All this is meaningless drivel. Unprimordial, filtered puritanically. If you want to really understand civilisation you have to go further back to its roots. What took the barbarian Enkidu from the outskirts to the city? Copulation with Shamhat, under the light of Ishtar. The holy copulation measured the man and founded his honour within the traditions of Uruk; creating that unquantifiable debt we all hold to the community and each other.

It is no coincidence that honour, debt and sex are so intertwined; they are a divine triad in which each pair finds expression in the other. Sex and honour express themselves in debt; honour became its price, a subtlety lost by you false heralds of Western civilisation, cherrypicking little myths to suit your heathen agenda. Honour and debt find their expression in the blood-price, the virgin-pawn, the immeasurable excess of the unmarried woman; dealt with through sacred fucking, worth deriving from the promise thereof. Sex and debt mark the power of a man, his greater worth and the influence he has over the reserve of pawns and holy prostitutes; the true purpose of women.

↪fdrake Oh yeah, blabbering your mouth is certainly an argument. Again, all those people who just blabber their mouths in this thread provide no arguments as to why adultery should not be a crime. — Agustino

Marriage can be between many woman and one man, as spoken by God. If a man needs sex but is unmarried, he should obtain a holy concubine. Animals have sex for reproduction alone, we humans can have it for pleasure. The soul of the prostitute is the soul of the adulterer, and what can be more holy than exercising the divine gift of mutual pleasure.

Each woman who provides this service is a protected, respected wife of the community, and should be treated as such. Monogamy is an impoverished form of our true relationship to the divine, and breeds hate for the wife as property. The commercialisation of this is a sickness, money should not buy people, yet the institution of marriage is compatible with this sickness.

The woman and the man who embody the relationship with the divine the best are the prostitute and the adulterer, the essence of Western civilisation since the Greeks. Moreover, the essence of civilised life from the time of Sumer.

I think I'm going to convert to an ancient Sumerian religion or ancient Islam to worship God by having sex with prostitutes and concubines. Then I'll come back and be as draconian and sanctimonious as @Agustino is over sex with just as much religious fervour, the damn heathen.

When looking at a scenario with randomness in it, it's quite interesting to look at what sources of randomness there are and how they interact.

(1) In Michael's analysis, the only thing random is the value in the other envelope. The sample space of the random variable is {X/2,2X}

(2) In every analysis which deals with many possible values for what is contained in an envelope, the amount for one envelope is random, then it is random whether that envelope is X or 2X. The sample space for X is (0,infinity).

(3) In Jeramiah's analysis, the random thing is which envelope contains which amount. The sample space is {X,2X}.

(4) In my analysis, which pair of envelopes consistent with an amount U was random, then whether the envelope was U=X or U = 2X was random. The sample space for the first distribution was { {X,2X}, {X,X/2}} and the sample spaces for the conditional distributions were {X,2X} and {X/2,X}.

These analyses are related through some conditioning operations. In principle, any distribution on the value of X as in (2) can be conditioned upon. This yields case (1) if what Michael is doing with the sample spaces is vindicated or case (3) or (4) if it is not.

Analysing whether to switch or not comes down to questions on what distribution to consider. This distribution can be different from the one considered in cases (1)->(4).

In case (1), Michael believes P(X|U=10) distributes evenly over his sample space. In case (2) as far as the thread is considered a diffuse uniform distribution is placed on (0,M) where M is large or the flat prior (y=1) is placed on (0,infinity); this translates to (something like) an equiprobability assumption on the possible values of X - this can be combined with an equiprobability assumption on which envelope the person receives to (wrongly) achieve (1) or to achieve (4) by conditioning. In case (3) an equiprobability assumption is used on which envelope the person receives but not on which set of envelopes the person receives. The conditional distributions for scenario (4) given {X,X/2} or {X,2X} are equivalent to those considered by Jeramiah in case (3).

The choice to switch in case (2) makes the choice based on an intuition of P(received amount is the smaller value) where the amounts are uniformly distributed in [0,M]; for sufficiently large M this suggests a switch. The choices in (1),(3),(4) instead base the decision on the probability of envelope reception. (1), (3) and (4) are fully consistent with (2) so long as the appropriate P(received amount is smaller value||specific envelope amount) (envelope reception distribution) is considered rather than P(received amount is the smaller value) (amount specification distribution).

There aren't just disagreements on on how to model the scenario, there are disagreements on where the relevant sources of randomness or uncertainty are. The fundamental questions distinguishing approaches detailed in this thread are (A) where does the randomness come from in this scenario? and (B) what model is appropriate for that randomness?

The thought that everything ends is a haunting one. It comes most persistently in times of personal transition. When things change, when there's conflict in yourself or between yourself and your environment, the thought questions us as a mocking whisper; 'why should I do anything if it all ends?'. It promises a hopeless life; devoid of significance and greater meaning. Irrespective of how you choose to live your life, it says, it will have been for nothing.

It seems that you're coming to the realisation that a 'good college', a 'good career' and all that are extrinsic motivators. They are undifferentiated, amorphous aims which are imposed on you by the circumstances of your life. Perhaps the same can be said of the juxtaposition of these goals to the life of sensory pleasure you're considering pursuing. That life is the very model of fleeting happiness; a life lived in the comedown between highs. It is a poor substitute for what is supposed to give you contentment and higher purpose.

Luckily, fleetingness can be felt as a sign; something feels wrong or out of place in the composite of what you think you ought to be, what you think you are and what you really are. Finding the conflicts there gives insight into another kind of goal; an intrinsic motivator. Something which comes from you and is right by you.

That might seem small compared to the highlights of history; the Eisenhowers, the Einsteins, the prophets; in short, the greats that are handed down to us as if history has a plot and main characters. But all they did is follow their nose like you might learn how to; the stories crop up in their wake, and all stories are told by mere people, fumbling along like you (and all of us).

You can be one of these storytellers, you can find and follow your own goals, you can experiment with your own life to find fulfilment. There's always plenty of time until it's gone.

This is really good. It's not very documentary-ish but it's definitely... very informative...

This is a Whitehead question. @Janus might be able to help.

I agree. I think it is most convincing to present multiple angles. But it isn't (just) the sample space that is the issue. It is the probability space which includes: — JeffJo

Yeah. In my analysis this shows up when you ask what the raw probability of receiving X is, not the conditional probability of receiving X given that you just looked in the envelope. Michael correctly intuited that this is unknown. Some unknowns are benign, this one makes the probability space wrong.

'Uninformative' priors usually aren't uninformative, and sometimes they aren't even probability functions. For example, the equiprobability 'prior' you use on the probabilities is actually infinitely informative for the probability! What you gain in 'uninformativeness' on the level of outcomes you spend in infinite informativeness on the level of the (in the problem unspecified) probability parameter. There's uncertainty you're not modelling with equiprobability.

I suppose I'll stop antagonising you now.

Random sampling's good. p's also random since it's uncertain. This is the point.

If you assume equiprobability as a prior then you're only not influencing the results if your prior is a pointmass on 0.5 and the actual value is 0.5. This is a terrible prior, as it fixes the posterior to have p=0.5 despite anything and everything. In a situation of uncertainty and ambiguity, assuming infinite certainty (no prior variance) is completely nuts.

That's a principle of indifference then. Equally possible things are equally probable. A rule for assigning subjective probabilities.

Why is equiprobability simple but other priors aren't?

There's no simple reason to assume equiprobability. Doing so requires appealing to more sophisticated mathematical arguments with assumptions that probably don't hold anyway. Things like limiting distributions of independent draws or entropy maximisation.

How I would actually approach the problem:

If I deem the amount of money in my envelope tiny (say <100) I'd switch to be able to afford some nice things.

If I deem the amount of money in my envelope around some threshold that would give me some more financial security, I'd walk away with the money in my envelope. I think 10,000 is about right. That's a down payment on a house, over a year's rent etc.

If it's a ridiculously large amount of money, I'd switch again out of greed and that the loss wouldn't be felt so much because I'd use it for financial security/savings. I'd still be able to do what I wanted with the middle amount or the smaller amount.

Which is why you need to stop and ask yourself, "Is this correct?" You need to make sure what you are doing is applicable. It is math, not magic, you need to make sure your approach fits. God forbid I question the legitimacy of this approach. — Jeremiah

This is exactly why andrewk and Pierre's comments are on point. They found ambiguities in the problem (the equiprobability assumption, which as stated is not part of the problem!), then showed another way of filling in the blanks in andrewk's case, and gestured towards the general difference in approach in Pierre's case.

You could pick up a bad habit like charitable reading or obtain some appreciation towards the other half of statistics, or worse even, philosophy in general - which you regularly express contempt for.

As soon as you apply an expectation operator to the random variable in the envelope, you're already required to apply the context of long term behaviour and the independent repeatability of the trial. Regardless of the interpretation of the sample space and sampling mechanism, the concordance of the expected gain calculation with the long run sample gain echos the assumption. It's literally set up that way. It's in the algebra. If you terminated the simulation when you still have a small sample, you can analytically calculate the finite sample properties - which don't have to coincide with the expectation as you well know, and rarely exactly do.

Michael's calculations are actually correct if his assumptions are, his algebra is right, it's the modelling of the situation that differs. It's just algebra.

ITT Jeremiah forgets that frequentist expectation calculations are asymptotic.

There's an infinity of well justified approaches if you relax both the equal probability assumptions I used. But yes, probability's just a modelling tool most of the time.

We can ask whether a probability space appears to help towards achieving its aim, but it makes no sense to ask whether a probability space is correct. — andrewk

But so are sample spaces. They're a usually neglected part of model formation since it's usually obvious what all possible values of the measurement are; extreme values of measurements are also pretty easy to eyeball (EG, the probability that an adult human is taller than 3 meters is close to 0) even if more general tail behaviour isn't.

Getting the sample space right in interpretation 2 is an important part of the solution, even when you're using priors to incorporate different information than what's available here from the relationship of the observed value in the envelope to whatever contextual information you deem relevant.

You could also use different loss functions rather than raw expected loss to leverage other contextual information, but I don't see any useful way of doing that here.

As an aside, I think we're saying the same thing from different angles. The fundamental error Michael's making is losing track of what the sample space is and how conditioning works in the problem set up. I've been trying (as have some other people in the thread) to communicate the idea behind:

Now, if we don't see a value in an envelope, we know that v-w must be either +x, or -x, with a 50% chance for either. So switching can't help. The point to note is that we don't know what value we might end up with; it could be anything in the full range of V.

but I'm stumped on how to explain it better. It's pretty clear that Michael thinks conditioning on X makes a difference, which yields an impossible sample space for the problem set up (hence my question on what the raw probability of P(X=the starting value) was, and Michael's answer that it was 'unknown' - which destroys the probability measure he's using). But yeah, don't know how to get it across.

Would you agree that my analysis would work if you didn't look in the envelope? Or that looking in the envelope was irrelevant?

If you're in case A, you don't know whether you're in case A. This means you don't know if you gain or lose by switching. Same with case B.

In case A you're assigned either X or 2X.
In case B you're assigned either X or X/2.

I think you're getting hung up on the idea that because you've observed a specific value for X, the other is known. Informally: you have X, yes, but you don't know whether it really is X or 2X. Or in my previous set up. You see U, you don't know whether it is [X or 2X] or [X/2 or X], so you don't know if you gain by switching.

But if the host selects the value any other way then the objective probabilities will differ. However, assuming the participant doesn't know how the values are selected, he'll just apply the principle of indifference and assume a probability of 0.5 each. — Michael

Right. So there are two cases.

A: One envelope is filled with X and the other 2X.
B: One envelope is filled with X and the other X/2.

Knowing that your envelope has X=10 doesn't let you distinguish between the two cases, right? Same for any value of X.

The values in the envelopes are entirely determined by the case. The cases can't be mixed to obtain a sample space of {X/2,2X} since that's not a possible assignment by the question.

It's also entirely random whether you're observing the thing or its multiple. This means switching has to be averaged within the case (with equal probability assignments for each outcome) and then over the cases (with equal probability assignments for each case).

Right. Let's try another thing.

What's the probability that the two envelopes are filled with {5,10} and the probability that the two envelopes are filled with {10,20}? You know that you're in one of these states given that you just saw 10.

In your set up. There are three possible values for envelopes, {5,10,20}. Imagine that all envelopes are still closed. And you are given an envelope. What's the probability of having 10?

Then what's the probability of receiving an envelope containing 10? Not the probability of having 10 given that you just looked,

If 5 and 20 are permissible values for the envelopes, does it stand to reason that I could receive an envelope which is 4 times another envelope?

Would you agree that {5,10,20} are the possible values for things in the envelopes?

Ok. This is from how expectations work.

Imagine that we have two envelopes, one filled with X and one with 2X.

If I have an envelope containing X, and switch to an envelope containing 2X, I gain X from the switch.
If I have an envelope containing 2X, and switch to an envelope containing X, I lose X from the switch.

Does that make sense to you?

You always observe X in your envelope. What was the probability of obtaining U=X? Not the conditional probability of getting X given that you opened the envelope containing it.

X is a fixed dollar amount. Set it equal to 1 if you want, the reasoning proceeds the same way.

Ok. I'll say it without the abuse of notation.

There are 2 envelopes.
You open an envelope and see the value U.
So your random variable is U.
The envelopes could have been filled with either X and 2X, or X and X/2. These are jointly exhaustive possibilities. These described the possible states, of either having an envelope twice the value of the other or half the value of another.
This gives two cases to analyse. Which are labelled (X,2X) and (X,X/2). These correspond to the case when one envelope has X in it and the other envelope has 2X in it. And the case where one envelope has X in it and the other envelope has X/2 in it. The probability distribution here is on the sample space:

A=(one envelope contains X and the other contains 2X)
B=(one envelope contains X and the other contains X/2)

Which means that we will analyse the cases of the expectation of switching given A, the expectation of switching given B, then use those to form the overall expectation of switching.

Having an envelope means being in case A or being in case B. Opening the envelope does nothing to inform you of whether you are in case A or case B. Let's assume case A and case B are equally likely. That is, it's equally like that A (your envelope contains X and the other contains 2X) or B (your envelope contains X and the other contains X/2).

If you are in case A, having an envelope means that the envelope you hold is either X or 2X. That is, U=X or U = 2X.
If you are in case B, having an envelope means that the envelope you hold is either X or X/2. That is, U = X or U = X/2.

Case A and Case B are jointly exhaustive. Note that if one envelope is filled with X/2, the other cannot be filled with 2X, it must be filled with X. Similarly, if one envelope is filled with 2X, the other must be filled with X. This is because one envelope must be twice the value of another (or equivalently, one is half the value of the other).

Assume you're in case A. If you have the lower valued envelope (U=X) and switch, you gain X (you now have U=2X). If you have the higher valued envelope U=2X and switch you lose X (since you had 2X and switched to X). It is equally likely that you have X or 2X, so the expected gain conditional on being in case A is:

0.5(X-X)=0

Assume you're in case B. If you have the lower valued envelope X/2 and switch, you gain X/2. If you have the higher valued envelope X and switch, you lose X/2. So the expected gain conditional on being in case A is:

0.5( 0.5X - 0.5X) = 0


Now, it is random whether you are in case A or case B. They are equally likely. This means that the overall expectation of switching is the average of the expectations of switching in each case.

So the average of 0 from case A and 0 from case B is 0.5(0+0)=0.

With your approach, you have the sample space (2X,X/2) from switching. Please answer the following question:

What is the probability of obtaining the envelope containing X? Not the conditional probability of the envelope you have containing X given that you just opened it. This is a trick, your approach does not answer this question - since it cannot be answered using only 2 envelopes unless the probability of obtaining an envelope containing X is 0...

Better?

Where did I switch between stuff?

There are 2 envelopes.
You open an envelope and see X.
The envelopes could have been filled with either X and 2X, or X and X/2. These are jointly exhaustive possibilities.
This gives two cases to analyse. Which are labelled (X,2X) and (X,X/2). These correspond to the case when one envelope has X in it and the other envelope has 2X in it. And the case where one envelope has X in it and the other envelope has 2X in it. The probability distribution here is on the sample space:

A=(one envelope contains X and the other contains 2X)
B=(one envelope contains X and the other contains X/2)

Having an envelope means being in case A or being in case B. Opening the envelope does nothing to inform you of whether you are in case A or case B. Let's assume case A and case B are equally likely. That is, it's equally like that A (your envelope contains X and the other contains 2X) or B (your envelope contains X and the other contains X/2).

If you are in case A, having an envelope means that the envelope you hold is either X or 2X.
If you are in case B, having an envelope means that the envelope you hold is either X or X/2.

Assume you're in case A. If you have the lower valued envelope X and switch, you gain X. If you have the higher valued envelope 2X and switch you lose X. It is equally likely that you have X or 2X, so the expected gain conditional on being in case A is:

0.5(X-X)=0

Assume you're in case B. If you have the lower valued envelope X/2 and switch, you gain X/2. If you have the higher valued envelope X and switch, you lose X/2. So the expected gain conditional on being in case A is:

0.5( 0.5X - 0.5X) = 0

Now, it is random whether you are in case A or case B. They are equally likely. This means that the overall expectation of switching is the average of the expectations of switching in each case.

So the average of 0 from case A and 0 from case B is 0.5(0+0)=0.

With your approach, you have the sample space (2X,X/2) from switching. Please answer the following question:

What is the probability of obtaining the envelope containing X? Not the conditional probability of the envelope you have containing X given that you just opened it. This is a trick, your approach does not answer this question - since it cannot be answered using only 2 envelopes unless the probability of obtaining an envelope containing X is 0...

It seems to me that most of the disagreements aren't actually about how to calculate probabilities given a scenario, but rather on which scenario we're in. In the hope of disambiguating it, I'll consider a generalised envelope problem. Instead of the amounts being X and 2X, we'll generalise to U and V which are assumed to be positive real valued.

This makes the problem, per Jeramiah's OP:

You are playing a game for money. There are two envelopes on a table.
You know that one contains $U and the other $V, [but you do not
know which envelope is which or what the numbers U or V are]. Initially you
are allowed to pick one of the envelopes, to open it, and see that it
contains $Y . You then have a choice: walk away with the $Y or return
the envelope to the table and walk away with whatever is in the other
envelope. What should you do?

The first interpretation consists in:
(1) Being given an envelope.
(2) Opening it doesn't do anything to the sample space, conditioning is irrelevant.



Now for when the card is revealed and the sample space is... to be determined, as it's more complex. This is dealing with the original problem as stated by Jeramiah:

You are playing a game for money. There are two envelopes on a table.
You know that one contains $X and the other $2X, [but you do not
know which envelope is which or what the number X is]. Initially you
are allowed to pick one of the envelopes, to open it, and see that it
contains $Y . You then have a choice: walk away with the $Y or return
the envelope to the table and walk away with whatever is in the other
envelope. What should you do?

The confusions go away when you actually realise there are two possible sample spaces given that you receive X, and there is no information on which you're in.

The second interpretation consists in:
(1) Opening the envelope to receive X.
(2) There is no information over whether you're in the case (X,X/2) or (X,2X).



You end up with expectation 0 from switching and not switching so long as you're attending to the sample space consistently, independent of the interpretation.

Why is this different from the Monty Hall problem? When you're asked 'whether you would like to switch', the probability distribution changes to a distribution yielding 0.5 probability to each door, giving 2/3 probability of winning by Bayes' theorem. In the second interpretation, being told you receive X gives you no information about the resultant gain. Monty Hall does.

's enduring confusion throughout the thread consists in sneaking in some assumption that, in the second interpretation, the subject knows which case they're in. If instead the subject does not, the sample space is not (X/2,2X), it's (X/2,X,2X) if you receive a random envelope (which you need three of for this to make sense). This doesn't resemble anything like the original problem in which there are 2 envelopes.

TLDR: you're never actually getting to the sample space (X/2,2X) from what you're given. If you actually do end up in that case, then the result's an obvious (even deterministic) choice given that you know X. If you interpret everything carefully, there's no preference for switching because there's never any information on which case you're in.

Edit2: Also, this is pretty much the analysis in Jeramiah's linked paper, though it's analysed in terms of conditional expectations rather than probabilities.

Actually, it might do you good to climb up on this cloud with me and take in the bigger picture for a change. — frank

I've been through a very similar discussion on here before, with @SophistiCat. If you're interested give it a read.

That question suggests that you believe there are situations where you'd give a thumbs up to the death of an innocent. — frank

Exactly. I wear mostly black so that it doesn't show any of that disgusting Bangladeshi or Vietnamese blood from their dirty little hands on it. Then I sit in Starbucks, get fairtrade, and post mostly about metaphysics on here in my hard earned free time.

If that sounds ridiculous, horrible, sociopathic to you - it should. It's because you're not realising the consequences of your own issue framing. You're deliberating about what's moral or immoral without reference to any choices which are actually made. This comes equipped with its dual - an ethics of ideas without actions, in which disembodied propositions float above the world as impossible maxims. Ethical dilemmas, in all their ambiguity and sacrifice, are beneath this perspective.

Really? Certain parties find themselves being invaded and it's just incidental that they're sitting on top of a lot of oil? Yes there are complexities to our world that didnt exist back when, but our prioritiea are fundamentally the same. — frank

Wars for land control have economic features of motivation and strategy, yes. Big difference between that and a city being sacked for its accumulated wealth. Dispossession's always been a thing, accumulation by dispossession and economic risk management through military force are more than a bit different. The barbarians don't want your stuff, they want to secure their interests or develop what's now their property, or both at once.

In what context is the death of an innocent bystander ok with you? — frank

Since I'm the moral arbiter of the entire world why don't you come up with some situations and I'll tell you once and for all if they're good or evil.

Possibly. Russians have historically robbed invaders of any prize by abandoning their cities. But if we stipulate that pacifism encourages invasions, that doesn't address the claim that any coordinated military action is immoral.

Have you read Augustine's City of God? He points to the storage of excess wealth that goes on in cities as the real cause of invasions. — frank

I've not read Augustine's City of God. Perhaps it's insightful in other ways, but demonstrably that's not why most military conflict takes case now. The barbarians sacking the city for economic resources is somewhat incidental nowadays, when its concomitant, terrorist depravity is legitimised through propaganda and noise. It's an antiquated picture.

Anyway, it doesn't address whether coordinated military action is immoral if you construe morality as an intellectual exercise towards good conduct out with any context of decision. That's exactly what I'm criticising; military resistance to military oppression and terror is a no-brainer but not a blank cheque. Morality without politics is empty, politics without morality is blind.

Eh, this pacifism as an ideal gives a comparative advantage to immoral violence as a strategy. When a coordinated military shows up, you cede all ground to them no matter the strength of their imposition (so long as it is much greater) and no matter what their goals are. It's a very convenient moralism for any coordinated oppressor.

Methodologically, it's an ideal imposed from on high to a world too conflicted for it to work as intended. As soon as military power differentials show up, the targets of intervention become immoral if they resist.

Does pacifism allow violent resistance to violence in your book?

It feels strange to have my red hat on again.