Economics and similar, for the sleep-deprived

A subtle change has been made to the comments links, so they no longer pop up. Does this in any way help with the problem about comments not appearing on permalinked posts, readers?

Update: seemingly not

Update: Oh yeah!


Thursday, April 22, 2010

 
Thursday Music Link

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 always wrong wrong with probability 1 (thanks, Larry) because a stopped clock says T=X for some particular X and for any particular X, P(T=X) is probability zero.
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.

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