= 27d567bc2d48d9f40e9ccc20ea370b15)
Shouldn't the expected value be $1000 (10p)*(1-p^10) or $1000 (10p - 10p^11) ? (p now maximised at 0.7868... giving EV $7.15K)
Currently I plan to strongly advise my future sons against marriage.
Does this imply that you favor (or at least are neutral about) long term relationships, but are opposed to marriage?
Do you think marriage itself is a bad deal for men, or do the problems mostly show up with divorce?
Altering the structure of divorce alters the payoff-matrix for behaviors inside the marriage itself.
In response to
Stuff That Makes Stuff Happen
p(a,b,c) = p(a)p(b)p(c) isn't a statement of uncorrelatedness but of independence. Using the term "uncorrelated" with that meaning might be defensible but probably merits mention as something not-mainstream.
It's helpful to go a bit further for these corrections. What's the reason not to use "uncorrelated" here?
In ordinary English, "uncorrelated" is indeed used for this (and a host of other things, because ordinary English is very vague). The problem is that it means something else in probability theory, namely the much weaker statement E(a-E(a)) E(b-E(b)) = E((a-E(a)(b-E(b)), which is implied by independence (p(a,b) = p(a)p(b)), but not does not imply independence. If we want to speak to those who know some probability theory, this clash of meaning is a problem. If we want to educate those who don't know probability theory to understand the literature and be able to talk with those who do know probability theory, this is also a problem.
(Note too that uncorrelatedness is only invariant under affine remappings (X and Y chosen as the coordiantes of a random point on the unit circle are uncorrelated. X^2 and Y^2 are perfectly correlated. Nor does correlated directly make any sense for non-numerical variables (though you could probably lift to the simplex and use homogeneous coordinates to get a reasonable meaning).)
In response to
A probability question
I think the first order of business is to straighten out the notation, and what is known.
- A - measurement from algorithm A on object O
- B - measurement from algorithm B on object O
- P(Q|I) - The probability you assign to Q based on some unspecified information I.
Use these to assign P(Q | A,B,O,I).
You have 2 independent measurements of object O,
I think that's a very bad word to use here. A,B are not independent, they're different. The trick is coming up with their joint distribution, so that you can evaluate P(Q | A,B,O,I).
The correlation between the opinions of the experts is unknown, but probably small.
If the correlation is small, your detectors suck. I doubt that's really what's happening. The usual situation is that both detectors actually have some correlation to Q, and thereby have some correlation to each other.
We need to identify some assumptions about the accuracy of A and B, and their joint distribution. A and B aren't just numbers, they're probability estimates. They were constructed so that they would be correlated with Q. How do we express P(QAB|O)? What information do we start with in this regard?
For a normal problem, you have some data {O_i} where you can evaluate P(A), your detector, versus Q and get the expectation of Q given A. Same for B.
The maximum entropy solution would proceed assuming that these statistics were the only information you had - or that you no longer had the data, but only had some subset of expectations evaluated in this fashion. I think Jaynes found the maximum entropy solution for two measurements which correlate to the same signal. I don't think he did it in a mixture of experts context, although the solution should be about the same.
If instead you have all the data, the problem is equally straightforward. Evaluate the expectation of Q given A,B across your data set, and apply on new data. Done. Yes, there's a regularization issue, but it's a 2-d -> 1-d supervised classification problem. If you're training A and B as well, do that in combination with this 2-d->1d problem as a stacked generalization problem, to avoid over fitting.
The issue is exactly what data are you working from. Can you evaluate A and B across all data, or do you just have statistics (or assumptions expressed as statistics) on A and B across the data?
The usual situation is that both detectors actually have some correlation to Q, and thereby have some correlation to each other.
This need not be the case. Consider a random variable Z that is the sum of two random independent variables X and Y. Expert A knows X, and is thus correlated with Z. Expert B knows Y and is thus correlated with Z. Expert A and B can still be uncorrelated. In fact, you can make X and Y slightly anticorrelated, and still have them both be positively correlated with Z.
Mandatory drug testing?
That's the big one I can think of, and this usually arises in a very different context where it's easy to dehumanize those forced to take such tests: alleged criminals and children.
(Even in these contexts, peeing in a cup or taking a breathalyzer is quite a bit less severe than enduring a forced pregnancy. Mandatory blood draws for DUIs do upset a signifianct number of people. How you feel about employment tests and sports doping might depend on how you feel about economic coercion and whether it's truly "mandatory".)
We don't generally require that people give up their bodily autonomy to support the life of others.
We don't?
In what situation, exactly, do we fail to do this? I can't think of any other real-world situation. I can imagine counterfactual ones, sure, but I'm fairly certain most people see those as analogies for abortion and respond appropriately.
We don't, for instance, require people to donate redundant organs, nor even blood. Nor is organ donation mandatory even after death (prehaps it should be).
What are some cases where we do require people to give up their bodily autonomy?
The example of stochastic evidence is indeed interesting. But I find myself stuck on the first example.
If a new reasoner C were to update Pc(X) based on the testimony of A, and had an extremely high degree of confidence in her ability to generate correct opinions, he would presumably strongly gravitate towards Pa(X).
Alternatively, suppose C is going to update Pc(X) based on the testimony of B. Further, C has evidence outlining B's apathetic proclivities. Therefore, he would presumably only weakly gravitate towards Pb(X).
The above account may be shown to be confused. But if it is not, why can C update based on evidence of infomed-belief, but A and B are precluded from similarly reflecting on their own testimony? Or, if such introspective activity is not non-normative, should they not strive to perform such an activity consistently?
They essentially have already updated on their own testimony.
I believe you mean "you will have incomplete information about any system you could really have."
Operationally, it's a distinction without a difference.
Is your answer any different for identical twins, who of course only separate after fertilization? How about chimeras?
Yes; if I had a twin, my obvious answer would be when I separated from my brother. Were I a chimera, I suspect I would have researched the issue more extensively than I have now, but at my present level of understanding it still seems like there's a discontinuous event- when the cells fuse together to form one organism.
It seems to me that you can find a discontinuous event for most person precursors, and the discontinuity is important for that question (because the components were continuous beforehand, and the composite is continuous afterwards). The main counterexample I can think of is clones- if I create a thousand copies of my DNA and implant them in embryos scrubbed of DNA, then they seem fungible in a way that a thousand unique fertilized embryos are not. And then, because they are fungible, I would ascribe to the group of them the specialness of a single fertilized embryo, and would only have qualms about destroying the last one (or perhaps last few). Note that as soon as they begin to develop, they begin to lose their fungibility (and we could even quantify that level of fungibility/uniqueness), and could eventually become unique people (that share the same genes).
Likewise, the position "every sperm is sacred" seems mistaken because sperm are by nature fungible (and beyond that, we can complain about the word sacred).
Likewise, the position "every sperm is sacred" seems mistaken because sperm are by nature fungible (and beyond that, we can complain about the word sacred).
In what way are sperm fungible? There is usually a wide variety of difference between two random ones from the same person. After all, half the genetic variability of two siblings is due to the difference in sperm.
It's true that differences are such that we can't easily tell much difference between any two sperm (of the same sex and chromosome number) -- but the same is true of a just fertilized zygote or just divided embryo, which you appear to count as non-fungible when you say that "I can't think of a situation where I would be willing to accept the death/murder of a fetus or infant where I wouldn't be willing to accept the death/murder of an adult."
It seems that "fungibility" needs to be treated as a continuum. I think that just about all reasonable criteria for deciding this turn out on closer inspection to be fairly continuous.
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A video of Daniel Dennett giving an excellent talk on free will at the Santa Fe Institute: https://www.youtube.com/watch?v=wGPIzSe5cAU It largely follows the general Less Wrong consensus, but dives into how this construction is useful in the punishment and moral agent contexts more than I've seen developed here.