it's a continuous function of X. Which might or might not have the value zero but it's not the number zero.
ReplyDeletesorry, the words "when evaluated at any particular value" dropped off that last comment - P(T=X) has to integrate to 1 because it's a probability density function - ie, unless you're Doctor Who, you have to believe that it is some time, somewhere in the universe.
ReplyDeleteP(a stopped clock says T=X) is a pdf with all of its probability mass concentrated on the time that the clock says and 0 elsewhere.
And finally, I clearly am extracting something useful out of it - as Ajay and Alex say, the proof that an unchanging signal doesn't change your posterior belief is a fundamental building block of information theory. It's not something original but what did you expect. Anyone wanting to claim that a stopped clock gives you information is going to have to reinvent a version of information theory different from Shannon's.
ReplyDeleteIt almost looks like Dennis is on the cusp of that claim, but looking carefully at it he's making a statement about forecasting, and something doesn't have to convey information in order to make a good forecast.
Sorry, I thought we were still in the realm of pretending that pdfs don't exist. (See label for why that's not my fault.)
ReplyDeleteYes, ok, I'll buy it. Final comment, in conclusion 4), not if W(T)=T, which opens up the possibility of a perfect watch.
2) is a stipulation or definition: a clock that is stopped may convey information by the fact of its stoppedness and the time at which it stopped.
ReplyDelete3) rather depends upon f but there are a class of functions for which it is true with a reasonable definition and an estimation regime, that is the estimate W(T) is not necessarily the best estimate of T using W and some other observations as an information set. There is an even larger class of functions that will appear to provide a useful estimate until they don't.
4) is most interesting. As Larry Teabag says allowing idealised watches and Real time there could be a perfect watch where W = T but the finite speed of light, relativistic considerations and neural computation lags mean this is still true (or would be if watches weren't typically finite range output (because ticking) devices and that for these purposes time has effectively one of 7200 values) A continuously rotating watch would therefore indeed be almost always wrong and could be made without expensive tourbillons and escapements. With that I'm going to send in a mockup of "The most accurate watch in the world that is almost always wrong" to How To Spend It and make my fortune. It's not quite as good as a watch made from steel recovered from the Titanic that cannot be exposed to normal air for fear of corrosion but that's been done and it does sound almost as useless.
(BTW in a simple, therefore almost certainly misspecified, VAR with lags selected by the Akaike criterion, pride does indeed Granger cause "the Fall")
a clock that is stopped may convey information by the fact of its stoppedness and the time at which it stopped.
ReplyDeleteyes, but not information about what time it is.
Since there is so much reader interest in this topic, I plan to venture next week into the actual philosophy of time, addressing such thoughts as that on a Heraclitean basis, a stopped clock with the hands pointing to 1000, is actually indicating the time 10am on the day that it stopped, which is numerically distinct from all the other 10ams there have been or will be. And perhaps more ...
It might well convey information about what time it is, for example if the clock has batteries that you know last for 5 days and you know when they were last replaced, you have reason to believe that it is more than 5 days since you replaced the batteries but we can rule out such cases as beside the point.
ReplyDeleteNumerically distinct is exactly what the 10 o'clocks are not. It doesn't necessarily mean am or pm. Or is a 12 hour clock without a date function also not telling the time?
Zeno might also be quite good on stopped clocks especially if we aren't being clear about the continuity of time.
Since there is so much reader interest in this topic
ReplyDeleteOh gawd. Me! Me! I'm not interested in this topic - I'm not interested at all. I didn't do calculus*, and as much as I would like** to plug this gap in my knowledge I'm not about to do it now. Can't we talk about Iraq casualties, or blog moderation, or favourite Welsh breweries***, or something?
*Or whatever the hell it is that you people are doing with the brackets and everything and no please don't explain thanks.
**I really would. I did A Level English, French & Latin, and I've been regretting not doing English, Maths and Latin more or less ever since. Just not right now.
***Conwy by a nose over Felinfoel. Apparently Buckley's 'merged' in 1997 with Brain's, who closed the brewery the next year. Shame - bloody nice pint that was. Conwy are producing some good stuff, anyway.
God, do they have a brewery in Conwy these days? Shows how long it's been since I've been to the henwlad.
ReplyDeleteFWIW, this is all building up to a massive troll post, designed to test out a hypothesis of mine that taken as a group, Dr Who fans are the worst-tempered people on the internet.
yes, but not information about what time it is.
ReplyDeleteOK, it looks as though we're settling on stopped clocks being stopped by stipulation. The thing is, if you stipulate that something gives no information about 'the time' then it's surely not meaningful to talk of that thing being right or wrong about 'the time'. The argument is just wheel spinning.
It's nice to see that epistemic relativism still has the power to annoy!
ReplyDeleteDaniel is pulling a standard philosophy-of-science gambit whereby plain propositions about the world (in this case, whether a clock is right) are reinterpreted as being about our knowledge of the world (in this case, whether a clock provides information about the time). This technique turns a boring truth ("a stopped clock is right twice a day") into an interesting paradox ("stopped clocks are always wrong"). By quick-stepping back and forth between these interpretations you can keep trolling as long as you like.
the correct answer to "What time is it?" is "Time you got a watch".
Who wear a watch these days?
Being a shut-in, and in constant great pain, I'll settle for any love I get, even troll love.
ReplyDeleteI love my commenters sometimes. I have a set of Dr Who DVDs on eBay if you want to buy them, Charlie.
ReplyDelete