I think you should just expect this, and the answer is not to deny the possibility of St Petersberg lotteries in objective quantities, but to figure out a good way to deal with them (e.g. ignore small enough probabilities, use a bounded utility function, use commitments, use bracketing of some form), or accept that they raise difficult normative problems.

Why do you think I should expect all distributions to have infinite or undefined expected value instead of rejecting such distributions?

Are you saying you believe in the existence with 100% credence in anything (such as ghosts) that is not ruled out by current evidence, as long as its effects couid be arbitrarily small and are so far indistinguishable from its nonexistence? Or must it also have no important normative implications (under classical utilitarianism?)?

I corrected my answers in my past comment. You can see what I crossed out, and wrote after "Edit after Michael's comment just below".

Some could be vengeful and want harm to fall upon those that have caused them harm in their lives (whether or not they enact it themselves).

For the "ghosts [I mentioned in my last comment] which have exactly 0 measurable effects on the world", the benefit and harm they could cause would be sufficiently small to be practically negligible.

Hi Alistair. Thanks for asking about the effects on soil animals. I think they may be much larger than the effects on farmed animals as I illustrate in the post you linked to. At the same time, I do not know whether decreasing/expanding agricultural land increases or decreases the welfare/suffering/happiness of soil invertebrates.

What do you think about the St Petersburg problem now?

Thanks for pushing me to think about this more. I had only looked into your post a few days after you published it around 3 years ago, but just had a look again. I agree the money pump you described there does not require prospect which could have an infinite value. It only requires prospects with infinite expected value as you have been saying.

I think there is exactly 0 empirical evidence for distributions with infinite expected value for the same reasons I believe there is exactly 0 empirical evidence for infinities. As far as I can tell, exactly 100 % of the empirical evidence that could ever be gathered in principle could be exactly 100 % explained by distributions with finite expected value. Do you agree? I agree distributions should not have a maximum because one cannot be exactly 100 % confident there are not higher values. However, a lack of maximum does not imply infinite expected value.

If there's no finite upper bound on what the value the specific probability distribution can take, how can you be 100% confident it is not a probabilistic mixture with a distribution with infinite or undefined EV (but finite for every actual value), like 0.0001% probability to it being drawn from something like a St. Petersburg lottery?

This proves too much? One could argue there is a probability above exactly 0 of any given quantity being a probabilistic mixture involving a distribution with infinite or undefined expected value. In this case, all distributions would have an infinite or undefined expected value? For me this is a bitter bullet to bite than fully rejecting distributions with infinite or undefined expected value.

I think your arguments re infinities conflict with Occam's razor, and the principle of indifference should be applied across models within the same complexity (or submodels), otherwise you will assign 0 or too little credence to simpler models that are special cases, e.g. parameter value=0 or effectively eliminating some type of feature.

I would apply the principle of indifference to models which explain the same empirical evidence. If a curvature of 0 had a probability above 0, and the values of the curvature just above 0 followed a continuous distribution, the curvature of 0 would be infinitely more likely than a positive curvature arbitrarily close to 0. This is very counterintuitive to me because the curvatures would have an arbitrarily close explanatory power. I would rather concede all universe models are wrong with probability 1 while acknowledging simpler ones are more useful for further scientific progress all else equal.

Would you assign exactly 0 probability to photons having exactly 0 mass?

Yes. I think there will always be infinitely many values arbitrarily close to 0 which explain exactly the same empirical evidence as a value of 0.

Exactly 0 probability to there being no additional fundamental force, because it could just be vanishingly weak? And shouldn't this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say it's very weak.

Yes. Edit after Michael's comment just below. I would assign a probability of exactly 0 to any physical law because there are arbitrarily many physical laws arbitrarily close to any physical law. So I would also assign a probability of exactly 0 to any set of physical laws, including the set of laws involving any given number of fundamental forces.

Exactly 0 probability to there not being ghosts [ghosts existing with probability of exactly 1], because their effects could just be very small or rare?

Yes, but the effects of the ghosts would have to be sufficiently small or rare to be unfalsifiable. I assume the existence of ghosts is falsifiable under some typical definitions. Likewise for some defitions of God. Edit after Michael's comment just below. I would not assign a probability of exactly 1 to something falsifiable.

I think consistently applying your arguments suggest you should assign 0 probability in these cases, and so your models blow up in complexity and you become too credulous, contrary to Occam's razor.

Do you see any undesirable implications of believing in ghosts which have exactly 0 measurable effects on the world? I think this is effectively the same as not believing in such ghosts.

On the scientific question, yes: under a categorical account of sentience, arbitrarily small changes near the relevant biological threshold could in principle place otherwise extremely similar organisms on opposite sides of it.

Would it make sense to run WFF's affective capacity analysis for only one of 2 organisms which only differ infinitesimally?

I do not regard this [the sentence quoted above] as incoherent. Continuous underlying variables can produce categorical system-level events; a neuron, for example, may or may not fire an action potential as its membrane potential crosses a threshold. This analogy does not establish that sentience is categorical, but it shows that gradual biological change does not rule out a categorical onset.

I found this example quite thought-provoking. Yet, I do not think neuronal firings are categorical. Membrane potential and neuronal firings are represented in the mid and bottom plots of Figure 9A below from section 9.1 of the book Neuronal Dynamics. Membrane potential varies continuously. It is not binary. So humans have to come up with a convention for which types of variations in it count as a firing. In principle, 2 different time series of the membrane potential could differ infinitesimally, and one have 0 firings while other has many firings which fall just short of counting as such. If one was interested in a property like the maximum membrane potential, it would make sense to analyse both time series

There is also no fact of the matter about whether a given membrane potential corresponds to a neuronal firing or not. The number of neuronal firings is a property of the time series of the membrane potential. It is not a property of a neuronal state. In contrast, I understand sentience (existence of affective experiences) is supposed to be a property of a state (of an entity at a given time). Membrane potential is a property of a state, and it is continuous/graded (not binary/categorical/sharp).

The physical world varies continuously. So I think all properties tracking it have to be continuous too. People point out phase transitions as evidence of categorical changes, but they involve continuous changes.

Would you keep the sentience gate in WFF if you thought it was a matter of human convention instead of scientific fact?

I think falsifiability for a Bayesian could just mean we can imagine evidence that would warrant assigning very very low credences to the hypothesis.

Makes sense. I cannot imagine any evidence that would update me. However, if I did, there would be falsifiability.

Actual infinities are often falsifiable in principle in this way, in specific cases. I think the probability you should assign to you being infinite in spatial size is extremely small. Similarly for the Earth being infinite in size.

The way I see it, the probability of Earth having a radius larger than X tends to 0 as X goes to infinity. So I would say the probability of Earth having an infinite radius is exactly 0.

I think our credences that the universe is infinite in spatial extent should increase with our credences that the universe is globally flat (0 global curvature), which has been measurable.

Do we really have any evidence that the universe is globally flat? From Wikipedia's page on the shape of the universe:

Final results of the Planck mission, released in 2018, show the cosmological curvature parameter, 1 − Ω = Ω_K = −Kc²/a²H², to be 0.0007±0.0019, consistent with a flat universe.^[47] (i.e. positive curvature: K = +1, Ω_K < 0, Ω > 1, negative curvature: K = −1, Ω_K > 0, Ω < 1, zero curvature: K = 0, Ω_K = 0, Ω = 1).

We have evidence that the universe is close to flat. However, there are infinitely many values arbitrary close to exactly 0. So applying some sort of principle of indifference results in a probability of exactly 0 of curvature being exactly 0 (or any other sharp value)?

As an aside, there have been many cases where quantities in physics were assumed to be 0, but then turned out to be just small values, like the mass of neutrinos. It often makes sense to round a quantity to 0 for simplicity, but a sufficiently small value would explain exactly the same empirical evidence.

Hi Matthew. Thanks for crossposting this. I am sharing below the comment I left on the original post.

Hi Matthew. Nice post.

I estimate going to the protest is worth donating a few hundreds to dollars to the Shrimp Welfare Project (SWP). For 30 billion insects raised per year (you say "This single farm, Protix, raises tens of billions of farmed insects"), your assumption that "Protix’s continued operation will lead to an extra 300 billion farmed insects" implies each person increases by 0.01 pp the probability of 10 years of counterfactual impact. Maybe 3 years is more accurate considering the rest of the industry could increase their production to offset the loss of Protix. Even then, each person would save 3 M insects (= 30*10^6*3/10). I estimate SWP's Humane Slaughter Initiative (HSI) helps 15 k shrimps per $ (https://forum.effectivealtruism.org/posts/EbQysXxofbSqkbAiT/cost-effectiveness-of-shrimp-welfare-project-s-humane). Say saving 1 insect from Protix is as beneficial as HSI helping 1 shrimp, and that donations to SWP are roughly as cost-effective as HSI has been. In this case, going to the protest would be as beneficial as donating 200 $ (= 3*10^6/(15*10^3)) to SWP.

Given the above, going to the protest may still be worth it for people close by, but it is not obvious. For no financial cost, and a time cost of 20 $/h, it would have to take less than 10 h (= 200/20). On the other hand, there may be significant indirect benefits from going to the protest. In particular, connecting with people who care about neglected animals.

My exoplanet example was intended only to distinguish graded evidence from the underlying fact being investigated. Whether a candidate celestial body exists is different from whether an already known body satisfies one of several proposed definitions of a planet. Margot’s figure addresses the latter question. My analogy concerned the former.

I was analogising celestial bodies to organisms, and being a planet to being sentient. However, I also think whether a celestial body exists or not is a matter of human convention. I assume a clear definition would involve coming up with some properties like a minimum mass and density. In this case, one could come up with some amount of matter slighly too light to be a celestial body which would become a celestial body after adding it one more atom of hydrogen.

So yes: under my hypothesis, two organisms could in principle be extremely similar while affective experience was present in one and absent in the other. I do not find that paradoxical.

Even if the difference between the organisms can be arbitrarily small (not just extremely small, but arbitrarily close to a difference of exactly 0)? Supposing sentience is categorical/sharp, 2 organisms could be equal in all regards except that one differs 10^-100 % along a single metric, and therefore is the only one of the 2 which is sentient.

A small difference in a complex system can sometimes make a new causal process possible. The supporting architecture may vary continuously even though the process we are asking about either occurs or does not occur in a particular organism at a particular time.

I agree for a "small difference", but not for an arbitrarily small difference, and I think this is what distinguises sentience from lack of it if it is categorical/sharp.

Scientifically, you favor treating sentience itself as graded, whereas I regard its minimal presence as categorical while treating our evidence—and subsequent affective range, resolution, and intensity—as graded.

I believe sentience should be treated as graded if it means having a moral value above exactly 0. This is because I do not think an arbitrarily small difference should make an entity go from having exactly 0 moral value to having some moral value.

I am open to sentience being treated as binary if it does not mean having a moral value above exactly 0. However, in this case, lack of sentience would not imply lack of moral value.

I like the principle of indifference. However, I think infinities are unfalsifiable in principle. In this case, does it make sense for me to attribute probabilities to them? If I did, they would be just metaphysical priors that can never be updated by any evidence.

I also believe 2 actions can have infinite expected cost-effectiveness considering all future effects, and still be comparable. Imagine actions A and B have expected cost-effectiveness of CE_A(t) and CE_B(t) considering effects in the next t years (after a given date), and that CE_A and CE_B tend to positive infinity as t tends to infinity. If the ratio R(t) = CE_A(t)/CE_B(t) tends to:

• 0, B is infinitely more cost-effective than A.
• N, A is N times as cost-effective as B.
• Positive infinity, A is infinitely more cost-effective than B.

I am sceptical of the 1st and last cases in practice. However, in any of the above cases, there would be a clear answer about which action has the highest expected cost-effectiveness considering all future effects.

There would be a dilemma if the expected cost-effectiveness growing faster kept oscillating forever between positive and negative forever. For example, if CE_A(t) = t, and CE_B(t) = t^2*sin(t), R(t) = 1/(t*sin(t)), which tends to 0 as t goes to infinity, but keeps oscillating between an infinitesimally positive and negative value. However, in the real world, one could reasonably assume actions grow similarly fast sufficiently far into the future when there would be exactly the same evidence about the effects of A and B? In this case, R always tends to a finite single number as t goes to infinity.

Thanks for the clarifications.

An analogy would be an exoplanet candidate. Astronomers may have evidence of varying strength that a planet exists around a distant star. Their confidence may be 20%, 70% or 99%, but the planet itself either exists or does not. Graded evidence does not require the underlying fact to be graded.

Looking into this example is helpful to explain my point. Humans have to come up with a defition for what is a planet, and then test whether a given celestial body satisfies it. Margot (2015) illustrates this. I present below their Figure 1 whose lines represents 3 potential definitons. They are analogous to the sharp evidential surface I mentioned in my last comment. Increasing the mass of a celestial body which is slightly too light to be a planet (just below the definitional line) by just 1 kg without practically changing its semi-major axis could make it a planet (cross the definitional line). I think this is fine. At the same time, I would not see any deep difference between 2 celestial bodies which are practically the same except one is 1 kg heavier, and therefore is a planet unlike the other.

My own personal view is that the biological capacities associated with sentience can vary continuously, while the minimal presence of affective experience [sentience] is categorical.

If sentience was categorical, there could in principle be 2 organisms which only differed infinitesimally in biological capacities, but one was sentient, and the other was not. This would be fine if sentience was similar to being a planet. However, sentience is often assumed to be necessary for moral consideration. In this case, one organism would get some moral value, and the other would get exactly 0 moral value. I would find this problematic if the organisms only differed infinitesimally.

This is also why the tool uses a sentience-plausibility gate followed by a separate affective-capacity analysis. The first asks whether there is sufficient reason to infer that anything can feel good or bad for the organism. The second asks, conditional on that, how differentiated and intense those states could be. The first is treated categorically in the underlying hypothesis but uncertain in our knowledge; the second is explicitly graded.

My take is that it would be better to have everything graded, and assume all organisms have more than exactly 0 moral value.

Hi Saulius. I think this is such a great post. I just reread it because it was shared in the last EA Forum newsletter.