I admit that I’ve learned about the KL divergence just now and through the wiki-link, and that my math in general is not so profound. But as it’s not about calculation but about the reasoning behind the calculation, I suppose I can have my word:
The wiki-entry mentions that
Typically P represents the “true” distribution of data, observations, or a precise calculated theoretical distribution. The measure Q typically represents a theory, model, description, or approximation of P.
So P here is 10^-18 and Q is either 0 or 0.5.
What your epistemic rationalist has done seems like falling pray to the bias of anchoring and adjusting. The use of mathematical equations just makes the anchoring mistake look more fomal; it’s not less wrong in any way. So while the instrumental rationalist might have a reason to choose the arbitary figure of 1⁄2 (it makes his decisions be more simple, for example) the epistemic rationalist does not.
If the epistemic rationalist is shown the two figures of 0 and 1⁄2 and is asked what approximation is “better” he would probably say 0. And that’s for several reason:
First of all, if he is an epistemic rationalist and thus trueseeking, he wouldn’t use the KL equation at all. The KL takes something accurate (or true) P and makes it less accurate (or less true) KLD, and that’s exactly against what he is seeking—having more accurate and true results.
But you tell me he has to choose between either “0“ or “1/2”. Well, if he has to chooce between one of these numbers he will still not choose to use the KL equation. The wiki mentions that the Q in the equation typically stands for ”… a theory, model, description, or approximation of P” while the number “1/2” in your example is none of these but an arbitary number—this equation, then, does not fit the situation. He will use a different mathematical method, let’s say, subtraction, and see the absolute value of what difference is smaller, in which case it will be 0′s.
Also, since 1⁄2 and 0 are arbitary numbers, an epistemic rationalist would know better than use any of these numbers in any equation, since it will produce a result that is accurate just as if would use any other two arbitary numbers. He would know that he should do his own calculations—ignoring the numbers 0 and 1⁄2 - and then compare his result to the numbers he is “offered” (0 and 1⁄2) and choose the closest number to his own calculation. Since he knows that the “true” probability is 10^-18 he will choose the closest number to his result which seems to be 0.
Of course, everything that I said about “1/2” above holds true about “0″.
(I’m sorry in advance if my mathematical explentation are unclear or clumsy. If I explain arguments through math badly, then I explain arguments through math in English much worst as I was studying mathematics in a different language)
I admit that I’ve learned about the KL divergence just now and through the wiki-link, and that my math in general is not so profound. But as it’s not about calculation but about the reasoning behind the calculation, I suppose I can have my word:
The wiki-entry mentions that
So P here is 10^-18 and Q is either 0 or 0.5.
What your epistemic rationalist has done seems like falling pray to the bias of anchoring and adjusting. The use of mathematical equations just makes the anchoring mistake look more fomal; it’s not less wrong in any way. So while the instrumental rationalist might have a reason to choose the arbitary figure of 1⁄2 (it makes his decisions be more simple, for example) the epistemic rationalist does not. If the epistemic rationalist is shown the two figures of 0 and 1⁄2 and is asked what approximation is “better” he would probably say 0. And that’s for several reason: First of all, if he is an epistemic rationalist and thus trueseeking, he wouldn’t use the KL equation at all. The KL takes something accurate (or true) P and makes it less accurate (or less true) KLD, and that’s exactly against what he is seeking—having more accurate and true results. But you tell me he has to choose between either “0“ or “1/2”. Well, if he has to chooce between one of these numbers he will still not choose to use the KL equation. The wiki mentions that the Q in the equation typically stands for ”… a theory, model, description, or approximation of P” while the number “1/2” in your example is none of these but an arbitary number—this equation, then, does not fit the situation. He will use a different mathematical method, let’s say, subtraction, and see the absolute value of what difference is smaller, in which case it will be 0′s. Also, since 1⁄2 and 0 are arbitary numbers, an epistemic rationalist would know better than use any of these numbers in any equation, since it will produce a result that is accurate just as if would use any other two arbitary numbers. He would know that he should do his own calculations—ignoring the numbers 0 and 1⁄2 - and then compare his result to the numbers he is “offered” (0 and 1⁄2) and choose the closest number to his own calculation. Since he knows that the “true” probability is 10^-18 he will choose the closest number to his result which seems to be 0.
Of course, everything that I said about “1/2” above holds true about “0″.
(I’m sorry in advance if my mathematical explentation are unclear or clumsy. If I explain arguments through math badly, then I explain arguments through math in English much worst as I was studying mathematics in a different language)