And ridiculously chauvinistic — Akanthinos

Why do you think you feel that way?

I'm not a Huguenot, by the way. I'm not even a Christian any more. But I was brought up RC and know the religion pretty well.

You are misreading my statement. I am not denying that some Christians believe the Ten Commandments are still applicable to them. What I am denying is that accepting that claim is a necessary part of being a Christian. The better class of Christian (IMHO) rejects the Ten Comms in favour of the Two. The best of all (again IMHO) focus almost exclusively on the second of the two.

You cannot claim that the Decalogue is not Christian if it is Catholic. — Akanthinos

I do though. Being a Christian is about following the ideas or teachings that one believes to have been given by Jesus of Nazareth, not about following a bunch of dogmas and rules written by a beastly bureaucracy of bishops.

(I am assuming that the game only allows for amounts in whole dollars, for simplicity — Pierre-Normand

Interestingly, if the game only allows whole dollars - or even whole cents, and the player knows that, they can use it as the basis for another strategy: if the number is odd, switch, otherwise don't. That's because if the number is odd it cannot be the doubled value, so the other one must be.

To remove that option, I recast the problem with the envelopes containing IOUs rather than cash, for an amount that is a real number of cents, with an arbitrary but large number of decimal places shown. The amount is only rounded to the nearest cent (or dollar) when the IOU is cashed in.

Matthew 19:16-30, Mark 10:17-31, and Luke 18:18-30 — Akanthinos

The devil can cite scripture for his purpose.

That applies to me as well as to you of course.

The Decalogue has no positive content but is merely negative. — Agustino

The Decalogue is for the Jews, not the Christians. There are only two Christian commandments, and they are both positive - exhortations to love.

lying itself becomes a virtue, as the necessary result of the exertion of one's inalienable rights. — Agustino

I don't know of any rights campaigners that assert that it is always virtuous to exercise a right. Indeed one of the main points of free speech is that you should not be legally punished for speaking even when what you say is the opposite of virtuous. That goes right back to Voltaire's original famous one-liner about free speech.

Right. The paper Jeremiah linked talks about this too.

That paper appears to put forward the same position as mine: that always-switching delivers no expected gain, even if the envelope has been opened, but that a strategy based on switching only if the observed amount is less than some pre-selected value delivers a positive expected gain.

Probability is, essentially, a measure of our uncertainty about a result. — JeffJo

In broad terms I do not disagree with that characterisation. But there is often more than one way to represent uncertainty, and these lead to different probability spaces. I have referred previously to the observation that in finance many different, mutually incompatible probability spaces can be used to assign a value to a portfolio of derivatives. To try to mount an argument that a particular probability space is the sole correct probability space for analysing a problem, one would have to make a bunch of assumptions to start and, as we see from the length of this interesting thread, those assumptions are rarely uncontroversial.

And what you seem to be avoiding with that attitude, is that the expectation formula (v/2)/2 + (2v)/2 is already assuming: — JeffJo

I am not an advocate for that expectation formula, so I don't see why you'd think I am avoiding those objections to it.

$\small \begin{align} E(N\mid M=a) &= (2S)P(M=S\mid M=a)+(S)P(M=2S\mid M=a)\end{align}$

Isn't this the same as: — Michael

$\small \begin{align} E(N\mid M=a) &= (2a)P(S=a\mid M=a)+(\frac{a}{2})P(2S=a\mid M=a)\end{align}$

Provided that all outcomes in the probability space have either M=S or M<=2S*, Yes it is.

In that case both are applications of the law of total probability, which says that:

$E[N|A] = E[N|A\wedge B]P(B|A) + E[N|A\wedge \sim B]P(\sim B|A)$

In this instance event A is the set of all outcomes where M=a and, for srap's version, event B is the set of outcomes where M=S. So event ##\sim B## is the set of outcomes where ##. For your version B is the set of outcomes where 2S=a.

* I haven't been following closely enough to know whether that is the case with this notation. I haven't seen letters N, S or M used before. Last time I was reading closely it was X, Y and U

No, it gives you a strategy that works on your assumed prior, not necessarily on reality. — JeffJo

Where I differ from that perspective is that I reject the notion that there is such a thing as a 'real' probability (aka 'true', 'raw', 'correct', 'absolute' or 'observer independent' probability).

If you right-click on a TeX formula and select 'Show Math As...' then 'TeX commands', then you can copy and paste the code for that in between [ math ] and [ /math ] delimiters (without the white space inside the brackets) and, all going well, it will appear as nicely displayed TeX in your post.

Unfortunately I don't think there is a single-click facility to quote a post including its TeX properly. It has to be done equation by equation, which is a bother.

That critical point is going to be the highest XX that can (will?) be selected by the host, correct? — Michael

Yes, if we assume a uniform distribution for X on the interval [1,M]. If we assume a more shaped distribution that decays gradually to the right then it will be something different. A gradually changing distribution would be more realistic because it would be strange to say that the probability density of choosing X=x is constant until we reach M and then suddenly plunges to zero. The calculations get messier and hard to discuss without long equations if we use fancy distributions (such as beta distributions or truncated lognormals) rather than a simple uniform distribution. But they can be done.

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. — fdrake

Indeed, and that's where utility curves come in. If a parent has a child who will die unless she can get medicine costing M, and the parent can only access amount F, the parent should switch if the observed amount is less than M-F and not switch otherwise.

Even in that more general case, the Bayesian approach can give a switching strategy with a positive expected net gain. Based on our knowledge of the world - eg how much money is likely to be available for putting in envelopes - we adopt Bayesian priors for U and V that are iid. We can use the priors to calculate the expected gain from switching as a function of the observed amount Y. That gain function will be a function that starts at 0, increases, reaches a maximum then decreases, going negative at some critical point and staying negative thereafter. The strategy is to switch if the observed amount Y is less than that critical point.

A simple way to think of this is that it's using the observed value Y to calculate an updated estimate of the Bayesian-updated probability that we have the larger of the two amounts. But because of nonlinearities, the strategy must be based on expected gains rather than just on whether that probability is greater than 0.5.

I like that your post encompasses multiple interpretations. This highlights an important point - that probability is simply a tool, not a fundamental truth about the universe. There is no such thing as 'the absolute probability (or expected value) of Z'. Probabilities are calculated in the context of a probability space, and we are free to set up whatever probability space we like to solve a problem. It is meaningless to say that one probability space is 'correct' and another is 'incorrect'. In finance one uses multiple different probability spaces, with different probabilities assigned to a single event, to analyse risk (eg we often speak of 'risk-neutral' vs 'historical' probabilities). Each probability space serves a different purpose. 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.

I feel like Buridan's ass right now.

Please help — TheMadFool

Buridan's ass was prey to indecisiveness that - hypothetically - caused it to starve.

Do you think that your opinion on the topic in the OP will cause you to starve?

Are you seriously considering abandoning the use of logic? If so, how would you go about doing that?

If not, why do you feel you need help? It seems to me you're getting along just fine.

She doesn't have to estimate it accurately. There are only three possibilities.

A. X<2X<c. In this case the expected gain from switching is zero.
B. c<X<2X. In this case the expected gain from switching is zero.
C. X<c<2X. In this case the expected gain from switching is X.

So for the competitor to decide not to use the strategy, she would have to be absolutely certain that case C is impossible.

I gave it above. The player would use it. They would estimate L, the maximum possible payout (eg the budget of the game show). The simplest approach is to then set c=L/2 and switch iff Y<c. That strategy delivers a positive expected gain under perspective 2.

I think the issue is that even if you know Y from opening the initial envelope, the expected gain from switching is still zero if you don't also know c. — Andrew M

c is not an observer-independent item that can be known or not. It is a feature of the Bayesian prior distribution the player adopts to model her uncertainty.

From the God's-eye (ie omniscient) point of view, which is perspective 1 from the quoted post, there is no c, because there is no non-trivial probability distribution of X. X is a fixed quantity, known only to God and to the game show host.

So we cannot talk meaningfully about 'the real value of c'.

Right, so if your object, or intent, in relation to a particular idea (that something is inherently unintelligible) is to falsify this idea, then doesn't it seem contradictory, or at least hypocritical to adopt this idea as an opinion? To hold as an opinion implies that you believe the idea. To work towards falsifying it implies that you do not believe it, and are skeptical. If you hold it as an opinion you will not be skeptical of it, and you will not work toward falsifying it. — Metaphysician Undercover

Yes, I agree. That is why I do not try to falsify opinions that seem to work well for me. I am open to others' suggestions when they think they have found a falsification, and sometimes they convince me and I change the opinion. But I don't personally set out to try to falsify it.

to say that the universe is intelligible to God does not render the word useless, it just denies that there is anything which is truly unintelligible, in an absolute sense. — Metaphysician Undercover

If there is nothing that is 'unintelligible' then the word has no use, because it cannot apply to anything. In everyday life the word 'intelligible' is useful because some things are and some are not, when we take it to mean 'capable of being understood by an intelligent human'. What would be the point of changing the meaning of the word to something that is different from how ordinary people use it, AND has no application?

As for angels, if they are finite, non-omniscient beings then 'intelligible' can make sense if some things are intelligible to them but not to humans. We just change the definition slightly to 'capable of being understood by one of the most intelligent finite beings'. The same can apply to hyper-intelligent alien species in another galaxy. It's only when you change it to 'capable of being understood by an omniscient, omnipotent being' that it becomes meaningless.

Therefore there is no reason to believe that anything is unintelligible to all intellects. — Metaphysician Undercover

It is my opinion that there is good reason to believe that the world is unintelligible to all finite intellects. And in the usual way 'intelligible' is used, that is the same as saying there's good reason to believe the world is unintelligible tout court.

Let's name the envelopes Y and Z (note, they do not denote amounts). The expression "if Y = X then Z is 2X or X/2" only adds up to 3X in one instance, the rest results in false conclusions as it contradicts the premise that the total should always be 3X. Knowing that Y is either X or 2X, we get four possibilities:

If Y = X then Z = 2X for a total of 3X is true.
If Y = X then Z = X/2 for a total of 1.5X is false.
If Y = 2X then Z = 2X for a total of 4X is false.
If Y = 2X then Z = X/2 for a total of 2.5X is false.

This suggests that replacing the variable of one envelope with a fixed amount or a fixed placeholder messes up things. I'm not sure why. Maybe andrewk can tell me. — Benkei

Hi Benkei. Nice to see you join this discussion.

Since X denotes the smaller of the two amounts, the first statement is true and the other three are false. But the player cannot use the statements because she only knows Y. She doesn't know what X is, and unless she adopts a Bayesian prior distribution for X she doesn't know the probability that Y=X either, so she can't use conditionals.

I understand that people feel discomfort with the use of Bayesian priors and feel that your expectation of gain is then based on your own guessed distribution, which you know to be wrong, but that's how Bayesian methods work. For all their limitations, they are all we have (except for a complicated exception that I'll mention further down, and which I doubt anti-Bayesians will feel any more comfortable with).

If we refuse to use a Bayesian prior then what can we say? The value X is a definite value, known to the game host, not a random variable. We have know the value Y, having seen it in the opened envelope.

If we insist on modelling the situation from a full-knowledge perspective rather than our limited-knowledge perspective, then the gain from switching is X with certainty if Y=X and -X with certainty if Y=2X. But we can't know which is the case so the calculation is useless. We can't introduce probabilities of one or the other cases being actual because we are modelling from a position of full knowledge and only one of them is true.

Some of the calculations adopt a half-way position where they make X a fixed, non-random amount, and the coin flip result a Bernoulli(0.5) random variable, such that Y=X if B=0, otherwise Y=2X. Such a calculation reflects the state of knowledge of the game host, if we assume the host knows X but doesn't know which envelope has the larger amount in it.

Under such an approach the expected gain from switching is zero.

But we ask, why is it reasonable to model the knowledge limitation of the host by randomness, but not to do the same for the player? If that approach is reasonable for the host, it is reasonable for the player, and it is more appropriate to use a Bayesian approach, since it is the player's expectation that we have been asked about.

On the other hand, if modelling knowledge limitation by randomness is not considered reasonable then we are forced back to where everything is modelled as known and the expected gain is either X or -X with certainty, but we don't know which applies.

Now for that exception. If one doesn't like that Bayesian approaches model gains by assuming the prior distribution is correct, then we could introduce a distribution for errors in the distribution.

Say our prior is a lognormal distribution. That has two parameters mu and sigma, for which we assume values to do our calcs. We could reflect our lack of certainty about the parameters by making those two parameters themselves random variables, to reflect our uncertainty about those. Then the calculation will reflect our uncertainty about the prior.

But guess what happens! It turns out that this approach is identical to assuming a Prior that is the convolution of the original prior with the distributions of the two parameters. So all we've done by modelling the fact that our prior is a guess is change to a different prior. We can repeat that process as often as we like - modelling the uncertainty in the uncertainty in the uncertainty in .... - and we'll still end up with a Bayesian prior. It'll just be more dispersed than the one we started with.

We might reject the parametric approach as too constrained and instead model uncertainty directly on the CDF of the prior. That gets messy but it will still end up with the same general outcome - a single but different Bayesian prior.

Summary

We have three options:

1. Treat X and Y as both known. Then there is no randomness and the switch gain is either X with certainty or -X with certainty, so the expected gain is equal to that certain gain but the player doesn't know the amount, so this approach is useless to her.

2. Treat Y as known and model X using a Bayesian prior. This leads to a rule under which the player can calculate a value c such that her expected switch gain is positive if Y<c and negative if Y>c.

3. Treat X as known and Y as unknown. Then the switch gain has a distribution of X or -X with even odds, so the expected switch gain is zero. This is the approach defended by srap. The approach is coherent but it begs the question of why it is valid to model lack of knowledge about Y/X by randomness, but not lack of knowledge about X.

Note that approaches 2 and 3 both predict zero as expected gain from blind switching. The difference between them is that 2 gives a strategy for switching based on the observed value of that, when followed, gives a positive expected gain from switching.

I can see how, for pragmatic reasons, one might proceed from such a premise, a proposition which could never be proven true, but could in principle be proven false. But any intent, other than the intent to prove that premise false, would be misguided. Conclusions derived from this premise would be very unsound, and therefore misleading. — Metaphysician Undercover

As Popper showed us, this is how science in particular, and almost all knowledge, works. We can prove almost nothing true, but we can falsify it. We act as if the theories that are useful and have survived many attempts at falsification are true, and use them to cross roads, send rockets to Mars and cure plague. All while we know that they could be falsified one day.

The point being that "unintelligible" means something different in theology than what it means to the atheist. — Metaphysician Undercover

Maybe to Aquinas, but he is only one person writing in theology. To say that the universe is intelligible because it is intelligible only to God renders the word useless because the Christian definition of God includes that he knows everything, which entails that She knows the reason for everything, so it is by definition intelligible to Her. That definition renders a useful word useless and it would take a great deal of evidence to back up a claim that it is the standard use of 'intelligible' in theology.

In short, to say that something is intelligible if it is intelligible to God is to say nothing at all.

we claim to know that such and such aspects of reality are inherently unintelligible

I think it would be a mistake to claim to know that things are inherently unintelligible, because it is hard to see how one could obtain sufficient confidence in that opinion to call it knowledge. On the other hand I find it entirely reasonable to hold an opinion that things are inherently unintelligible. I would definitely not call an opinion knowledge, or even a claim.

I don't think it's philosophical laziness, or unphilosophical. It's just saying 'I see no way to proceed in that direction so I won't try'. Instead one focuses one's philosophical efforts on other things like ethics, politics and finding meaning in life, that are likely to be useful. To me that just looks like a judicious allocation of limited resources.

It's a bit like how scientists won't entertain ideas about perpetual motion machines. We cannot prove they're impossible. We have a scientific law that says they are, but scientific laws have been revised many times over the years, so it's not inconceivable that that one could be revised too. But the prospects of that happening seem so slim that scientists choose to spend their efforts in more promising fields.

I wish the best of luck to those that like to speculate in metaphysics. I doubt they will ever come up with something that is not hotly contested. But if they do I will be delighted to read about it, and will give great kudos to those that came up with the innovation.

By the way, the first place I came across a suggestion that the universe was unintelligible was in Stella Gibbon' book "Cold Comfort Farm", in which Flora, the protagonist, reads a book by the Abbé Fausse-Maigre - a RC priest - which is described as proclaiming the fundamental unintelligibility of the world. Suggestions of unintelligibility are not particularly associated with atheists.

there's a 50% chance that the other envelope contains £20 and a 50% chance that the other envelope contains £5. — Michael

The resolution of the apparent paradox is that the probabilities are not 50:50 for most values of Y.

Either the player has not adopted a Bayesian prior distribution for X, in which case she has no basis for assigning any probabilities to the options of U=X and U=2X, or she uses the prior distribution to calculate the probabilities. (U is the value in the unopened envelope)

It is fairly straightforward to show (and I did so in my note) that when she does that, regardless of the prior distribution adopted, the probability of U=2X depends on the observed value Y and will be more than 50% up to a certain calculable critical point after which it will be less than 50%.

The case where the prior distribution is X=5 or 10 with equal probability demonstrates this. If Y=5 then it is certain that U=2X=10. If Y=20 it is certain that U=X=10. If Y=10 then the odds are 50:50 that U=2X, ie 5 or 20.

There are many different ways the calculations for this can be approached, and we've seen several of them in this thread. But whatever approach one is using, one should subject it to a hard critical eye when a 50:50 assumption is made because in many cases, and possibly in all cases when it's about what's in the unopened envelope, that assumption will not be justified.

The atheist however, is justified in claiming that unintelligibility is a feature of reality itself, that there are aspects of reality which are purely random or some such thing, which by their very nature are impossible to be understood — Metaphysician Undercover

Yes that is approximately my position, although (1) I would replace 'claiming' by 'speculating' and (2) it would be overly simplistic to describe me as an atheist tout court. But I do know people who strongly self-identify as atheists that, like me, expect reality is ultimately unintelligible to humans or to any finite being.

Perhaps we are not in disagreement then.

OK then, let's undismiss your first post. I don't agree that I did 'dismiss' it but let's undismiss it just in case. That post says:

The atheist tends to believe that all reality can be brought into human understand, — Metaphysician Undercover

I can see no logical connection between lacking a belief in God and believing that everything can be understood. I know know-it-all theists and mystical, I-know-nothing atheists, as well as know-it-all atheists and mystical, I-know-nothing theists. The two dimensions are orthogonal.

At best there could be a correlation but I don't even see any sign of that. Do you have any evidence for this claim other than a throwaway line here or there from a celebrity atheist?

To say "God is unintelligible to the human intellect" is to say something meaningful about God. — Metaphysician Undercover

OK, then let's use a version of Socrates' famous dictum. The humble theist says 'the only thing I know about God is that I know nothing else about God'. It is, in my experience, a rare theist that exhibits that humility. It seems that Kant may have been one, and some mystics.

It seems like there's a point you want to make in your two posts, but I can't see what it is. Are you able to explain the point? It might be easier to understand if it's more than just a curt one-liner.

What doesn't resemble theology? Also, how is theology relevant? You are the first in this thread to mention it.

One way out of the dilemma is to discard the idea that there is a sharp divide between objectivity and subjectivity, and replace it by the recognition that we all assign different degrees of belief to different propositions. We tend to call those propositions in which we have a high degree of belief 'objective' and those with a low degree 'subjective'.

I would say, given the information available to me, that it's possible that there's a blue ball in the second box and that it's possible that there's a green ball in the second box, with a 50% probability of each. The sample space for the other box is [blue, green]. — Michael

Probability is used to model uncertainty, and nearly all uncertainty in our world is epistemological. If one is a Hard Determinist (a term that I think is not well-defined, but let's leave that aside for now) then ALL uncertainty is epistemological.

One constructs a probability space based on one's knowledge, so there is no absolute probability space that models a game but rather a probability space that models a particular stakeholder's perspective on the game.

In the above case the probability space you describe that is blue and green, each having 50% probability, is an appropriate probability space for the player of the game. For the game host however, who knows whether the second box has blue or green, the probabilities are either 0 and 1, or 1 and 0.

A minor technical point: the sample space is the set of all conceivable outcomes (called 'events'). I deliberately say 'conceivable' rather than 'possible' because it can contain events that the person knows to be impossible. A probability space is a sample space together with an assignment of probabilities to each event, as well as some other technical stuff (sigma algebras) that we needn't go into here. A probability space can assign zero probabilities to some events, as long as the sum of all assigned probabilities of events in the sample space is 1. Events that the space's 'owner' knows to be impossible will be assigned probability zero.

So the game host and the player can have the sample space. But they will assign different probabilities to events in it, so they have different probability spaces. If the second box holds green, the host will assign 0 probability to the event 'blue' while the player will assign 0.5 to it.

Does the answer depend on whether or not one is a Bayesian? — Michael

I'm not sure. I feel the answer may be 'perhaps', but the definition of the Bayesian vs Frequentist divide seems to be very fuzzy. I think a hard-line Frequentist may reject the epistemological interpretation, but that would seem to render them unable to use most methods of modern statistics. EIther I've misunderstood what frequentism is, or there are very few hard-line Frequentists in the world.

BTW, Mary being "immaculate" just means she was born without original sin (see this) — Relativist

Yes I know. As a former RC I find myself constantly being tempted to correct people that say or imply that the doctrine of the Immaculate Conception says that Mary conceived Jesus without having sexual intercourse - a temptation I often fail to resist :snicker:. The question I'm raising there is whether proponents of the free will theodicy defence are cornering themselves into saying Mary had no free will, since removing a tendency to commit 'sins' (shielding her conception from the taint of original sin) sounds to me like depriving them of free will. After all, if God could do that for Mary, why didn't She just do it for everybody, and that way make sure that everybody goes to heaven?

D'oh! You are right. What a silly mistake.

Here is a proof that addresses the actual question being asked. We need to introduce another variable z because when swapping two things completely, we need to use a third as a transit point.

@SophistiCat This error of mine may have contributed to some of the crossed wires on this. Sorry about that.

1. ∀x.∀y.p(x,y) ⊢∀x.∀y.p(x,y)
2. ∀x.∀y.p(x,y) ⊢∀x.∀y.p(x,y) → ∀y. p(z,y) [Q5 with x:=z]
3. ∀x.∀y.p(x,y) ⊢∀y.p(z,y) [Modus Ponens on 1, 2]
4. ∀x.∀y.p(x,y) ⊢∀y.p(z,y)→ p(z,x) [Q5 with y:=x]
5. ∀x.∀y.p(x,y) ⊢p(z,x) [Modus Ponens on 3, 4]
6. ∀x.∀y.p(x,y) ⊢∀z.p(z,x) [Generalisation metatheorem on 5]
7. ∀x.∀y.p(x,y) ⊢∀z.p(z,x)→ p(y,x) [Q5 with z:=y]
8. ∀x.∀y.p(x,y) ⊢p(y,x) [Modus Ponens on 6,7]
9. ∀x.∀y.p(x,y) ⊢∀x.p(y,x) [Generalisation metatheorem on 8]
10.∀x.∀y.p(x,y) ⊢∀y.∀x.p(y,x) [Generalisation metatheorem on 9]
11. ⊢∀x.∀y.p(x,y)→∀y.∀x.p(y,x) [deduction theorem (metatheorem) on 10]

We can then switch y and x and follow the same steps to prove that

⊢ ∀y.∀x.p(y,x)→∀x.∀y.p(x,y)

and putting the two together we get

⊢∀x.∀y.p(x,y)↔∀y.∀x.p(y,x)

I think that's right but if I've mixed up any letters while rearranging it, please let me know.

I am not sure but I have a feeling you are referring to this, the semantic interpretation of a logical language.

To make a logical theory meaningful, we usually need need to have interpretations to which it applies and that are important to us. There are a few exceptions such as Godel's incompleteness theorem, but generally theories are only interesting if they have interpretations. In the fourth para of that linked section it talks about how quantified expressions involving variables can be interpreted in the model. Does that relate to the concerns you outline above?

For the case of the OP, it is purely a syntactical exercise, devoid of any interpretation.

The theist recognizes the vast reality which is beyond the capacity of human understanding, and that the unintelligibility of God is a reflection of this. — Metaphysician Undercover

My experience is quite the contrary of this. Most theists I've encountered do not recognise that at all. Instead they write and speak at length about alleged properties of God - what She can do, what She wants, what She thinks, what She has said, what books She has dictated.

The theist that agrees that God is unintelligible and we can say nothing meaningful about Her is a rare beast indeed - but all the more admirable for that.

But the caricatures which Dawkins makes out of God - the Flying Spaghetti Monster, orbiting teapot, celestial potentate - are indeed the figments of his own imagination.

Whoa there! The teapot is from Bertrand Russell who, I think it should be acknowledged, engaged charitably with Christianity while rejecting its claims, at least it seems that way from his discussions with Father Coplestone. I'd be pretty confident that Russell had read Aquinas and understood the claims of classical theism.

And the celestial potentate - Alan Watts was discussing that in the sixties, long before Dawkins got involved. Watts had a typically nuanced perspective on that. Although he pointed to how comic that myth was, and how ironic it was that America was so proud of being a republic yet favoured a monarchic theology, he also stood up for it, saying that it was no less idolatrous than Protestant idolisation of the Bible, Paul Tillich's notion of Undifferentiated Ground of Being or an Enlightenment atheist's reverence for Progress. Watts was very pro-myth. He saw it as poignant that Protestant ministers would become enthused by all sorts of interesting notions of God in the seminary but then be expected to trot out simplistic Stern-Daddy-God platitudes to their congregations, because they paid his wage.

If the player adopts this as the sample space then they cannot use a uniform distribution as their Bayesian prior for X, because there cannot be a uniform distribution on the natural numbers. Hence a 'tailing-off' distribution would need to be selected, such as P(X=n)=2^-n for n=1, 2, ....

Given the adoption of a Bayesian prior distribution, the higher the observed value of Y (the value in the opened envelope), the greater the chance that it is 2X rather than X, and hence the less the expected gains from switching. Since a strategy of compulsory blind switching delivers expected gains of zero, a strategy of choosing an amount H and switching only when Y<H must have an expected positive gain.

for a single trial, your guess would have to be awfully lucky to be any help. — Srap Tasmaner

Why? It's not about helping, it's about her expected gain, which is a pure calculation and nothing to do with luck. And my analysis is for a single trial.

I have a lot of sympathy with the argument, but I'll have a go at Devil's Advocate:

I think what the argument might be is that what matters to God is that every being has had the opportunity to make free decisions. So the beings in heaven have had that ability and, by the decisions they chose to make, have qualified for entry to heaven. The arguments that are used to say that, for instance, a being that is shielded from temptation, or created immaculate like Mary, would not have free will, can be equally applied to say that, once in heaven, a being no longer has free will (they are now shielded from temptation, or rendered immaculate). But perhaps it is enough for an apologist that the qualified being once had free will.

I think Nietzsche would be a natural philosopher to turn to for such arguments. His philosophy seems to me to be about embracing your passions rather than controlling them. I say that as somebody that is much influenced by Stoicism but also is fascinated by Nietzsche.

We'll label the envelopes L and R. — Srap Tasmaner

Please Nooooooo! Let's not do that. I reserved the symbol L many pages ago for the maximum possible value ('Limit') in the player's prior distribution for 2X. If we start using it for something else, like one of the envelopes, we'll end up a terrible muddle. :joke:

I'd be interested in your thoughts on this post, and whether you've tried doing a simulation using the simple strategy outlined there. The discussion has been very fluid but I think I'm agreeing with you that the player's expected gain from blind switching is zero, but her expected gain from switching based on comparing the value of Y to L/2 is strongly positive (here L has the original meaning of Limit, not Left).

Thank you for that. Now I can check my progress in PHP interpretation by comparing the two.

It's a .25 gain in my program. — Michael

That's a different metric than the ones I quoted. Mine is in dollars while yours, if I'm reading the PHP correctly, is the dollars divided by the total of all Ys observed, minus 1.

Measured on the same basis as yours, my model gives 0.5. That is more because mine uses L from the beginning, rather than progressively adapting it based on observations. My suggestion is that we set L to be the budget of the game show, because we know they can't pay more than that. If we don't know that we could use a suitably small proportion of the net assets of the company running the game show. Or if that's unknown, a suitably small proportion of the GDP of the country it comes from (or of the world, if we don't know the country).

Which approach is appropriate may depend on the exact setup of the repetitions. Yours appears to be a serial one, where the info from previous trials is available to the player in subsequent trials. Alternatively, we could have a parallel one, where a team of 10,000 players do one trial each simultaneously, and aim to maximise their team winnings.

There's also the question of whether we use the assumed prior distribution of X to model the possible values of X. To not do that requires having an additional distribution that models the uncertainty in our prior distribution, which is a bit too meta- for me at this time of night. I'll sleep on it. If we start with L estimated as above, in the serial case we could update it (improve it) in a Bayesian fashion based on each new observation of 2X.