Note that if you divide a random variable with units by its variance, the result will not be unitless (itâll have the reciprocal units of the random variable), and so you would need to make sure the units match before adding.
I agree, but I do not expect this to be a problem:
A priori, I would expect any theory of consciousness to produce a mean moral weight of poultry birds relative to humans in pQALY/âQALY [or QALY/âpQALY].
Moreover, if this is not the case, it seems to me that weighting the various moral weight distributions by the reciprocal of their standard deviations (or any other metric, with or without units) would also not be possible:
As you point out, the terms in the numerator would both be unitless, and therefore adding them would not be a problem.
However, the terms in the denominator would have different units. For example, for 2 moral weight distributions MWA and MWB with units A and B, the terms in the denominator would have units A^-1 and B^-1.
Dividing by the standard deviation or the range or some other statistics with the same units as the random variable would work.
As explained above, I do not see how it would be possible to combine the results of different theories if these cannot be expressed in the same units.
E[(Vc,tMc)/(Vh,tMh)] isnât generally useful for this unless, without further assumptions that are unjustified and plausibly wrong, e.g.(Vc,tMc)/(Vh,tMh) and Vh,tMh are independent.
In order to calculate something akin to (Vc,tMc)+(Vh,tMh) instead of (Vc,tMc)/(Vh,tMh), I would compute S_PH = T*PH*Q + H instead of R_PH = T*PH*Q/âH (see definitions here), assuming:
All the distributions I defined in Methodology are independent.
All theories of consciousness produce a distribution for the moral weight of poultry birds relative to humans in QALY/âpQALY.
PH represents the weighted mean of all these distributions.
Under these assumption (I have added the 1st to Methodology, and the 2nd and 3rd to Moral weight of poultry), E(R_PH) is a good proxy for E(S_PH) (which is what we care about, as you pointed out):
S_PH = (R_PH + 1) H.
I defined H as a constant.
Consequently, the greater is E(R_PH), the greater is E(S_PH).
Normalizing PH (or HP) by its variance on each theory could introduce more arbitrarily asymmetric treatment between animals, overweight theories where the variance is lowest for reasons unrelated to the probability you assign to them (e.g. on aome theories, capacity for welfare may be close to constant), and is pretty ad hoc. I would recommend looking into more general treatments of moral uncertainty instead, and just an approach like variance voting or moral parliament, applied to your whole expected value over outcomes, not PH (or HP).
As I discussed in other comments and the other links discussing the two envelopes problem, H should not be defined as constant (or independent from or uncorrelated with PH) without good argument, and on any given theory of consciousness, it seems pretty unlikely to me, since we still have substantial empirical uncertainty about human (and chicken) brains on any theory of consciousness. You can estimate the things you want to this way, but the assumptions are too strong, so you shouldnât trust the estimates, and this is partly why you get the average chicken having greater capacity for welfare than the average human in expectation. Sometimes PH is lower than on some empirical possibilities not because P is lower on those possibilities, but because H is greater on them, but youâve assumed this canât be the case, so may be severely underweighting human capacity for welfare.
If you instead assumed P were constant (although this would be even more suspicious), youâd get pretty different results.
I would recommend looking into more general treatments of moral uncertainty instead, and just an approach like variance voting or moral parliament, applied to your whole expected value over outcomes, not PH (or HP).
I will do, thanks!
You can estimate the things you want to this way, but the assumptions are too strong, so you shouldnât trust the estimates, and this is partly why you get the average chicken having greater capacity for welfare than the average human in expectation.
Note that it is possible to obtain a mean moral weight much smaller than 1 with exactly the same method, but different parameters. For example, changing the 90th percentile of moral weight of poultry birds if these are moral patients from 10 to 0.1 results in a mean moral weight of 0.02 (instead of 2). I have added to this section one speculative explanation for why estimates for the moral weight tend to be smaller.
If you instead assumed P were constant (although this would be even more suspicious), youâd get pretty different results.
I have not defined P, but I agree I could, in theory, have estimated R_PH (and S_PH) based on P = âutility of poultry living time (-pQALY/âperson/âyear)â. However, as you seem to note, this would be even more prone to error (âmore suspiciousâ). The two methods are mathematically equivalent under my assumptions, and therefore it makes much more sense to me as a human to use QALY (instead of pQALY) as the reference unit.
Michael, once again, thank you so much for all these comments!
I agree, but I do not expect this to be a problem:
Moreover, if this is not the case, it seems to me that weighting the various moral weight distributions by the reciprocal of their standard deviations (or any other metric, with or without units) would also not be possible:
As you point out, the terms in the numerator would both be unitless, and therefore adding them would not be a problem.
However, the terms in the denominator would have different units. For example, for 2 moral weight distributions MWA and MWB with units A and B, the terms in the denominator would have units A^-1 and B^-1.
As explained above, I do not see how it would be possible to combine the results of different theories if these cannot be expressed in the same units.
In order to calculate something akin to (Vc,tMc)+(Vh,tMh) instead of (Vc,tMc)/(Vh,tMh), I would compute S_PH = T*PH*Q + H instead of R_PH = T*PH*Q/âH (see definitions here), assuming:
All the distributions I defined in Methodology are independent.
All theories of consciousness produce a distribution for the moral weight of poultry birds relative to humans in QALY/âpQALY.
PH represents the weighted mean of all these distributions.
Under these assumption (I have added the 1st to Methodology, and the 2nd and 3rd to Moral weight of poultry), E(R_PH) is a good proxy for E(S_PH) (which is what we care about, as you pointed out):
S_PH = (R_PH + 1) H.
I defined H as a constant.
Consequently, the greater is E(R_PH), the greater is E(S_PH).
Normalizing PH (or HP) by its variance on each theory could introduce more arbitrarily asymmetric treatment between animals, overweight theories where the variance is lowest for reasons unrelated to the probability you assign to them (e.g. on aome theories, capacity for welfare may be close to constant), and is pretty ad hoc. I would recommend looking into more general treatments of moral uncertainty instead, and just an approach like variance voting or moral parliament, applied to your whole expected value over outcomes, not PH (or HP).
As I discussed in other comments and the other links discussing the two envelopes problem, H should not be defined as constant (or independent from or uncorrelated with PH) without good argument, and on any given theory of consciousness, it seems pretty unlikely to me, since we still have substantial empirical uncertainty about human (and chicken) brains on any theory of consciousness. You can estimate the things you want to this way, but the assumptions are too strong, so you shouldnât trust the estimates, and this is partly why you get the average chicken having greater capacity for welfare than the average human in expectation. Sometimes PH is lower than on some empirical possibilities not because P is lower on those possibilities, but because H is greater on them, but youâve assumed this canât be the case, so may be severely underweighting human capacity for welfare.
If you instead assumed P were constant (although this would be even more suspicious), youâd get pretty different results.
I will do, thanks!
Note that it is possible to obtain a mean moral weight much smaller than 1 with exactly the same method, but different parameters. For example, changing the 90th percentile of moral weight of poultry birds if these are moral patients from 10 to 0.1 results in a mean moral weight of 0.02 (instead of 2). I have added to this section one speculative explanation for why estimates for the moral weight tend to be smaller.
I have not defined P, but I agree I could, in theory, have estimated R_PH (and S_PH) based on P = âutility of poultry living time (-pQALY/âperson/âyear)â. However, as you seem to note, this would be even more prone to error (âmore suspiciousâ). The two methods are mathematically equivalent under my assumptions, and therefore it makes much more sense to me as a human to use QALY (instead of pQALY) as the reference unit.
Michael, once again, thank you so much for all these comments!