Is/does she paid/pay this $1 on both days, or on Wednesday after the experiment is over? In the latter case, can she choose not to have amnesia, and then choose "Heads" if she recalls no other waling but change that to Tails if she does? — JeffJo

I simplified my thought experiment to say that she loses $1 on Monday if heads comes up, otherwise she loses nothing. No stake is involved, and she isn't asked to bet on the outcome during the course of the experiment.

It boils down to the following question. Should she risk a single draw from

P(Flip = Heads | Today = Monday) = 1/2 (no amnesia)

Or should she risk a single draw from

P(Flip = Heads | Today = Monday Or Tuesday) = x (amnesia)

According to halfers, x = 1/2, implying that she gains nothing by choosing amnesia.

According to thirders, x = 1/3, indicating that she gains from choosing amnesia.

Say I draw a card. After I look at it, I tell Andy that it is a black card, Betty that it is a spade, Cindy that its value is less than 10, and David that it is a seven (all separately). I ask each what they think the probability is that it is the Seven of Spades. Andy says 1/26, Betty says 1/13, Cindy says 1/32, and David says 1/4. All are right, but that does not affect my draw. I had a 1/52 chance to draw it. — JeffJo

If you are referring to a context involving repeated trials, then all of your probabilities are physically meaningful posterior probabilities , for we are confident on the basis of our past experience with card games that the frequencies will roughly obtain. In which case none of your above probabilities are appealing to the controversial principle of indifference.

In stark contrast, Elga (for example) invokes the principle of indifference on the basis of ignorance to assert

P(Monday | Tails) = P(Tuesday | Tails)

Leading to him to a conclusion the conflicts with his knowledge of fair coins and that encourages avoidance coping. What he ought to have done, is to represent his ignorance with the maximal set of permissible distributions and to assign confidence bounds, until as and when he has reason to whittle the set down to a smaller subset and make more specific predictions.

Or if he really must assign a single distribution (i can't think of a good reason), to choose one whose deductive implications cohere with his broader state of knowledge.

1/3 would be her prior upon awakening and before being informed that the day is Monday. Upon being informed of this, Sleeping Beauty would update her credence to 1/2. — Pierre-Normand

That is true according to thirder's logic, but it doesn't alter their fallacious implication that consenting to amnesia before the experiment improves SB's chances of getting tails. If SB accepts that amnesia isn't a relevant causal factor of the coin's outcome, then she must avoid using a self-location credence that has that implication.

Similar magical thinking is demonstrated in avoidance coping. A poor person who has financial anxiety might raise their credence that their bank balance is in the black by choosing not to read their financial statements. Having accurate credences can be painful and reduce one's ability to cope in the short term, so there are short-term psychological benefits in choosing ignorance.

If she has amnesia she should guess heads and will will 2/3 of the time.

If she doesn't have amnesia she should guess either on the first wake up (1/2 probability so doesn't matter which she guesses) and she should guess tail with absolute certainty if she remembers having woken up before (ie on her second wake up). Again she will win 2/3 of the time.

So having amnesia or not does not change the probability that she will win, but the tactics she should use are different. — PhilosophyRunner

In my argument, SB isn't asked to guess anything during the experiment. To make things really simple, let us only suppose that SB will lose $1 if the coin lands heads, without a wager being involved.

According to thirders, if she has amnesia then

P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday) = 1/3

So, according to thirders her probability of losing $1 when awakened on monday in a state of amnesia is 1/3.

On the other hand, if she doesn't have amnesia when waking up on Monday then thirders will agree that her probability of losing $1 is 1/2.

So according to thirder logic, it is rational for SB to consent to having amnesia before the experiment begins, in order to bias the coin's outcome towards tails.

Here's another criticism of the thirder position: Their reasoning implies that self-induced amnesia is a valid strategy for controlling outcomes:

Suppose that SB gets paid $1 if the coin lands tails, otherwise she must pay $1. Furthermore, suppose that before the experiment begins she is given the choice as to whether or not she will have amnesia during the course of the experiment. According to thirder reasoning, she should choose to have amnesia in order to raise the probability of tails to 2/3

There's no inconsistency here. It's precisely because the premise holds that the coin is equally likely to land heads or tails when tossed, and the fact that tails would result in twice as many awakenings, that Sleeping Beauty deduces that she could accept 2 to 1 odds on each awakening and break even in the long run. — Pierre-Normand

Let P(Flip | Today ) represent the conditional probability of the state of coin, relative to the value of SB's indexical "Today". Then

2/3 = P(Flip = Tails | Today = Mon Or Tue) = P(Flip = Tails) = 1/2

What is wrong with this conclusion of inconsistency?

But indifference with respect to the joint outcomes (Day, Flip) implies not only

(1) a "posterior" probability for heads that differs from it's prior probability in spite of not learning anything upon awakening and before interacting with the world upon waking up.

but also

(2) different credence assignments for marginals one is equally ignorant of.

So even if (1) is acceptable for Principle of Indifference advocates, how can they justify (2) that flagrantly disregards their beloved principle?

From the premises

P(Flip) = 1/2
P (Day = mon | Flip = heads) = 1
P (Flip = tails | Day = tue) = 1

If a PoI advocate assigns the credence

P(Day = mon, Flip = tails ) = P(Day = mon , Flip = heads) = P(Day = tue, Flip = tails) = 1/3

Then his unconditioned credence that it is Monday is

P(Day = mon) = 2/3

which is clearly not the expression of indifference.

Then for sake of consistency he must ignore the actual information he is given about the coin, by using

P(Flip = tails) = 2/3

Not only does this credence assignment ignore the physical probability of heads that is known to him in advance, but it also indicates different credence assignments for outcomes of Flip when assuming ignorance of Flip! - in flagrant contradiction to PoI.

In general, when a PoI advocates uses PoI for specifying an unknown distribution, they don't possess the epistemic resources to determine the credences that they are implying regarding the values of related variables. And when their implied credences are pointed out to them, they are likely to withdraw their initial credence estimates.

IMO, there isn't a good reason for using the principle of indifference, not even in the case of describing credences. For handling ignorance, it is always better to assign probability intervals than to assign precise probabilities.

I would say that neither the halfers nor the thirders position concerns credence. Plus the concept of "credence" is a minefield of unresolved philosophical difficulty, which is usually overlooked by advocates of probabilism.

The halver's position is based on the common-sense postulate that the physical properties and propensities of objects remain unchanged when their states are conditioned upon irrelevant information. Given acceptance of that postulate plus the premise of a fair coin, why shouldn't the halver insist that they are talking about reality, as opposed to their mere mental state?

The thirder's position is indeed a ratio of possible words, but there is scant evidence to support the idea that credences are accurately represented by taking ratios over possible worlds.

For example, if one were to ask the average person to express their credences regarding the outcome of a two horse race that they know absolutely nothing about, they will simply say "I don't know who will win" and refrain from assigning any odds, equal or otherwise. They will also tend to accept bets in which they have knowledge that the physical probabilities are 50/50 over bets that they are totally ignorant about.

Fundamentally, it is difficult to even know what the object of a credence or belief is, which relates to an overlooked problem of epistemology: If every belief is fully explained by it's natural causes, then how are false beliefs possible and where does the error reside? In the mind of the speaker who expresses their beliefs, or in the mind of the listener who interprets the speaker?

Whatever SB says when interrogated upon awakening, if she is assumed to have amnesia then how are her interrogators supposed to interpret her "credence estimate"? Why should they consider her uttered "credence estimate" as being qualitatively different from yawning?

I had a hard time understanding what you were conveying, as I think we just use terminology differently, so let me ask some questions pertaining thereto.

Subjective Idealism and solipsism aren't ideas

By “idea”, I was meaning it in the colloquial sense of the term. Technically, those are metaphysical theories. One is a sub-type of idealism that does not hold there is an objective reality but, rather, that all that exists is to perceive or to be perceived (e.g., the tree doesn’t exist other than an image within your perception). The other is the theory that all that exists is one’s own mind, or, epistemically speaking, one can only know the existence of their own mind. — Bob Ross

The question here concerns whether realism and idealism are truth-apt synthetic propositions, with each representing competing theories that describe different and incompatible possible worlds, or whether they are analytic tautologies that are referring to different aspects of the logic of language and are merely talking past one another. Notably, realists and idealists each consider their stances to be irrefutable parts of commonsense, and are concerned with understanding the semantics of language in opposite directions from opposite starting points.

"Being is perception" is an unavoidable tautology of non-representational idealism that is necessarily appealed to whenever an observer interprets a physical proposition in terms of his personal experiences

I don’t see how this is true. For example, both physicalists and analytic idealists hold that being is more than perception. No one inevitably speaking in terms of their experiences forcing “being” to be perception. Why would that be the case? — Bob Ross

If your "analytic" idealism abandons "esse is percipi" how does it differ from representational realism?

Berkeley's subjective idealism was already "analytic" in the sense that he postulated that observation and conception is tautologically equivalent to existence. Many philosophers misunderstand this principle.

For example, they take the principle to imply that unobserved items disappear from existence. But this doesn't follow from the principle, for according to the principle it isn't false that unobserved objects exist, but nonsensical.

What Berkeley's principle is actually saying, is rather trivial ; that only what is observed or conceived can be thought or talked about. If a realist asserts that "unperceived objects such as quarks exist", Berkeley wouldn't contradict the content of the assertion but remind the realist that his use of "unperceived" requires elaboration until it refers to something thought or perceived, for the assertion to become sensical.

I'm not redefining y, the switching argument is. I'm showing you what it covertly does. — Michael

I haven't redefined y, and still derive the switching argument from it's premises.

Notice that in E(z) the variable y stands for 3 different values. In one case it stands for the value of the smaller envelope (10), in another case it stands for the value of the larger envelope (20), and in the final case it stands for a different value entirely (12). — Michael

Yes, I see that. So why are you redefining y?

Your two definitions of E[z] aren't equivalent. The first one is implicitly referring to
E[ z | y], the conditional expectation of z given the value of envelope y. Whereas the second definition refers to E[z | x] the conditional expectation of z given the value of the smallest envelope x.

I believe it does, as I showed above. It covertly redefines y
such that when it concludes E(z)=54y is no longer the value of the chosen envelope. — Michael

No covert redefinitions of y are happening, just covert pathological expectations over infinity. For the problem implicitly assumes

$P(x) = c$

This is an improper prior that has a divergent mean.

$E [x] = \infty$

The distribution of envelope values conditioned on the smallest value is symmetric, consisting of two spikes on the infinite plane at (x,2x) and (2x,x)

$P(y , z| x) = 0.5 \delta(x,2x) + 0.5 \delta (2x,x)$

Summing out, say, z gives

$P(y | x) = 0.5 \delta(x) + 0.5 \delta (2x)$

The two previous formulas imply the conditional

$P(z | y, x) = 0.5 \delta(2x)^{y=x} + 0.5 \delta(x)^{y=2x}$

since $P(x)$ is improper, one cannot "integrate out" x from the last equation and get a well-defined distribution. Instead one gets a function defined up to proportionality.

$P(z | y) \propto 0.5 \delta(2y)+ 0.5\delta((1/2)y)$

Taking the expectation of $y$ over this function as if it were a distribution gives

$E [ z | y ] = \frac 5 4 y$

Hence by symmetry

$E [ y | z ] = \frac 5 4 z$

No redefinitions, covert or otherwise, of any variable were involved here.

But I perhaps should correct myself a bit; the expectation values aren't contradictory in the sense of

$E [ y | z ] \gt E [ z | y ] \gt E [ y | z ]$

rather, the expectations imply contradictory strategies in the context of utility maximisation.

Yes, my previous criticism was directed more towards the confusing use of notation with respect to expectation values. It is clearer to condition expectation values on the variables and information they assume.

But the switching argument isn't invalid; It's contradictory expectation values don't appeal to faulty reasoning given acceptance of the premises. Rather the switching argument is unsound, for among it's premises is an improper prior distribution over x, the smallest amount of money in an envelope. And this premise isn't possible in a finite universe.

Intuitively, it's contradictory conclusions makes sense; if the smallest amount of money in an envelope could be any amount of money, and if the prior distribution over the smallest amount of money is sufficiently uniform, then whatever value is revealed in your envelope, the value of the other envelope is likelier to be higher.

I think you're conflating two different expectations. I think your post should read :

Let $y$ be the value of the chosen envelope and $z$ be the value of the unchosen envelope.

1. Let $y = x$ or $y = 2x$

2. $E[z | y ] = {5\over4}y$ and $E[y | x] = {3\over2}x$

MacTaggart's remarks concerning the A and B series are relevant here. The premises of the SB paradox are tenseless and so refer only to the B series , whereas Sleeping Beauty's question is tensed and so refers to her ability to correlate her A series to her B series.

The problem's premises can be written

$\begin{array} {|r|r|}\hline Sample\ Space & Awoken\ Mon & Awoken\ Tue & Joint\ Probability \\ \hline Heads & True & False & 1/2 \\ \hline Tails & True& True & 1/2 \\ \hline \end{array}$

According to this parameterisation, it doesn't make sense to ask SB "what Day is Today?" for "Today" isn't a random variable of the sample-space. (Thirders implicitly ask this question). But all that can be talked about, according to this parameterisation, is the state of the coin and whether SB is awoken on both monday and tuesday, and not "what day is today?".

To ring home the point, suppose that if the coin lands tails then the experimenters change SB's location from London on Monday to Paris on Tuesday. in which case, we can eliminate the notions of time and date from the premises of the the problem, by changing "Monday" to "London" and "Tuesday" to Paris.


Recall that MacTaggart decided the A series to be "unreal" and contradictory. From the B series point of view, "today" is subjective and refers to SB's mental state that comprises a variable that is separate and independent of the "calendar" events in the B series, such as wakening a subject up in London and then in Paris. Under the assumption of SB undergoing amnesia however, her mental state is uncorrelated with her understanding of B series events, and hence uncorrelated with the state of the coin.

No, the question has nothing whatsoever with her mental state.

She is being asked, given that she is awakened, what is the probability of heads. If she is awakened 1000 times for every tails and once with heads, given enough coin flips you can see that it is overwhelmingly likely to be tails, even though the probability of heads remains 50%. This is independent of her mental state. — hypericin

Her "mental state" M refers to her epistemic state upon waking. In the context of repeated trials, it would include her knowledge of previous trials. But the paradox doesn't assume repeated trials. If it did in the manner of your example, then

P ( C = tails |M) = 1000/1001

In which case, her mental state has proven to be a very reliable indicator of the state of the coin.

But this isn't part of the question's premises. We are told that in the context of a single trial, that her mental state is conditioned so as to have no previous memories of waking up. This condition implies that her mental state is statistically independent to the state of the coin, i.e.

P(C | M ) = P(C) = 1/2

Here i have assumed that she is told that the physical probability of the coin landing heads is 1/2. If she isn't told this, then she should refrain from assigning any distribution to P(C | M)

Good question: no. Solipsism is the idea that everything is in my mind, whereas analytical idealism is the idea that both our minds are in a universal mind. — Bob Ross

In my view :

Subjective Idealism and solipsism aren't ideas, but a tautological understanding that the meaning of all propositions is ultimately reducible to whatever is perceived or thought in the first-person at the end of the day.

Naturalism isn't an idea, but an understanding that the meaning of inter-subjectively valid propositions, such as those concerning the properties of natural kinds, cannot be identified with particular thoughts and experiences of the first person. For this reason, scientific naturalists talk about meaning in terms of potential experiences through the use of conditionals, counterfactual analysis , and perspectivally invariant abstract properties.

But this shouldn't be taken to imply that naturalism discounts experience as being it's semantic foundation - after all, naturalists pride themselves for judging the validity and soundness of their theories in terms of empirical evidence. And every naturalist must determine for themselves how they should privately cognize the inter-subjective propositions expressed by their fellow community. The assumption that naturalists can relate to their subject matter implicitly appeals to the existence of semantic "bridging" rules for converting the inter-subjective representations of naturalistic language into phenomenal first-person understanding.

So i don't consider Naturalism and solipsism or idealism to be incompatible per-se. I see them as comprising different semantic aspects of thought and language. Nonetheless, their logics are radically different, lending to the false impression of conflict.

E.g "Being is perception" is an unavoidable tautology of non-representational idealism that is necessarily appealed to whenever an observer interprets a physical proposition in terms of his personal experiences (regardless of whether he self-identifies as an idealist) .

On the other hand, "perception is representation" is an unavoidable tautology of naturalism for universalising intersubjective semantics in an abstract fashion that isn't dependent upon the perceptual judgements of any particular observer.

Taken together, "Being is Perception" and "Perception is Representation" don't necessarily imply that "Being is Representation", as is often naively assumed by materialists, if one understands these principles as referring to different and non-overlapping aspects of semantics.

Here's another analysis that only refers to credences , i.e subjective probabilities referring to the mental state of a believing agent - as opposed to physical probabilities referring to the physical tendencies of "mind-independent" reality.

According to this interpretation of the paradox, the paradox is only psychological and concerns the mental state of an agent who derives contradictory credence assignments that conflict with his understanding of his mental state. So this interpretation isn't adequately analysed by appealing to a physical model.

Suppose the participant called Bob, before opening either envelope, tells himself that he knows absolutely nothing regarding the smallest quantity of dollars S that has been inserted into one of the two envelopes:

Before opening either envelope, Bob reasons that since he knows absolutely nothing about the value of S, that he should appeal to Laplace's principle of indifference (PoI) by assigning equal credence to any of the permissible values for S. He justifies this to himself by arguing that if he truly knows nothing about the value for S, then he doesn't even know the currency denominations that is used to describe S. So he assigns

P(S = s) = P(S = 2s) = P(S = 3s) = P(S = 4s) ..... for every positive number s.

There is only one "distribution" satisfying those constraints, namely the constant function P(S) = c ,
that cannot be normalised, where c is any positive number which can therefore be set to c = 1. This is called an 'improper prior', and it's use often results in conflicting credence estimates, as shown by other paradoxes, such as Bertrand's Paradox.

Having chosen this so-called "prior", Bob reasons that when conditioned on the unknown quantity S, the unknown quantity X in his unopened envelope has the value S with a subjective probability p, else the value 2S with subjective probability (1 - p):

P(X | S) = p Ind (X,S) + (1- p) Ind (X, 2S) (where Ind is the indicator function)

He again appeals to PoI and assigns p = 1/2 (which merely a non-informative proper prior)

Substituting his choices for P(S) and p, Bob realises that the unnormalised joint distribution P(S,X) describing his joint credences for S and X is

P (S , X) is proportional to 0. 5 Ind (X ,S) + 0.5 Ind (X , 2S)

Summing over S, he derives his credences for X, namely P(X) that he realises is also an improper prior.

P(X) is proportional to 1

Consequently, his subjective 'unnormalized' posterior distribution (which does in fact sum to 1, but is nevertheless the ratio of the two unnormalised distributions P(S,X) and P(X) ) is described by

P (S | X) 'is proportioanal to' 0. 5 Ind (X ,S) + 0.5 Ind (X , 2S)

Bob wonders what would happen if he were to naively compute expectations over this 'unnormalised' distribution. He decides to compute the implied expectation value for the unopened envelope V conditioned on the value of his unopened envelope:

P (V = 2x | X = x ) = P(S = x | X = x) = 0.5
P(V = 0.5 x | X = x) = 1 - P(S = x | X = x) = 0.5

E [V | X ] = 5/4 X

Bob decides that he cannot accept this expectation value, because it contradicts his earlier credences that are totally agnostic with regards to the states of S and X. However, Bob also knows that this conditional expectation value is a fallacious value, due to the fact that his subjective probability distribution P(S | X) isn't really normalised, in the sense of it being the ratio of two unnormalised distributions P(S,X) and P(X).

Bob therefore knows how to avoid the paradox, without needing to revise his earlier credences.

Crucially, Bob realises that his 'unnormalised' subjective distribution P(S | X) should only be used when calculating ratios of P(S | X) .

So instead of strongly concluding that E [ V | X ] = 5/4 X that involved averaging with respect to an unnormalised posterior distribution P(S | X), he reasons more weakly to only conclude

P(V= 2x | X = x) / P( V = 0.5x | X = x) = 1

Which merely states that his credences for V=2X and V=0.5X should be the same.

So if Bob is mad enough to reason with subjective probability distributions (which IMO should never be used in science, and which can be avoided even when discussing credences by using imprecise probabilities), Bob can nevertheless avoid self-contradiction without revising his earlier credences, simply by recognising the distinction between legitimate and non-legitimate expectation values.

Since you're an R user, you might find it interesting to define a model in RStan, using different choices for the prior P(S) for the smallest amount S put into a envelope. Provided the chosen prior P(S) is proper, a sample from the posterior distribution P( S | X) , where X is the observed quantity of one of the envelopes, will not be uniform, resulting in consistent and intuitive conditional expectations for E [ Y | X] (where Y refers to the quantity in the other envelope)

No, the question is what is the probability SB experiences an awakening with the coin being heads — hypericin

Unless additional premises are included in the problem, I cannot tell the difference between how I phrased the question and your phrasing of the question.

Upon being awoken, SB is asked: "What is your credence now for the proposition that the coin landed heads?" "

She is therefore being asked "What is P(C | M) , where M is your current mental state?"

But the premises don't specify anything regarding the correlation of SB's mental state M to either the state of the coin C or the day of the week D.

So we can only interpret P (C | M ) as referring to P(C) .

Yes, we're in full agreement. By "non-informative" I was referring to the distributional conditions of both uniformity of probability mass and of infinite support . But you're right in pointing out that a "non informative prior" is often used by Bayesians to refer only to uniformity of probability mass, in which the range of the support is considered to be a separate independent hyper-parameter.

Have you ever tried experimenting with psychedelics?

You might be suffering from a biologically rare type of depression that isn't treatable by usual methods.

Maybe a different example. I have a red ball hidden in one hand and a blue ball hidden in my other hand. You point to one of my hands at random. What is the probability that you pointed to the hand holding the red ball? It's 1/2. — Michael

It depends on what interpretation of probability you are appealing to. For those of us who reject Laplace's principle of indifference, the answer is to refrain from asserting a subjective probability.

In any case, it isn't relevant to the two envelopes problem, for It can be reproduced by appealing to a causal interpretation of probability in which a person observes the contents of an envelope he is given, without him making any decisions. A Bayesian analysis reveals that the culprit of the paradox is the assignment of a non-informative prior to the distribution that generates the envelopes contents.

Without that assumption, the conditional expectations involved behave sensibly and the paradox dissolves.

That’s not what happens in this example. I am shown two envelopes, one containing £10 and one containing £20, and I freely choose one at random. I don’t open it. The probability that I picked the one with £10 is 1/2. — Michael

I'll agree for sake of argument . I think the problem is how we are fitting our shared understanding of the problem to probability calculus.

In my preferred description, one of the envelopes is opened to reveal a quantity A, but It isn't known as to whether the other envelope is more than or less than A.

In your preferred description, the quantities of both envelopes is known a priori, but neither of the envelopes are opened.

The problem with your description, is that it runs contrary to how conditional probabilities and expectations are normally interpreted. For the information upon which a probability or expectation is conditioned, is normally interpreted as observed information, rather than possessed information that isn't revealed, compensated by assumed knowledge of other quantities.

The paradox is premised on not knowing the value of any. — Michael

That is flat out contradicted by the switching argument. Furthermore, without the premise of knowing the value of one of the envelopes, the paradox disappears.

To have an expectation value containing A on the one hand, and insisting that the envelope isn't opened on the other, is a bait and switch. Wikipedia's article isn't written very well, which might be part of the confusion.

But do you agree that the probability in my example situation is 1/2? — Michael

It is a half if you assume it to be 1/2, but not necessarily. Consider for instance someone sending you the smaller of two envelopes through the post, according to a probability that they have decided. You open the letter and are informed that if you return the envelope and it's contents, you will receive another envelope that has half as much or twice as much.

My argument with you is over the assigned probabilities. So ignore the expected value. I just want to know an answer to this:

1. One envelope contains £10
2. One envelope contains £20
3. I pick an envelope at random
4. I don't open my envelope
5. What is the probability that I picked the envelope containing £10?

My answer is 1/2. What is yours? — Michael

The paradox doesn't apply in that scenario, since the values of both envelopes are given.

To my understanding , the paradox requires,

1) Knowledge of the value of only one of the envelopes.

2) The assumption of a non-informative prior for P(M) , where M is the smallest (or greatest) money in the envelopes.

In particular, the puzzle is not solved by finding another way to calculate the probabilities that does not lead to a contradiction. — Michael

The contradiction vanishes when a normalised prior is used for P(M), since in that instance knowledge of the value of one envelope is indicative of the value of the other.

There is no opened envelope: — Michael

The switching argument begs to differ :

" 1. Denote by A the amount in the player's selected envelope."

The argument's computed expectation value of (5/4) A is a conditional expectation with respect to the variable A denoting the amount of money in the player's selected envelope.

Not opening the envelope and taking an expectation over the envelope's value means that A cannot be mentioned in the expectation value, even as a variable with an unspecified value.

The unconditional expectation of the players envelope value is 0.5 x M + 0.5 x 2M = 1.5M , where M is the mean of the unspecified distribution F for the smallest amount of money in an envelope. No paradox arises from this calculation.

I've subsequently come to realise however, that my initial analysis wasn't quite right. If my calculations are correct (that are too tedious to post here), the switching dilemma is caused by assuming an un-normalised "non-informative prior" for F(M), which causes the conditional expectations for each envelope to diverge.

Intuitively, this can be explained as follows:

if there could be any amount of money in the two envelopes, and if all monetary values are equally likely, then whatever value you observe in one envelope, the other envelope is likely to have an even higher value.

To remove this bug, one has to replace the physically implausible non-informative prior for F with a normalised distribution, so that the conditonal expectation for the unopened envelope gets lower as the observed money in one's opened envelope gets higher.

Thanks.

The premises of the Sleeping Beauty, at least in some popular incarnations of the problem, specify an experimental design, but they don't specify or assume how temporal self-location is related to that design. The design only specifies a few constraints relating the physical outcome of a coin toss C to the actual days of the week D that a subject is awoken. But an awoken subject's mental state M comprises yet another variable that has no specified relationships to C or D in the question's premises.

The question is literally asking, on the basis of a partial definition of (C,D) what is the marginal distribution of P(C), which it's premises already state is 1/2. So if it is intending to probe the valid philosophical question pertaining to self-location it needs to be more candid and ask

What is the probability P (C = h | M) ?

But if nothing about M and it's relationships to C and D are specified or assumed, one isn't in a position to assign conditional probabilities, and can only respond with the vacuous answer " P(C = h | M) is between 0 and 1. "

Always start by writing down the probability of everything :

Variables :

C = Tossed coin (Binary variable in {head, tail} )
D = Awoken Day (Binary variable in {mon, tue} )

Unknown Probability Measure :

P (C, D) = P (C | D) P(D) = P(D | C) P(C)

Constraints given on the Probability Measure :

P (C = head) = 1/2
P (D = mon | C = head) = 1
P (C = tail | D = tue) = 1

Task :

Determine the marginal distribution P (C = head ) from the above premises

Answer :

P(C = head) = 1/2 (by premise)



Why was the scientific american wasting time on this?

What is the probability that your envelope contains twice as much money as the other? It's 1/2. — Michael

Again, that's an additional subjective premise that isn't objectively implied by the 'physics' of the two-envelope premise. Nevertheless we can assume it for sake of argument and continue the discussion as an exercise in subjective Bayesian probability estimation in which we hope for self-consistency.

There's no reason that the probability in the second case should be different to the probability in the first case. — Michael

Yes there is. That's why conditional probabilities should be used. The conditional probability

P ( B = b | A = a)

that refers to the amount of money in the unopened envelope B when conditioned on the amount of money in opened envelope A, is generally unequal to

P (A =a | B = b)

that refers to the amount of money in unopened envelope A when conditioned on the amount of money in opened envelope B.

Knowing the former conditional distribution upon opening envelope A generally says nothing about the latter distribution unless the ratios of the priors P(A=a) and P(B=b) is assumed to be 1, as indicated by Bayes Theorem.

Only if you take the ratio to be 1 do the calculated subjective conditional expectations come into conflict with respect to decision making.

Assuming your subjective premise:

- Recall the fact that the implied subjective expectation regarding the amount of money gained upon opening either envelope is 1.5x, where x is the smallest amount of money (as i showed in my first analysis).

- Also recall that unlike in my first analysis that referred to the envelopes by labels that were assigned to them prior to them being opened, here we are defining A to refer to the opened envelope. In which case the probability that we will open A is 1!

Doesn't the fact that you will definitely open A first, together with the fact that your expected gain is 1.5x imply that your subjective prior for P(B) should be different to your subjective prior for P(A)?

It just assumes that:

P(A = the smaller envelope) = P(B = the smaller envelope) = 1/2 — Michael

That expression is used to represent the same set of initial assumptions, but is less explicit with regards to its premises, such as the fact that some distribution is responsible for placing a certain amount of money in each envelope.

For debugging probabilistic arguments and resolving paradoxes it is better to use conditionals so that every premise is made explicit, so that problematic premises are more easily identified.

Therefore, within the same equation, A is referring to two different amounts. Am I correct in thinking that this is why the equation gives a false result. — RussellA

I would say that is a potential cause of the paradox, but isn't the paradox itself. The paradox is the fact that the switching argument consists of a logically inconsistent set of probabilistic assumptions.

The switching argument, which produces a contradictory strategy for solving the two-envelope problem, starts by subjectively assuming, without evidence, the following conditional distribution, with respect to envelopes A and B whose values are a and b respectively :

P (B = (1/2) a | A = a) = P(B = 2a | A = a) = 1/2 For all values a

which yields the conditional expectation value

E [ B | A = a] = (5/4) a For all values a

So far, terrible reasoning with respect to the two-envelope problem, but no inconsistency.

Next, the switching argument further assumes, without evidence, that

P (A = a | B = (1/2) a) = 1 / 2 for all values a

However, using Bayes Theorem gives

P (A = a | B = (1/2) a) = P (B = (1/2) a | A = a) x P(A = a) / P(B = (1/2) a)

Hence the switching argument assumes that ratio on the right hand side is always 1 :

P(A = a) / P(B = (1/2) a) = 1 for all values a

This is the most deranged part of the argument. It amounts to asserting "If we know nothing about the prior distribution of A and nothing about the prior distribution of B, then we can be certain that the ratio of their probabilities is 1 - for our ignorance cancels out! "

We then derive the contradiction

E [ A | B = b ] = (5/4) b For all values b

Relative to your use of probabilities, how do you distinguish knowing that an outcome has probability 1/2 from not knowing the likelihood of an outcome?

So by your logic we can't even talk about the probability of a coin toss landing heads being 12
1
2
? — Michael

In my view , epistemic probabilities are derived from causal knowledge or assumptions on the basis abductive reasoning and repeated trials. One cannot derive causal knowledge from mathematical concepts as is demanded in the two-envelopes problem.

In my strong opinion, Laplace's principle of Indifference and the principle of maximum entropy are grave misuses of probability calculus that only lead to erroneous inferences and the conflation of ignorance with information.

I don't think that's at all reasonable, or even relevant when we consider puzzles like this. For the sake of puzzles like this we assume a coin toss landing heads has a probability of 12
1
2
, and we assume that my choice of envelope is truly random. — Michael

It is fair to make probabilistic assumptions, but they shouldn't be arbitrary (unless for the sake of philosophical argument) and they must be explicitly stated, and any conclusion cannot go farther than what is explicitly assumed from the outset. The less one assumes, the less one can conclude.

Since the two-envelopes problem isn't well posed, more must be assumed and/or imprecise probabilities are needed.

The only rational response to the two-envelopes problem as it is traditionally stated without additional assumptions, is to reply

"The probability of getting a greater or lesser prize when opening the other envelope, is between 0 and 1"

Why not?

I know that one envelope contains twice as much as the other. I pick one at random. What is the probability that I picked the smaller envelope? It seems perfectly correct to say 12
1
2
. — Michael

By "meaningful probabilities" i am referring to epistemically meaningful probabilities that quantify how the real world is expected to behave on the basis of past experience, as opposed to purely mathematical probabilities that merely quantify mathematical properties such as combinations and symmetry.

For example, suppose that the surface area of "heads" on an unknown but symmetric coin is roughly half of it's total surface area. If nothing else is known and assumed about the coin , including how and where it is to be thrown, then one cannot deduce solely only on the basis of the coin's mathematical properties that it's likelihood of landing heads is 50/50. Nevertheless, probabilities are often assigned to such unknown objects purely on the basis of their mathematical properties, leading to the conflation of "mathematical probability" with "physical probability" or "epistemic probability".

The two-envelopes question is epistemic because it concerns decision making in the real-world; so more than mathematics must be appealed to when deciding whether or not a strategy exists for decision-making in that context.

Our conclusions might agree. I am saying that only the statements 1,3 4 and 5 are valid in the "switching argument". For any prior probability distribution over envelope choices or envelope contents is consistent with the premise

"Imagine you are given two identical envelopes, each containing money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. "

And a chosen prior probability distribution mustn't refer to a person's subjective beliefs about the envelopes, but to a logically consistent set of causal hypotheses concerning the generation of the envelopes and their contents, by which a contradiction isn't derivable.

All that the two-envelope premise concretely specifies is a relation from the cartesian product of booleans and naturals to the naturals

r :: B x N ---> N
r ( 0, a) = a
r ( 1 ,a) = 2a

From which we trivially obtain the inverse relation

ir :: N --> N x N
ir a = { (0,a) , (1 , a/2) }

There is literally nothing else that can be said, deductively or inductively, from the two envelope premise alone. One cannot extract a meaningful notion of probabilities, let alone expectations and averages in relation to that premise when it is stated without additional causal assumptions, which are necessarily to give physical meaning to probabilities in that context.

An optimal decision doesn't exist on the basis of the information provided, because the premises fail to specify a well-posed problem :

Let P ( r | x , e) denote the probability of obtaining a value r when opening an envelope labelled e, where x represents the smallest amount of money in the two envelopes. Both e and x are assumed to be hidden variables in the sense that they aren't deducible from a drawn value of r.

The premises of the problem allow the following physical characterisation of P:

P ( r | x , e = 0) := Ind (r ; x)
P ( r | x, e = 1) := Ind (r ; 2x)

Here Ind(r; y) is the indicator function that outputs 1 when r equals y and is otherwise 0. (i.e. we get whatever is in the chosen envelope).

To "objectify" the problem, suppose that instead of choosing an envelope, an envelope is automatically drawn from a prior probability distribution p(e) and then opened. Then we could use Bayes theorem to compute a physically interpretable probability that a given value of r is "caused" by a given envelope:

P (e | r , x) =

P( r | e, x) P( e)
---------------------
P (r | x)

where P(r | x) is obtained by summing over e in the numerator, i.e

P(r|x) = P(r | e= 0, x) P(e = 0) + P(r | e = 1 , x) P(e=1)

But the problem doesn't specify the prior probability P(e) , which implies that P (e | r , x) doesn't have a unique solution, which in turn implies that the decision problem isn't well-posed.

Instead, one might follow the OP and a number of authors, and make the arbitrary selection
P (e =1) = P(e =2) = 1/2. In which case Bayes rule gives

P (r | x) = 0.5 Ind (x ; r +2r)

whose expected value is E [ r | x] = 1.5x

However, the problem is still ill-posed due to the fact that P(r | x) has zero mass at it's expected value of 1.5x :

(r = x) => (r < 1.5x )
(r = 2x) => (r > 1.5x )

Unless we make additional assumptions about P(x) , such that we can gain knowledge of x from knowledge of r, we cannot know which of these cases is the most likely, and so cannot deduce anything about the envelope e from the assumption of a uniform prior for P(e).

Any idea how "idealism" can be used to solve "the hard problem"? Do share, Rogue. — 180 Proof

Deflationism, that rejects representational accounts of semantics , shares much in common with subjective idealism and logical positivism that both considered every proposition to be reducible to sense-data.

In so far as the hard-problem is considered to be a metaphysical problem that is an artifact of representationalism, idealism can be considered to be a metaphysical strategy for dissolving the hard-problem, even if such a strategy is regarded to be epistemically impractical for the inter-subjective purposes of science , as the positivists discovered.

Putting it cynically, your proposition and methodology of divine revelation isn't qualitatively different to the thoughts of the average physicist, who uses commonsense to argue for both the big-bang theory whilst simultaneously denying a beginning of time.

Yet there are alternative theories, such as the Hawking-Hartle proposal that time is finite but lacks a boundary, that consider the question as to whether the universe truly has a beginning or not to be a topological question whose answer is relative to perspective.

Also, the subjective nature of phenomenological time concerning the intimate world of experience, is a distinct question whose relationship to the theoretical time of physics isn't decided. So a person could conceivably be a psychological presentist who denies the existence of a beginning/end with respect to their world of experience, who nevertheless believes physical time to have a beginning/end.

Perhaps the hard problem is inconceivable for phenomenolgists, but I'm not a phenomenologist. — Luke

So what is your definition of unconsciousness? Is it a pure postulate, or something that reduces to empirical criteria?

Granting this, how does it imply that the hard problem is inconceivable? — Luke

We can take the hard-problem in it's broadest sense, as asking what grounds the existence of first-personal phenomenological criteria that are used to understand propositions?

For phenomenologists who consider first-personal phenomenological criteria to be the very essence of meaning, the question is circular and makes no sense from their perspective. Which is what i was getting at above.

On the other hand, scientists working in the natural sciences will either side with the phenomenologist or not, depending on whether they believe the inter-subjective empirical criteria that they use to understand scientific theories to be ultimately grounded in first-person phenomenology or in pure reason. (e.g whether they are ultimately empiricists equipped with a deflationary understanding of truth and an anti-representationalist understanding of their own minds, or whether they are ultimately rationalists equipped with a correspondence understanding of truth and a representational understanding of themselves).

As for Dennett, he sometimes sounds like a rationalist who agrees with the phenomenologist that the question is meaningless, but for opposite reasons, namely due to a narrow interpretation of the natural sciences as denying the question on the basis of it being materially inconsequential (as opposed to be phenomenologically inconsequential)