Richard_Kennaway

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Here's the general calculation.

Take any probability distribution defined on the set of all values  where  and  are non-negative reals and . It can be discrete, continuous, or a mixture.

Let  be the marginal distribution over . This method of defining  avoids the distinction between choosing  and then doubling it, or choosing  and then halving it, or any other method of choosing  such that .

Assume that  has an expected value, denoted by .

The expected value of switching when the amount in the first envelope is in the range  consists of two parts:

(i) The first envelope contains the smaller amount. This has probability . The division by 2 comes from the 50% chance of choosing the envelope with the smaller amount.

(ii) The first envelope contains the larger amount. This has probability . The extra factor of 2 comes from the fact that when the contents are in an interval of length , half of that (the amount chosen by the envelope-filler) is in an interval of length .

In the two cases the gain from switching is respectively  or .

The expected gain given the contents is therefore .

Multiply this by , let  tend to 0 (eliminating the term in ) and integrate over the real line:

The first integral is . In the second, substitute  (therefore ), giving . The two integrals cancel.

And causally-connected panpsychism is just materialism where we haven't discovered all the laws of physics yet.

Materialism, specifically applied to consciousness, is also just materialism where we haven't discovered all the laws of physics yet — specifically, those that constitute the sought-for materialist explanation of consciousness.

It is the same as how “atoms!” is not an explanation of everyday phenomena such as fire. Knowing what specific atoms are involved, what they are doing and why, and how that gives rise to our observations of fire, that is an explanation.

Without that real explanation, “atoms!” or “materialism!”, is just a label plastered over our ignorance.

To follow a maxim of Edwin Jaynes, when a paradox arises in matters of probability, one must consider the generating process from which the probabilities were derived.

How does the envelope-filler choose the amount to put in either envelope? He cannot pick an "arbitrary" real number. Almost all real numbers are so gigantic as to be beyond human comprehension. Let us suppose that he has a probability distribution over the non-negative reals from which he draws a single value , and puts  into one envelope and  into the other. (One could also imagine that he puts  into the other, or tosses a coin to decide between  and , but I'll stick with this method.)

Any such probability distribution must tail off to zero as  becomes large. Suppose the envelope-chooser is allowed to open the first envelope, and then is allowed to switch to the other one if they think it's worth switching. The larger the value they find in the first envelope, the less likely it is that the other envelope has twice as much. Similarly, if they find a very small value in the first envelope (i.e. well into the lower tail of the distribution), then they can expect to profit by switching.

In the original version, of course, they do not see what is in the envelope before deciding whether to switch. So we must consider the expected value of switching conditional on the value in the first envelope, summed or integrated over the probability distribution of what is in that envelope.

I shall work this through with an example probability distribution. Suppose that the probability of the chosen value being  is  for all positive integers , and no other value of  is possible. (Taking  would be simpler, but that distribution has infinite expected value, which introduces its own paradoxes.)

I shall list all the possible ways the game can play out.

1. $2 in the envelope in your hand, $4 in the other. Probability  for selecting the value , and  for picking up the envelope containing , so . Value of switching is , so the contribution of this possibility to the expected value of switching is .

2. $4 in your hand, $2 in the other. Probability , value of switching  , expectation .

3. $4 in your hand, $8 in the other. Probability , value of switching , expectation .

4. $8 in your hand, $4 in the other. Probability , value of switching  , expectation .

And so on. Now, we can pair these up as , etc. and see that the expected value of switching without knowledge of the first envelope's contents is zero. But that is just the symmetry argument against switching. To dissolve the paradoxical argument that says that you should always switch, we pair up the outcomes according to the value in the first envelope.

If it has $2, the value of switching is .

If it has $4, the value is .

If it has $8, the value is .

The sum of all of the negative terms is , cancelling out the positive one. The expected value is zero.

The general term in this sum is, for  
  , which is negative. The value conditional on the value  having been drawn is just this divided by , which leaves it still negative. If we write  and , this works out to  and . The expected value given  is then . Observe how this weights the negative value three times as heavily as the positive value, but the positive value is only twice as large.

Compare with the argument for switching, which instead computes the expected value as   , which is positive. It is neglect of the distribution from which  was drawn that leads to this wrong calculation.

I worked this through for just one distribution, but I expect that a general proof can be given, at least for all distributions for which the expected value of  is finite.

Brace yourself. Whatever they say will likely be painful to hear.

But will it be true?

Even given the earnest request, are they going to take it all at face value and give you the information you are asking for as best they can (and how good is their best, anyway?), or are they going to either be nicey-nicey, or use it as an open goal to offload their own "stuff" into? I am seeing a lot of ways this could go wrong. It would not be good to acknowledge and accept ugly falsehoods.

Something something Ask vs. Guess culture.

After getting a cohort of monks killed, the Buddha

Can you point me to the specific story being referenced?

No, it’s just a sense of the word I’m familiar with. But I find that googling “mere autodidact” gives a wealth of examples.

what's the key difference? What is the essence of autodidact-ness?

There are no essences of words. This word, "autodidact", in general means someone who has taught themselves some subject without a teacher. In the anecdote about Alice and Bob, this is true of both of them regarding chemistry, since Alice's lecturer might as well be a rubber duck.

However, the word is often loaded with a pejorative air, to suggest that the autodidact has learned only by studying books, without the guidance of personal interaction with experts or even other students of the subject, and has thereby obtained an understanding that is distorted or superficial.

However2, there are many who have achieved distinction in an area while having no formal education in it.

However3, everyone must at some point take responsibility for their own learning, and treat books, teachers, and fellow students all, as resources to use, not nipples to suck.

Pro tem you can edit your post to delete its text, then delete the post. I am assuming that old versions of posts no longer exist.

If the mechanic does shoddy work I won't come back.

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