I have no useful information here, so a uniform prior seems reasonable, in which case the analysis from a linear model holds.
I’m not sure what the uniform prior means in this case and how the conclusion follows—can you expand?
But anyway, granting this for the moment, in an actual real-life situation when you contemplate actual charities, you do have all sorts of useful information about them, for example the information that allows you to estimate their effectiveness. This information will probably also throw some light on how the effectiveness changes over time, and so let you determine whether the linear approximation is good.
This seems especially true when the differences in first derivatives between different charities is large, such that a second order correction would have to also be very large in order to sway the analysis.
I agree that when first derivatives are wildly different according to your utility function, it’s a no-brainer (barring situations with huge second order effects that’ll show up as very weird features of the landscape) to put all your budget into one of them. What I object to is slam-dunk arguing along the lines of “Landsburg has a solid math proof that the rational thing to do is to take first derivatives, compare them, and act on the result. If you don’t agree, you’re an obscurantist or you fail to grok the math”.
But anyway, granting this for the moment, in an actual real-life situation when you contemplate actual charities, you do have all sorts of useful information about them, for example the information that allows you to estimate their effectiveness. This information will probably also throw some light on how the effectiveness changes over time, and so let you determine whether the linear approximation is good.
If you have additional information beyond the first derivatives then by all means use it. Use all the information you have. However, in general you need more information to get an equally good approximation to higher order derivatives. Cross terms especially seem like they would be very difficulty to gauge empirically. In light of that I would be very skeptical of high confidence estimates for higher order terms, especially if they conveniently twist the math to allow for a desirable outcome.
Consider the simpler case with only two charities and total utility U(X,Y). For simplicity assume the second order derivatives are constant, and that the probability that
where R is some finite interval symmetric about 0. We can actually take R to be the whole real line, but the math becomes hairier. Now, each of these integrals is 0, because the uniform distribution is symmetric about each axis. The symmetry is all that is needed actually, not uniformity, so you could weaken the assumptions.
I’m not sure what the uniform prior means in this case and how the conclusion follows—can you expand?
But anyway, granting this for the moment, in an actual real-life situation when you contemplate actual charities, you do have all sorts of useful information about them, for example the information that allows you to estimate their effectiveness. This information will probably also throw some light on how the effectiveness changes over time, and so let you determine whether the linear approximation is good.
I agree that when first derivatives are wildly different according to your utility function, it’s a no-brainer (barring situations with huge second order effects that’ll show up as very weird features of the landscape) to put all your budget into one of them. What I object to is slam-dunk arguing along the lines of “Landsburg has a solid math proof that the rational thing to do is to take first derivatives, compare them, and act on the result. If you don’t agree, you’re an obscurantist or you fail to grok the math”.
If you have additional information beyond the first derivatives then by all means use it. Use all the information you have. However, in general you need more information to get an equally good approximation to higher order derivatives. Cross terms especially seem like they would be very difficulty to gauge empirically. In light of that I would be very skeptical of high confidence estimates for higher order terms, especially if they conveniently twist the math to allow for a desirable outcome.
Consider the simpler case with only two charities and total utility U(X,Y). For simplicity assume the second order derivatives are constant, and that the probability that
frac{partial2U}{partialx2}=z_0,frac{partial2U}{partialxpartialy}=z_1,frac{partial2U}{partialy2}=z_2
is given by
Then the second order contribution to
is given by the integral over all possible second derivatives
which equals
where R is some finite interval symmetric about 0. We can actually take R to be the whole real line, but the math becomes hairier. Now, each of these integrals is 0, because the uniform distribution is symmetric about each axis. The symmetry is all that is needed actually, not uniformity, so you could weaken the assumptions.