# Economics and similar, for the sleep-deprived

Update: seemingly not

Update: Oh yeah!

### Thursday, April 22, 2010

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.