This is simply wrong. Bayesian statistics is just Bayesian probability theory. As is Bayesian epistemology. Bayesian probabilities are epistemic probabilities.
But you'd have to be one really stupid correctional officer to get an order to disable the cameras around Epstein's cell the night he was murdered, and not know who killed him after he dies.
I assume you mean "who ordered him killed."
...Here's what a news report says happened:
A letter filed by Assistant US Attorneys Jason Swergold and Maurene Comey said "the footage contained on the preserved video was for the correct date and time, but captured a different tier than the one where Cell-1 was located", New York City media report.Prince Andrew spoke to the BBC i
But when you do assert that basically the entire U.S. government has collaborated on murdering Epstein
Isn't this a straw man? If someone powerful wanted Epstein dead, how many people does that require, and how many of them even have to know why they're doing what they're doing? It seems to me that only one person -- the murderer -- absolutely has to be in on it. Other people could get orders that sound innocuous, or maybe just a little odd, without knowing the reasons behind them. And, of course, there are always versions of "Will no one rid me of this troublesome priest?" to ensure deniability.
The context is *all* applications of probability theory. Look, when I tell you that A or not A is a rule of classical propositional logic, we don't argue about the context or what assumptions we are relying on. That's just a universal rule of classical logic. Ditto with conditioning on all the information you have. That's just one of the rules of epistemic probability theory that *always* applies. The only time you are allowed to NOT condition on some piece of known information is if you would get the same answer whether or not you condition...
You are simply assuming that what I've calculated is irrelevant. But the only way to know absolutely for sure whether it is irrelevant is to actually do the calculation! That is, if you have information X and Y, and you think Y is irrelevant to proposition A, the only way you can justify leaving out Y is if Pr(A | X and Y) = Pr(A | X). We often make informal arguments as to why this is so, but an actual calculation showing that, in fact, Pr(A | X and Y) != Pr(A | X) always trumps an informal argument that they should be equal.
Your "probability of...
But randomly awakening Beauty on only one day is a different scenario than waking her both days. A priori you can't just replace one with the other.
Yes, in exactly the same sense that *any* mathematical / logical model needs some justification of why it corresponds to the system or phenomenon under consideration. As I've mentioned before, though, if you are able to express your background knowledge in propositional form, then your probabilities are uniquely determined by that collection of propositional formulas. So this reduces to the usual modeling question in any application of logic -- does this set of propositional formulas appropriately express the relevant information I actually have available?
This is the first thing I've read from Scott Garrabant, so "otherwise reputable" doesn't apply here. And I have frequently seen things written on LessWrong that display pretty significant misunderstandings of the philosophical basis of Bayesian probability, so that gives me a high prior to expect more of them.
I'm not trying to be mean here, but this post is completely wrong at all levels. No, Bayesian probability is not just for things that are space-like. None of the theorems from which it derived even refer to time.
So, you know the things in your past, so there is no need for probability there.
This simply is not true. There would be no need of detectives or historical researchers if it were true.
If you partially observe a fact, then I want to say you can decompose that fact into the part that you observed and the part that you didn't, and say that...
I think you are correct that I cannot cleanly separate the things that are in my past that I know and the things that are in my post that I do not know. For example, if a probability is chosen uniformly at random in the unit interval, then a coin with that probability is flipped a large number of times, then I see some of the results, I do not know the true probability, but the coin flips that I see really should come after the thing that determines the probability in my Bayes' net.
path analysis requires scientific thinking, as does every exercise in causal inference. Statistics, as frequently practiced, discourages it, and encouraged "canned" procedures instead.
Despite Pearl's early work on Bayesian networks, he doesn't seem to be very familiar with Bayesian statistics -- the above comment really only applies to frequentist statistics. Model construction and criticism ("scientific thinking") is an important part of Bayesian statistics. Causal thinking is common in Bayesian statistics, because causal int...
I don't believe that the term "probability" is completely unambiguous once we start including weird scenarios that fall outside the scope which standard probability was intended to address.
The intended scope is anything that you can reason about using classical propositional logic. And if you can't reason about it using classical propositional logic, then there is still no ambiguity, because there are no probabilities.
You know, it has not actually been demonstrated that human consciousness can be mimicked by Turing-equivalent computer.
The evidence is extremely strong that human minds are processes that occur in human brains. All known physical laws are Turing computable, and we have no hint of any sort of physical law that is not Turing computable. Since brains are physical systems, the previous two observations imply that it is highly likely that they can be simulated on a Turing-equivalent computer (given enough time and memory).
But regardless of that, the Sleeping B...
In these kinds of scenarios we need to define our reference class and then we calculate the probability for someone in this class.
No, that is not what probability theory tells us to do. Reference classes are a rough technique to try to come up with prior distributions. They are not part of probability theory per se, and they are problematic because often there is disagreement as to which is the correct reference class.
When Sleeping Beauty wakes up and observes a sequence, they are learning that this sequence occurs on a on a random day
Right here is your error. You are sneaking in an indexical here -- Beauty doesn't know whether "today" is Monday or Tuesday. As I discussed in detail in Part 2, indexicals are not part of classical logic. Either they are ambiguous, which means you don't have a proposition at all, or the ambiguity can be resolved, which means you can restate your proposition in a form that doesn't involve indexicals.
What you are pro...
All this maths is correct, but why do we care about these odds? It is indeed true that if you had pre-committed at the start to guess if and only if you experienced the sequence 111
We care about these odds because the laws of probability tell us to use them. I have no idea what you mean by "precommitted at the start to guess if and only if..." I can't make any sense of this or the following paragraph. What are you "guessing"? Regardless, this is a question of epistemology -- what are the probabilities, given the information you have -- and those probabilities have specific values regardless of whether you care about calculating them.
Neal wants us the condition on all information, including the apparently random experiences that Sleeping Beauty will undergo before they answer the interview question. This information seems irrelevant, but Neal argues that if it were irrelevant that it wouldn't affect the calculation. If, contrary to expectations, it actually does, then Neal would suggest that we were wrong about its irrelevance.
This isn't just Neal's position. Jaynes argues the same in Probability Theory: The Logic of Science. I have never once encountered an academic bo...
Unfortunately, Ksavnhorn's post jumps straight into the maths and doesn't provide any explanation of what is going on.
Ouch. I thought I was explaining what was going on.
But the development of probability theory and the way that it is applied in practice were guided by implicit assumptions about observers.
I don't think that's true, but even if it is an accurate description of the history, that's irrelevant -- we have justifications for probability theory that make no assumptions whatsoever about observers.
You seemed to argue in your first post that selection effects were not routinely handled within standard probability theory.
No, I argued that this isn't a case of selection effects.
Certainly agreed as...
When Sleeping Beauty wakes up and observes a sequence, they are learning that this sequence occurs on a on a random day out of those days when they are awake.
That would be a valid description if she were awakened only on one day, with that day chosen through some unpredictable process. That is not the case here, though.
What you're doing here is sneaking in an indexical -- "today" is either Monday if Heads, and "today" is either Monday or Tuesday if Tails. See Part 2 for a discussion of this issue. To the extent that indexicals are...
From the OP: "honor requires recognition from others." That's not a component of the notion of honor I grew up with. Nor is the requirement of avenging insults.
This is a very, very different concept of honor than the one I grew up with. I was taught that honor means doing what is right (ethical, moral), regardless of personal cost. It meant being unfailingly honest, always keeping your word, doing your duty, etc. How others perceived you was irrelevant. One example of this notion of honor is the case of Sir Thomas More, who was executed by Henry VIII because his conscience would not allow him to cooperate with Henry's establishment of the Church of England. Another is the Dreyfus Affair and Colonel Georges P...
...the standard formalization of probability... was not designed with anthropic reasoning in mind. It is usually taken for granted that the number of copies of you that will be around in the future to observe the results of experiments is fixed at exactly 1, and that there is thus no need to explicitly include observation selection effects in the formalism.
1. Logic, including probability theory, is not observer-dependent. Just as the conclusions one can obtain with classical propositional logic depend only on the information (propositional axioms) availabl...
No, P(H | X2, M) is , and not . Recall that is the proposed model. If you thought it meant "today is Monday," I question how closely you read the post you are criticizing.
I find it ironic that you write "Dismissing betting arguments is very reminiscent of dismissing one-boxing in Newcomb's" -- in an earlier version of this blog post I brought up Newcomb myself as an example of why I am skeptical of standard betting arguments (not sure why or how that got dropped.) The point was that standard betting argu... ,,,,,
The point is that the meaning of a classical proposition must not change throughout the scope of the problem being considered. When we write A1, ..., An |= P, i.e. "A1 through An together logically imply P", we do not apply different structures to each of A1, ..., An, and P.
The trouble with using "today" in the Sleeping Beauty problem is that the situation under consideration is not limited to a single day; it spans, at a minimum, both Monday and Tuesday, and arguably Sunday and/or Wednesday also. Any properly constructed proposition us...
...the details of the experiment do provide context for "today." But as a random variable, not an explicit value.
You seem to think that "random" variables are special in some way that avoids the problems of indexicals. They are not. When dealing with epistemic probabilities, a "random" variable is any variable whose precise value is not known with complete certainty.
Still, there are ways to avoid using an indexical in a solution. I suggested one in a comment to part 1: use four Beauties, where each is left asleep under a diffe...
The situation with indexicals is similar to the situation with "irrelevant" information. If there is any dispute over whether some information is irrelevant, you condition on it and see if it changes the answer. If it does, the judgment that the information was irrelevant was wrong.
Same thing with indexicals. You may claim that use of an indexical in a proposition is unambiguous. The only way to prove this is to actually remove the ambiguity -- replace it with a more explicit statement that lacks indexicals -- and see that this doesn't chang... ,
You're right, my argument wasn't quite right. Thanks for looking into this and fixing it.
I think a variation of my approach to resolving the betting argument for SB can also help deal with the very large universe problem. I've taken a look at the following setup:
How do I do things like tables using the WYSIWYG interface? There doesn't seem to be any way to insert markdown in that interface. And once you've already been using WYSIWYG on an article, you can't really switch to markdown -- I tried, and it was a complete mess.
At any point in the history that Beauty remembers in step 2 of step 3, the proposition has a simple, single truth value.
No, it doesn't. This boils down to a question of identity. Absent any means of uniquely identifying the day -- such as, "the day in which a black marble is on the dresser" -- there is a fundamental ambiguity. If Beauty's remembered experiences and mental state are identical at a point in time on Monday and another point in time on Tuesday, then "today" becomes ill-defined for her.
In some instances of the expe...
Note the clause "in general."
Now you're really stretching.
And over the duration of when Beauty considers the meaning of "today," it does not change.
That duration potentially includes both Monday and Tuesday.
"Today" means the same thing every time Beauty uses it.
This is getting ridiculous. "Today" means a different thing on every different day. That's why the article lists it as an indexical. Going back to the quote, the "discussion" is not limited to a single day. There are at least two days invol...
It is true on Monday when Beauty is awake, and false on Sunday Night, on Tuesday whether or not Beauty is awake, and on Wednesday.
That's not a simple, single truth value; that's a structure built out of truth values.
The proposition "coin lands heads" is sometimes true, and sometimes false, as well.
No, it is not. It has the same truth value throughout the entire scenario, Sunday through Wednesday. On Sunday and Monday it is impossible to know what that truth value is, but it is either true that the coin will land heads, or false that it...
On the first read I didn't understand what you were proposing, because of the confusion over "If the two coins show the same face" versus "If the two coins are not both heads." Now that it's clear it should be "if the two coins are not both heads" throughout, and after rereading, I now see your argument.
The problem with your argument is that you still have "today" smuggled in: one of your state components is which way the nickel is lying "today." That changes over the course of the time period we ...
Your whole analysis rests on the idea that "it is Monday" is a legitimate proposition. I've responded to this many other places in the comments, so I'll just say here that a legitimate proposition needs to maintain the same truth value throughout the entire analysis (Sunday, Monday, Tuesday, and Wednesday). Otherwise it's a predicate. The point of introducing R(y,d) is that it's as close as we can get to what you want "it is Monday" to mean.
Are you really claiming that the statement "today is Monday" is not a sentence that is either true or false?
Yes. It does not have a simple true/false truth value. Since it is sometimes true and sometimes false, its truth value is a function from time to {true, false}. That makes it a predicate, not a proposition.
Or are you simply ignoring the fact that the frame of reference, within which Beauty is asked to assess the proposition "The coin lands Heads," is a fixed moment in time?
It is not a fixed moment in time; if it were, the SB probl... ,,,,,,,,,
By bayes rule, Pr (H | M) * Pr(X2 |H, M) / Pr(X2 |M) = Pr(H∣X2, M), which is not the same quantity you claimed to compute Pr(H∣X2).
That's a typo. I meant to write , not .
Second, the dismissal of betting arguments is strange.
I'll have more to say soon about what I think is the correct betting argument. Until then, see my comment in reply to Radford Neal about disagreement on how to apply betting arguments to this problem.
“probability theory is logically prior to decision theory.” Yes, this is the common view because probability... ,,,,,,,,
Yes, there is. I'll be writing about that soon.
Hmmm. I would be responsive to "that's a slur," but the follow-on "the preferred term is X" raises my hackles. The former is merely a request to be polite; the latter feels like someone is trying to dictate vocabulary to me.
None of this is about "versions of me"; it's about identifying what information you actually have and using that to make inferences. If the FNIC approach is wrong, then tell me what how Beauty's actual state of information differs from what is used in the analysis; don't just say, "it seems really odd."
As I understand it, FDT says that you go with the algorithm that maximizes your expected utility. That algorithm is the one that bets on 1:2 odds, using the fact that you will bet twice, with the same outcome each time, if the coin comes up tails.
The standard textbook definition of a proposition is this:
A proposition is a sentence that is either true or false. If a proposition is true, we say that its truth value is "true," and if a proposition is false, we say that its truth value is "false."
(Adapted from https://www.cs.utexas.edu/~schrum2/cs301k/lec/topic01-propLogic.pdf.)
The problem with a statement whose truth varies with time is that it does not have a simple true/false truth value; instead, its truth value is a function from time to the set .
As for the rest of ...
In regards to betting arguments:
1. Traditional CDT (causal decision theory) breaks down in unusual situations. The standard example is the Newcomb Problem, and various alternatives have been proposed, such as Functional Decision Theory. The Sleeping Beauty problem presents another highly unusual situation that should make one wary of betting arguments.
2. There is disagreement as to how to apply decision theory to the SB problem. The usual thirder betting argument assumes that SB fails to realize that she is going to both make the same decision and get th... ,,,,,
for decision-theoretic purposes you want the probability to be 1/3 as soon as the AI wakes up on Monday/Tuesday.
That is based on a flawed decision analysis that fails to account for the fact that Beauty will make the same choice, with the same outcome, on both Monday and Tuesday (it treats the outcomes on those two days as independent).
the mismatch with frequency
There are no frequencies in this problem; it is a one-time experiment.
Probability is logically prior to frequency estimation
That's not what I said; I said that probability theory is logically prior to decision theory.
If your "probability" has zero application because your decision theory uses "likeliness weights" calculated an entirely different way, I think something has gone very wrong.
Yes; what's gone wrong is that you're misapplying the decision theory, or your decision theory itself breaks ...
Classical propositions are simply true or false, although you may not know which. They do not change from false to true or vice versa, and classical logic is grounded in this property. "Propositions" such as "today is Monday" are true at some times and false at other times, and hence are not propositions of classical logic.
If you want a "proposition" that depends on time or location, then what you need is a predicate---essentially, a template that yields different specific propositions depending on what values you substitute i... ,,,,
It's a useless and misleading modeling choice to condition on irrelevant data
Strictly speaking, you should always condition on all data you have available. Calling some data irrelevant is just a shorthand for saying that conditioning on it changes nothing, i.e., . If you can show that conditioning on does change the probability of interest---as my calculation did in fact show---then this means that is in fact relevant information, regardless of what your intuition suggests.
even worse to condition on the assumption the unstated... ,,,
There are several misconceptions here:
1. Non-indexical conditioning is not "a way to do probability theory"; it is just a policy of not throwing out any data, even data that appears irrelevant.
2. No, you do not usually do probability theory on centered propositions such as "today is Monday", as they are not legitimate propositions in classical logic. The propositions of classical logic are timeless -- they are true, or they are false, but they do not change from one to the other.
3. Nowhere in the analysis do I treat a data point as &qu... ,,,,,,,,
The fact that people tend to specialize in one or the other does not mean that "the two have little to do with each other." Likewise, there are physicists who spend a lot of time working in foundations and interpretation of QM, and others who spend their time applying it to solve problems in solid state physics, nuclear physics, etc. They're working on different kinds of problems, but it's absurd to say that the two have "little to do with each other."