Which of these do you think is problematic (I have clarified above what I would do; see 2nd bullet)?
Giving weights to each of the theories of consciousness (e.g. as I described here).
Determining the overall moral weight distribution from the weighted mean of the moral weight distributions of the various theories of consciousness.
I think the first probably makes some unverifiable and unjustified assumptions. Why normalize by the variance in particular?
It seems similar to variance voting, although variance voting normalizes by the standard deviation instead of the variance, to ensure each has variance 1 (Var(aX)=a2Var(X), so Var(X/√Var(X))=Var(X)/Var(X)=1). It is one approach to moral uncertainty, but there are others, like the parliamentary approach. Why normalize by the variance or standard deviation and not some other measure, for example?
You are taking expected values over products of values, one of which is the moral weight, though, right?
Here is how I would think about it, with variables with units.
Mh = measured units of human welfare from the intervention, in QALYs (total, not per year or per capita, for simplicity)
Mc = measured units of chicken welfare from the intervention, in pQALYs (total, not per year or per capita, for simplicity)
Vh,t = value per measured unit of human welfare on the theory of consciousness t, in units valt/QALY
Vc,t = value per measured unit of chicken welfare on the theory of consciousness t, in units valt/pQALY
Vh,t/Vc,t, basically the relative moral weight of chickens wrt humans humans wrt chickens, in units pQALY/QALY
Vc,t/Vh,t, basically the relative moral weight of humans wrt chickens chickens wrt humans, in units QALY/pQALY
You’re trying to calculate, where T is a random variable for the theory of consciousness,
E[Vc,TMc+Vh,TMh]=E[Vc,TMc]+E[Vh,TMh]
but first, the above means taking expectations over values with units valt for different t, like adding values in Fahrenheit and values in Celsius (or grams), so you need to condition on a theory of consciousness T=t first. So, let’s look at, for each theory t,
E[Vc,TMc|T=t]+E[Vh,TMh|T=t]=E[Vc,tMc]+E[Vh,tMh]
Then, I think you’re effectively assuming Vh,t=1 and that it’s unitless, and so you infer the following:
Vc,t=Vc,t/Vh,tVc,tMc=Vc,t/Vh,tMc
But even on a fixed theory of consciousness, there could still be empirical uncertainty about Vh,t, so you shouldn’t assume Vh,t is fixed.
I mainly wanted to understand whether you tought the simple fact of attributing weights and then calculating a weighted mean might be intrinsically problematic. Weighting the various moral weight distributions by the reciprocal of their variances is just my preferred solution. That being said:
It is coherent with a bayesian approach (see here).
It mitigates Pascal’s Mugging (search for “Pascal’s Mugging refers” in this GiveWell’s article). This would not be the case if one used the standard deviation instead of the variance. For a distribution k X:
The mean E(k X) is k E(X).
The variance V(k X) is k^2 V(X).
Therefore the ratio between the mean and standard deviation is inversely proportional to k.
The standard deviation V(k X)^0.5 is k V(X)^0.5.
Therefore the ratio between the mean and standard deviation does not depend on k.
It facilitates the calculation of the weights (as they are solely a function of the distributions).
You are taking expected values over products of values, one of which is the moral weight, though, right?
I am calculating the mean of R = “negative utility of poultry living time as a fraction of the utility of human life” from the mean of R_PH, which is defined here.
Vh,t/Vc,t, basically the relative moral weight of chickens wrt humans, in units pQALY/QALY
I think you meant “humans wrt chickens” (not “chickens wrt humans”), as “h” is in the numerator.
Vc,t/Vh,t, basically the relative moral weight of humans wrt chickens, in units QALY/pQALY
I think you meant “chickens wrt humans” (not “humans wrt chickens”), as “c” is in the numerator.
But even on a fixed theory of consciousness, there could still be empirical uncertainty about Vh,t, so you shouldn’t assume Vh,t is fixed.
Let me try to match my variables to yours, based on what I defined here:
R_PH (= R_HP), which is what I am trying to calculate, is akin to (Vc,tMc)/(Vh,tMh), not Vc,tMc+Vh,tMh.
Mc is akin to T*Q, where:
T = “poultry living time per capita (pyear/person/year)”.
Q = “quality of the living conditions of poultry (-pQALY/pyear)”.
Mh is akin to H = “utility of human life (QALY/person/year)”.
PH = “moral weight of poultry birds relative to humans (QALY/pQALY)” is Vc,t/Vh,t.
I did not set Vh,t to 1, because my PH represents Vc,t/Vh,t, not Vc,t.
Note that if you divide a random variable with units by its variance, the result will not be unitless (it’ll have the reciprocal units of the random variable), and so you would need to make sure the units match before adding. In this case, with the notation I introduced, you’d have different theory-specific units you’re trying to sum across, and this wouldn’t work. Dividing by the standard deviation or the range or some other statistics with the same units as the random variable would work.
I think you meant “humans wrt chickens” (not “chickens wrt humans”), as “h” is in the numerator.
(...)
I think you meant “chickens wrt humans” (not “humans wrt chickens”), as “c” is in the numerator.
Woops, yes, good catch.
R_PH (= R_HP), which is what I am trying to calculate, is akin to (Vc,tMc)/(Vh,tMh), not Vc,tMc+Vh,tMh.
I think this is the problem, then. You should not take and use the expected value of the ratio (Vc,tMc)/(Vh,tMh), for basically the reasons I gave previously that you should not in general (except when you condition on enough things or make certain explicit and justified assumptions) take expected values of relative moral weights. Indeed, these are moral weights, just aggregates. When you’re interested in the impacts of an intervention on different individuals, you would sum the impacts over each individual, and then take the expected value (or sum expected individual impacts), i.e. E[Vc,tMc+Vh,tMh]. E[(Vc,tMc)/(Vh,tMh)] isn’t 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.
(You could estimate E[Vc,tMc]/E[Vh,tMh] instead, though, and that could be useful, if you also have an estimate of E[Vh,tMh].)
Note that if you divide a random variable with units by its variance, the result will not be unitless (it’ll 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/QALY [or QALY/pQALY].
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)] isn’t 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/H (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/pQALY.
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 shouldn’t 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 you’ve assumed this can’t 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), you’d 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 shouldn’t 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), you’d 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 = “utility of poultry living time (-pQALY/person/year)”. However, as you seem to note, this would be even more prone to error (“more suspicious”). 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 think the first probably makes some unverifiable and unjustified assumptions. Why normalize by the variance in particular?
It seems similar to variance voting, although variance voting normalizes by the standard deviation instead of the variance, to ensure each has variance 1 (Var(aX)=a2Var(X), so Var(X/√Var(X))=Var(X)/Var(X)=1). It is one approach to moral uncertainty, but there are others, like the parliamentary approach. Why normalize by the variance or standard deviation and not some other measure, for example?
You are taking expected values over products of values, one of which is the moral weight, though, right?
Here is how I would think about it, with variables with units.
Mh = measured units of human welfare from the intervention, in QALYs (total, not per year or per capita, for simplicity)
Mc = measured units of chicken welfare from the intervention, in pQALYs (total, not per year or per capita, for simplicity)
Vh,t = value per measured unit of human welfare on the theory of consciousness t, in units valt/QALY
Vc,t = value per measured unit of chicken welfare on the theory of consciousness t, in units valt/pQALY
Vh,t/Vc,t, basically the relative moral weight of
chickens wrt humanshumans wrt chickens, in units pQALY/QALYVc,t/Vh,t, basically the relative moral weight of
humans wrt chickenschickens wrt humans, in units QALY/pQALYYou’re trying to calculate, where T is a random variable for the theory of consciousness,
E[Vc,TMc+Vh,TMh]=E[Vc,TMc]+E[Vh,TMh]but first, the above means taking expectations over values with units valt for different t, like adding values in Fahrenheit and values in Celsius (or grams), so you need to condition on a theory of consciousness T=t first. So, let’s look at, for each theory t,
E[Vc,TMc|T=t]+E[Vh,TMh|T=t]=E[Vc,tMc]+E[Vh,tMh]Then, I think you’re effectively assuming Vh,t=1 and that it’s unitless, and so you infer the following:
Vc,t=Vc,t/Vh,tVc,tMc=Vc,t/Vh,tMcBut even on a fixed theory of consciousness, there could still be empirical uncertainty about Vh,t, so you shouldn’t assume Vh,t is fixed.
Thanks for the reply!
I mainly wanted to understand whether you tought the simple fact of attributing weights and then calculating a weighted mean might be intrinsically problematic. Weighting the various moral weight distributions by the reciprocal of their variances is just my preferred solution. That being said:
It is coherent with a bayesian approach (see here).
It mitigates Pascal’s Mugging (search for “Pascal’s Mugging refers” in this GiveWell’s article). This would not be the case if one used the standard deviation instead of the variance. For a distribution k X:
The mean E(k X) is k E(X).
The variance V(k X) is k^2 V(X).
Therefore the ratio between the mean and standard deviation is inversely proportional to k.
The standard deviation V(k X)^0.5 is k V(X)^0.5.
Therefore the ratio between the mean and standard deviation does not depend on k.
It facilitates the calculation of the weights (as they are solely a function of the distributions).
I am calculating the mean of R = “negative utility of poultry living time as a fraction of the utility of human life” from the mean of R_PH, which is defined here.
I think you meant “humans wrt chickens” (not “chickens wrt humans”), as “h” is in the numerator.
I think you meant “chickens wrt humans” (not “humans wrt chickens”), as “c” is in the numerator.
Let me try to match my variables to yours, based on what I defined here:
R_PH (= R_HP), which is what I am trying to calculate, is akin to (Vc,tMc)/(Vh,tMh), not Vc,tMc+Vh,tMh.
Mc is akin to T*Q, where:
T = “poultry living time per capita (pyear/person/year)”.
Q = “quality of the living conditions of poultry (-pQALY/pyear)”.
Mh is akin to H = “utility of human life (QALY/person/year)”.
PH = “moral weight of poultry birds relative to humans (QALY/pQALY)” is Vc,t/Vh,t.
I did not set Vh,t to 1, because my PH represents Vc,t/Vh,t, not Vc,t.
Note that if you divide a random variable with units by its variance, the result will not be unitless (it’ll have the reciprocal units of the random variable), and so you would need to make sure the units match before adding. In this case, with the notation I introduced, you’d have different theory-specific units you’re trying to sum across, and this wouldn’t work. Dividing by the standard deviation or the range or some other statistics with the same units as the random variable would work.
Woops, yes, good catch.
I think this is the problem, then. You should not take and use the expected value of the ratio (Vc,tMc)/(Vh,tMh), for basically the reasons I gave previously that you should not in general (except when you condition on enough things or make certain explicit and justified assumptions) take expected values of relative moral weights. Indeed, these are moral weights, just aggregates. When you’re interested in the impacts of an intervention on different individuals, you would sum the impacts over each individual, and then take the expected value (or sum expected individual impacts), i.e. E[Vc,tMc+Vh,tMh]. E[(Vc,tMc)/(Vh,tMh)] isn’t 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.
(You could estimate E[Vc,tMc]/E[Vh,tMh] instead, though, and that could be useful, if you also have an estimate of E[Vh,tMh].)
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/H (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/pQALY.
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 shouldn’t 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 you’ve assumed this can’t 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), you’d 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 = “utility of poultry living time (-pQALY/person/year)”. However, as you seem to note, this would be even more prone to error (“more suspicious”). 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!