The Quantum Pontiff

Bayesians Say the Cutest Things

The Dutch book argument of Bruno de Finetti is an argument which is supposed to justify subjective probabilities. What one does in this argument is gives probabilities an operational definition in terms of the amount one is willing to bet on some event. Thus a probability p is mapped to your being willing to make a bet on the event at 1-p to p odds. In the Dutch book argument one shows that if one takes this operational meaning and in addition allows for the person you are betting to take both sides of the bet, then if you do not follow the axiomatic laws of probability, then the person betting against you can construct a Dutch book: a set of bets in which the person you are betting against always wins. For the best explanation and derivation of this result that I know, consult the notes written by Carl Caves: Probabilities as betting odds and the Dutch book.

Now I have many issues with the Dutch book argument, the first and foremost being that it is a ridiculous setup. I mean how often do you place a bet in which you are willing to give both sides of the bet (buy and sell)? “Yes, I would like to either buy or sell a lottery ticket please?” Sure you can do it, but there are many reasons why money has a value outside of the single bet being placed, and therefore buying (giving someone your money and getting paid back if you win the bet) versus selling (recieving money and then having to pay off the bet if you lose) are not symmetric in any world where the unit being exchanged has a temporal value and the bet is placed before the event is resolved. I am, indeed, a one-sided Bayesian. I will leave it up to the reader to construct the axioms of probability by which I work.

Amusingly, at least to me, this objection does not seem to be raised much in the literature on the Dutch book argument. But the other day I found a great quote relevant to this objection which I just have to share. This is from Artificial Intelligence: A Modern Approach by Russell and Norvig. In this book they discuss but don’t prove the de Finetti’s argument. Then they say

One might think that this betting game is rather contrived. For example, what if one refuses to bet? Does that end the argument? The answer is that the betting game is an abstract model for decision-making situation in which every agent is unavoidably involved at every moment. Every action (including inaction) is a kind of bet, and every outcome can be seen as a payoff of the bet. Refusing to bet is like refusing to allow time to pass.

You heard it here first people: if you want to stop time all you have to do is not bet! Crap I have homework due tomorrow what should I do? Well certainly you should not bet, because we all know that refusing to bet is refusing to allow time to pass. ROFL Baysians are so cute when they try to justify themselves.

Comments

  1. #1 Markk
    October 21, 2009

    I don’t get it. The quote you give is saying that you CAN’T refuse to bet. Similar to Thermodynamics aphorisms like you can’t leave the game. What you laugh at is exactly the opposite of what they are implying to my eyes, so you are agreeing with their view. I could be way off, but that is the way it looks to me.

  2. #2 Brad Pitcher
    October 21, 2009

    What’s “Baysian”? Did you mean “Bayesian”?

  3. #3 Dave Bacon
    October 21, 2009

    My god now I have to argue about why this is funny. ;)

  4. #4 Dave Bacon
    October 21, 2009

    It’s a Australian spelling of the word, you know.

  5. #5 Matt Leifer
    October 21, 2009

    This is probably not the place to get into a serious discussion of Bayesian justifications of probability since you were just trying to make a joke. However, since we Bayesians are an evangelical creed, I am afraid I just can’t help myself.

    First of all, pretty much no serious Bayesian that I know of uses the Dutch book argument to justify probability. Things like the Savage axioms are much more popular, and much more realistic. Therefore, the scheme does not in any way rest on whether or not you find the Dutch book scenario reasonable. These days you should think of it as an easily digestible demonstration that simple operational decision making principles can lead to the axioms of probability rather than thinking of it as the final story. It is certainly easier to understand than Savage, and an important part of it, namely the “sure thing principle”, does survive in more sophisticated approaches.

    My second point probably diverges from mainstream Bayesians, but I say if you don’t accept the premises of the Dutch book argument then great. Go ahead and determine a set of principles for decision making that you find more reasonable and use a theory derived from them to make your decisions. Nobody ever said that Dutch book was the last word on rationality or indeed that probability theory itself was either. For me, the whole point of foundational investigations into things like probability (and to some extent quantum theory as well) is to provide a clear definition of terms that are confusing and to show how well-defined and easily understandable principles lead to the existing theory. Ideally, the definitions and principles should have operational content so that you are clear about what it means to accept the theory and what principles have to be changed if you want to give it up. The Dutch book does indeed do this and it is much more meaningful to talk about the implications of giving up the two-sidedness of betting than it is to just generalize the Kolmogorov axioms directly. In this way we can have a meaningful discussion about what our theory of rationality should be, which parts are logical necessities, and how to go about changing it. This is much better than just bandying about with mathematical theories that are mere formal generalizations of existing theories. For me, this is the main reason to prefer de Finetti over something like Cox, which in my view is a mere formal derivation with no operational content.

    In short, if, by design, your foundational investigation can have no other outcome than justifying the existing theory and if it gives no meaningful insight into how to change it, then you are doing it wrong. The whole point is to discover the limits of our theories and address them by advancing the theory.

    BTW, in case it isn’t obvious, most Bayesians would disagree with me on this point since they argue that Bayesian theory is “normative”, i.e. we know that most people don’t act in accordance with the theory, but when they don’t they are being irrational. On the other hand, I would stress that the theory is only “conditionally normative”, i.e. you are irrational if you believe all the premises of the theory and don’t act according to it, but there is no reason to believe all the premises. Also, I kind of hate this stress on the normative aspect of the theory since if the kind of people that we usually think of as being rational did not behave in similar ways to what the theory suggests then we would not have accepted it in the first place, regardless of any rhetoric about its normative value. The normative mantra is just a cop-out designed to cope with the fact that the theory is only approximate and to appease the mass of statisticians who would be unwilling to change the theory of probability in light of this fact. If you take the idea that quantum theory is a generalization of classical probability seriously, as I do, then it provides a case in point. In short, the normative Bayesian foundationalists do not realize that their goal should be to change probability theory rather than to merely justify it.

  6. #6 Robin Blume-Kohout
    October 21, 2009

    Warning: Dave, I’m going to pick on you. If you’re feeling unfairly picked on by Bayesians who appear to have had their senses of humor surgically removed, don’t read on.

    Yes, you are being asked to justify why this is funny, because (to me) it’s not. You’re misreading it in a funny way, but (intentional fallacy notwithstanding) this is a misreading.

    What would be funny is if the original authors had written/meant “If you refuse to bet, then time will not pass.” What they actually meant was “Trying to avoid betting is like trying to stop time from passing — you are constantly making bets whether you like it or not.” A reference to King Canute would probably be apropos here, albeit quite limited as a metaphor.

    Just to belabor the point, you are currently betting on the proposition “Within the next 10 seconds, a meteor will hit the spot where I am currently located.” You’re betting your life against whatever utility you derive from sitting in that spot (at rather good odds, though!)

    Anyway, that’s why I don’t find the statement amusing. But, hey, humor is subjective, right? So as a Bayesian, I’m okay with that… :)

  7. #7 Dave Bacon
    October 21, 2009

    Ha! So much fun. Yes stirring up the Church of Bayes is even more fun that attacking the Church of the Larger Hilbert Space. Plus the former has a whole new set of libretarian supporters who can be fun to argue with (Hilbert space people tend toward the left?)

    Matt: I agree with lots of what you say, especially the part about foundational arguments being important for moving the discussion forward, not making it stand still (sorry couldn’t resist the pun.) I would point, however, that in the real world the Dutch book argument is explicitly wrong: bid and ask prices in a market are only ever rarely equal. (This isn’t a function of money being used here either, the asymmetry is because you are betting resources which, unlike in the one-off arguments like the Dutch book, these do have a time value.)

    @Matt and Robin: Of course I’m misreading it. But neither of you have justified why “Refusing to bet is like refusing to allow time to pass.” Sure he says “like.” But I find not betting on the superbowl completely unlike not allowing time to pass. But maybe (obviously) that’s just me. I mean analogies are great and all, but this one is silly. And why the heck should we believe that time, as measured by clocks, has anything to do with probability? Myself I think it does, but ignoring this leads to analogies that even as a lit major I can’t justify (okay well a B.S. in lit can justify nearly everything as my fellow B.S.liters know.)

    @M and R: Also neither of you have explicitly commented on my first point which is that the Dutch book argument assumes that I will bet the same price for either side of the bet. What is the probability theory that arises when one gets rid of this?

    Oh I may be wrong and misguided and not as steeped in the philosophy or arguing about philosophy, but damn if you are going to push me off of thinking this is a hilarious passage :)

  8. #8 Ian Durham
    October 21, 2009

    Oh, this is a nice, meaty discussion. Just what I needed after a day-long administrative meeting and 75 minutes of freshman.

    Anyway, I have to agree with Matt that De Finetti’s argument is not reliant on any reference to a Dutch book. You can formulate the argument in terms of exchangeability (a term I believe De Finetti invented) which relates to permutations under mixing where the exchangeable random variables are independent.

    Basically, it’s in contrast to Carnap’s argument which is essentially frequentist – what is the rate at which one can learn from experience. By contrast de Finetti asks in which discrete interval between 0 and 1 does the probability for an event lie.

    > For me, the whole point of foundational investigations
    > into things like probability (and to some extent quantum
    > theory as well) is to provide a clear definition of terms
    > that are confusing and to show how well-defined and easily
    > understandable principles lead to the existing theory.
    > Ideally, the definitions and principles should have
    > operational content so that you are clear about what it
    > means to accept the theory and what principles have to be
    > changed if you want to give it up.

    @Matt: Great point. I wrote an essay on this for the latest FQXi essay contest. It could use a little more refining but, at its heart, it makes the point that we need to do more of this as physicists. We’ve gotten too sloppy in our language (definitions, etc.) and this is going to lead to problems, particularly in relation to studies of emergence and unification.

  9. #9 David
    October 21, 2009

    I haven’t read about the Dutch book yet, but off the cuff, futures traders make both sides of the book all the time, called selling spreads.

  10. #10 Rocky Humbert
    October 21, 2009

    Dave: Might there be some applicability here to professional options market-makers (who are always willing to take both sides of a bet)? I’m not sure….

    Remember: in addtion to “vanilla” American and European style put and call options — there are esoteric options including binary, knock-in, quantos, barriers, and so on. The common theme is they are all bets with associated probabilities…. and you can buy, sell or buy-and-sell these bets.

  11. #11 Robin Blume-Kohout
    October 21, 2009

    @Dave #7: I didn’t comment on the Dutch Book because… well, I don’t disagree. Not being a religious Bayesian, I don’t have any really compelling attachment to Dutch Book. It’s useful, but I’ve never taken the time to really study it. So while I don’t necessarily agree that it’s fundamentally flawed, I’m totally open to the idea, and appreciated that part of your post.

    There are at least two practical issues with Dutch Book. (1) People have nonlinear utility for money. (2) People are never “rational” in the strict sense (this includes asymmetry and hysteresis, as in the experiments where people will only pay $50 for concert tickets, but if they win them in a lottery, will not sell them for less than $200). Your more sophisticated Bayesians are not only aware of this but view it as a challenge. Savage’s “Elicitation of personal probabilities and expectations” is a wonderful paper in this vein.

    Now, as to the challenge to justify why “Refusing to bet is like refusing to allow time to pass”… ah, now I can get involved, because I do agree with this statement (though I find it resolutely unamusing).

    The authors’ point is that you are constantly faced with decisions — at the very least, you are constantly deciding whether to (a) do something, or (b) do nothing. Only if time does not pass (or you are asleep or paralyzed or something) do you avoid making decisions.

    You decide among N options, and the utility you will reap from your choice is a function of (a) your choice and (b) what state of nature occurs. The latter cannot be predicted with certainty, so every choice is a gamble. Moreover, every choice can be modeled as a bet.

    So these guys are not trying to say that you can’t avoid betting on the Super Bowl. They’re saying that you are constantly making decisions, and that they can model every decision as a bet. So you can choose to avoid betting on the Super Bowl, but that should be seen as choosing one of three options:
    1. Bet on the Patriots,
    2. Bet on the Dolphins,
    3. Don’t bet on any team.
    That choice is itself a bet — or, more precisely, Russell and Norvig can and do model it as a bet.

    If you’re now fuming at the apparent tautology of stating that the choice not to bet on the Super Bowl is itself a bet… consider the following. There is some [small] probability that if you were to go and place a bet on the Super Bowl, a $1000 bill would fall out somebody’s pocket onto your shoes while you stood in line. By choosing not to bet on the Super Bowl, you are betting that this will not happen. Of course, you and I both agree that the probability is vanishingly small… but the principle is still valid.

  12. #12 Jeff Watson
    October 21, 2009

    Al life is 6:5 against……Damon Runyon

  13. #13 Matt Leifer
    October 22, 2009

    @Ian – Huh, what are you talking about? The de Finetti theorem has nothing to do with the Dutch book. They serve completely different purposes.

    @Dave – I thought I *did* comment on the inapplicability of the Dutch book argument to real life. Firstly, I said that no serious Bayesian accepts it as the foundation of probability, preferring things like the Savage axioms. Even de Finetti himself switched to a loss-function approach later on in his life. Because of this, I said that the Dutch book should be viewed as a pedagogical demonstration that you *can* arrive at probability theory from simple decision-theoretic principles, rather than being the foundation for Bayesianism, so criticisms of the Dutch book largely miss the point. Thirdly, I said that you were free to accept any alternative set of decision theoretic principles you like and figure out the consequences.

    You are right that I didn’t comment on the Russell and Norvig quote, mainly because I don’t think it is very interesting. If you look at the more general post-Dutch book approaches to Bayesian foundations, e.g. Savage, you will see that things are not phrased in terms of bets and money. Any time you have to make a decision and you have a preference ordering over the consequences you can apply the theory. Their point, I guess, is that you are making decisions passively all the time, e.g. even if you are just staring into space and daydreaming you could instead be doing work and that might have consequences that you care about. Therefore, in principle, you could be applying Bayesian probability theory to every moment of your existence. I think I agree with this, although I would stress that nobody is saying that you *should* apply Bayesian probability to every moment of your existence, only that you *could* apply it to any given moment if you wanted.

  14. #14 Dave Bacon
    October 22, 2009

    @Robin: What I think is funny is that “time” as measured by a “clock” has anything to do with what the hell I’m choosing to do. I mean I sure hope that the world keeps turning independent of whether I am betting or not betting. I know what Russell and Norvig are trying to say, but the way they phrase it is patently absurd. Or at least really funny.

    Also I don’t understand why the argument you give about deciding to bet doesn’t lead to an infinite heirarchy of such decisions (and you know me, any infinity that is supposed to be useful is suspect.) Thus if the choice to bet is itself a bet, then isn’t the choice to chose whether to chose to bet or not a choice…ad infinitum? I am also shocked by the amount of free will necessary to get this whole thing to work.

    @Matt and Robin: And finally what no one has given me a good answer to, and this is a real problem with a real answer is ” I will leave it up to the reader to construct the axioms of probability by which I work.” Real Carl’s notes. How do you change them to if you require wagers (Carl’s “S”) to be positive (i.e. don’t allow the Dutch book to take both sides of the bet)? The answer is fun.

    Now take the next step: don’t just do the above for events which all resolve themselves in parallel, but give a temporal order to the events, and perhaps if you want a time value to money. I’d rather use these axioms when I’m thinking about most real world betting experiments independent of whether this has anything to do with the foundations of probability :)

  15. #15 Anonymous Coward
    October 22, 2009

    @RBK #11

    Three options for the Super Bowl? I think you’re undercounting. Why not bet on the Chicago Bulls? (If you’re interested, I’ll give you very good odds.)

    The idea that we’re making decisions for every possible option among the infinite (or just very large?) number of possible options for the infinite (or just very large?) number of questions we could be asking ourselves is a poor model for humans. Even if we’re considering non-human agents (and surely we are) it would seem that actually “choosing” for every one of those decisions is only possible for an agent of infinite capacity?

  16. #16 John Smolin
    October 22, 2009

    Um, Dolphins and Patriots are both AFC teams. Use that in picking your priors.

  17. #17 Neil B ♪
    October 22, 2009

    “subjective ….” Yeah, I know that’s relevant in context – but isn’t the essence of the Bayes approach really about what you have a right to expect, given a condition? Like, I got a “positive” X-test result, what is the chance I really have X, etc. That is based on calculations and not “expectations” of the mind, or willingness to bet etc.

    BTW, one of the coolest paradoxes in prob-thy is the two-envelope problem: There are two envelopes, and one has twice the other (eg $100, 200.) You open one, let’s say $10 inside. Should you switch? That seems silly, you could have picked it first anyway. But consider “expectation value” (like in QM!) If you say, 50% chance the other is $5 and 50% the other is $20, the EP is $12.50! It always seems you should switch, given that rather naive (due to having no clue how the “universe” of initial picks was set up ….) assumption. Somebody just came up with a new explanation of why it doesn’t matter, see http://en.wikipedia.org/wiki/Two_envelopes_problem (which offers a better puzzle but not the specific authors I think just published.)

    Then there’s the St. Petersburg/Martingale stuff, which is really weird. Say I have a chance below 1/3 of winning. I don’t even have to get the expected value minus a cut for the House, so let’s say twice back. So I can bet in a sequence like 1,3,9,27,… I may lose a long time, but whenever I do win, I always get back more than I already lost! This seems absurd, and even though I can keep playing with “unlimited” money (since any given pre-determined run gives a chance of losing), it still doesn’t involve “infinite quantities” per se. Playing with this I found you can’t really beat the odds long-term if you are indeed limited to some given n of runs, but it sure sounds like a wow thing.

    Finally, remember Marilyn vos Savant and “the three doors problem”? You can look it up – she turned out to be right and plenty of experts were wrong. It was a big flap, and makes me wonder about “expertise” some times. (Briefly, you had only a 1/3 chance of being right the first time, so switch!) In that case, it really did make sense to switch.

  18. #18 chris
    October 23, 2009

    You really don’t get this do you. Wow, I kind of just don’t get how you can not get it.

    It really isn’t very complicated. You have a choice how to act in the next second, you cant not choose how to act in the next second, because you will do something, maybe what you will do is sit there confused about the notion of choice, but that is a choice. There isn’t an infinite regress, there just is one bet, you cant choose not to bet, it “is” not “is like” “is” choosing time to stop, which you can’t do! That’s the whole point of pointing it out!

  19. #19 klil
    October 23, 2009

    It’s a Australian spelling of the word, you know.

    That’s silly. The Australian spelling is upside down.

  20. #20 Robin Blume-Kohout
    October 24, 2009

    Oi. I go away for a day or two to prepare a QIP abstract, and all this happens. I’m going back to my cave! :)

    @Dave #14: I think I must be missing something about how you’re reading this. (Which is really a bummer, ’cause I like funny jokes!) Why is it funny that time and choices are related? Of course, it’s absurd if you think that {making choices} ==> {time passing}, but they’re not saying that. They’re saying that {time passing} ==> {making choices}. Inexorably. And it’s clear from what you say that you do understand what they’re saying, yet find it funny… so I conclude that my sense of humor is defective.

    Your second point is insightful — but has an answer. Decisions — represented by sets of alternatives — are not ontological nor objective. They’re elements of a theory, not of reality. I’m describing your behavior using decision theory, and I’m describing your options at a given instant by using a particular coarse-grained partition of the set of all possible actions. I hope you’re not terribly bothered by the idea that the set of all possible actions is potentially infinite — that’s just a consequence of the continuum. You have infinitely many places (x) to put your coffee cup down. But I’m lazy, so I’m going to coarse-grain that set into two or three lumps. The important thing here is that the options have to be a partition, so I always have to include a catchall “none of the above”. Which bails me out of any infinite regress — I just say, “Nah, ain’t gonna regress any further.”

    As for free will — heck, we don’t need no stinkin’ free will! I can model you as a deterministic (or stochastic) agent, operating with finite computational power, without ever worrying about free will. Sort of like I can write quantum algorithms without ever worrying about collapse (which is good, because I don’t understand collapse or free will).

    Your last two paragraphs sound interesting, and I have no response or comment right now!

  21. #21 Robin Blume-Kohout
    October 24, 2009

    @15: You’re making reasonable points, but they don’t really pose major problems. Regarding number of options, see my response above (@20); we could include the Bulls, or the Twins, or Real Madrid… or we could implicitly leave them all in the “none of the above” bucket. Regarding cognitive overload, there are two answers. First, the potentially infinite set here is the set of options; we make one decision with [potentially infinitely] many options. So it’s not like I’m necessarily overwhelmed with decisions, just choices. Second, we sure as heck don’t make good choices all the time. We have bounded computational power, and for the vast majority of the “decisions” we’re faced with, we just adopt some kind of de facto option like “stand there and look stupid without actually doing anything.” So if the modeler changes his model so that (in his model) you’re making decisions every 0.5sec rather than every 1sec, then 99.5% of your “choices” will be “do nothing in particular”, instead of 99%.

    @John #16: Er, oops.

    @Neil #17: Most applications of Bayes’ rule, and most Bayesian statistical inference, are indeed of the noncontroversial sort to which you allude. Nonetheless, 95% of the arguments about Bayesian statistics are not concerned with any of that. Like this one, they’re about a definition: “Probability = betting odds.” It shows up implicitly in your post, when you say “…what is the chance I really have X.” Chance is a synonym for probability, and probability needs a definition. The ground rules of Bayesian probability define probability as the odds at which the agent would take a bet. If you don’t like this definition, you’re still allowed to use Bayes’ Rule! :)

  22. #22 Allison
    November 2, 2009

    Dave,

    You should read E T Jaynes’ book, Probability Theory, The Logic of Science. Amazon has it. He’s THAT Jaynes. Perhaps you wouldn’t laugh him off so easily. You do believe in MAXENT, right?

    Jaynes is right about probability theory, and when you realize that, you will realize that you have been making errors in your understanding of probability, and therefore, of qm.

  23. #23 Dave Bacon
    November 3, 2009

    Hey Allison,

    Yeah I’ve read Jaynes. He’s a funny curmudgeon!

    And no I don’t believe in MAXENT (Here is a fun read from a former believer: http://complementaryslackness.wordpress.com/2009/08/25/maxent/)

  24. #24 Allison
    November 10, 2009

    I read what you pointed me to. It’s a terrible misunderstanding of MaxEnt and of Bayes.

    In a nice paper detailing aspects of the constraint rule of MaxEnt, Uffink examines the two approaches for the case of rolling a die (in this context called the Brandeis dice problem). Suppose we roll the die many times and observe a mean number of 3.5. This is what one would expect for an unbiased die, i.e. one with probability 1/6 for each of the outcomes 1 to 6, and moreover this distribution has the largest entropy, so the MaxEnt probability is the uniform distribution. On the other hand, following the usual Bayesian prescription, if our prior distribution is that the die could be biased in any way, each equally-likely, then after going through a calculation similar to that of the rule of succession (after witnessing s in rolls, the probability of on the next toss is ) and adding up the probabilities for sequences of results which all have an average of 3.5, Uffink obtains the posterior distribution for the next roll. It’s not uniform; rather it gives more weight to 3 and 4 than to 1 and 6. (2 and 5 are somewhere in between.)

    What rubbish. To claim to be a Bayesian and to misunderstand the notion that Jaynes would be making any claim at all about the prior bias of the die.

    No, the issue is not with the physical world. It is with OUR KNOWLEDGE of it. The issue is not whether or not the die as all-equally-likely-probable-prior-biases or a different bias. It’s what we KNOW about the die.

    When we haven’t thrown it yet, we know nothing. Therefore, we can say nothing but “might as well say, since we KNOW Nothing, that we can prefer no face to any other; all faces are equally likely, AS FAR AS WE KNOW.” And that’s what Jaynes says. Of course posterior distributions would come up differently based on the actual results we see, since the results we see are the way to come to reality. But adding up probabilities of unperformed sequences is madness in this context, since they are unperformed.

    He understood more quantum than you do. Probabilities in quantum theory, like probabilities in classical theory, express uncertainty in human knowledge, not in the underlying system.

  25. #25 Dave Bacon
    November 10, 2009

    Ah Allison, why such bitterness? This is why I love posting about frequentists and bayesians: there is nothing that brings out the crazy more in smart people (except maybe discussing Ann Rand, but that’s a subject for another day.)

    I’ve no doubt Jaynes understood quantum better than me (that being a very low bar, you know.) But in my personal opinion, anyone who thinks they “know” what quantum theory is about is full of it, and needs to go learn more physics. Yep that’s right. A prereq for you, or the dead ghost of E.T. Jaynes, ever convincing me that you know what the hell you’re talking about is that you have to go and figure out how to reconcile general relativity with quantum theory.

    Here’s another fun one for you to bash: http://arxiv.org/abs/cond-mat/0410063 (I know the “hole” in that one.)

  26. #26 Dave Bacon
    November 10, 2009

    By the way, that second paragraph is a joke peoples, before you go off on me!

    For the record I’d settle for a derivation of quantum theory along the lines of the arguments for classical probability theory in Jaynes book. But I haven’t seen one that convinces me yet (please exclude pointing out any examples that invoke Gleason’s theorem.)

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