It looks like Theorem 1 can be improved slightly, by dropping the "only if" condition on . We can then code up something like Kolmogorov complexity by adding a probability transition from every site to our chosen UTM.
If you only want the weaker statement that there is no stationary distribution, it looks like there's a cheaper argument: Since is aperiodic and irreducible the hypothetical stationary distribution is unique. is closed under the action of , and (2) implies that for any , the map is an automorphism of the Markov chain
...That sounds a rather odd argument to make, even at the time. Astronomy from antiquity was founded on accurate observations.
Astronomy and epistemology aren't quite the same. Predicting where Saturn would be on a given date requires accurate observation, and nobody objected to Coperniucus as a calculational tool. For example, the Jesuits are teaching Copernicus in China in Chinese about 2 years after he publishes, which implies they translated and shipped it with some alacrity.
The heavens were classically held to be made of different stuff; quintessense (...
The precise phrasing is deliberately a little tendentious, but the issue of the epistemological status of the telescope was raised by loads of people at the time. For a modern review with heavy footnotes, see eg Galileo, Courtier: The Practice of Science in the Culture of Absolutism, pp 95-100, (though the whole chapter is good)
For example, the first anti-Galilean tract is by Horky in 1610 and focussed mostly on the lack of reliability of the telescope. For another, Magini's letters (confirmed in Kepler and Galileo) write of a "star party" in...
tl;dr: The side of rationality during Galileo's time would be to recognise one's confusion and recognise that the models did not yet cash out in terms of a difference in expected experiences. That situation arguably holds until Newton's Principia; prior to that no one has a working physics for the heavens.
The initial heliocentric models weren't more accurate by virtue of being heliocentric; they were better by virtue of having had their parameters updated with an additional 400 years of observational data over the previous best-fit model (the Alfonsine tab...
Thank you for that informed account of the history.
You mention three times, without attributing it to any contemporary of Galileo, that the telescope "distorted the vision", which is a tendentious description. Given that the military application of the telescope was grasped as soon as the instrument became known, who at the time made this criticism? Did they similarly eschew its terrestrial use for the improvement of vision?
So, my observation is that without meta-distributions (or A_p), or conditioning on a pile of past information (and thus tracking /more/ than just a probability distribution over current outcomes), you don't have the room in your knowledge to be able to even talk about sensitivity to new information coherently. Once you can talk about a complete state of knowledge, you can begin to talk about the utility of long term strategies.
For example, in your example, one would have the same probability of being paid today if 20% of employers actually pay you every da...
The substantive point here isn't about EU calculations per se. Running a full analysis of everything that might happen and doing an EU calculation on that basis is fine, and I don't think the OP disputes this.
The subtlety is about what numerical data can formally represent your full state of knowledge. The claim is that a mere probability of getting the $2 payout does not. It's the case that on the first use of a box, the probability of the payout given its colour is 0.45 regardless of the colour.
However, if you merely hold onto that probability, then if y...
Concretely, I have seen this style of test (for want of better terms, natural language code emulation) used as a screening test by firms looking to find non-CS undergraduates who would be well suited to develop code.
In as much as this test targets indirection, it is comparatively easy to write tests which target data driven flow control or understanding state machines. In such a case you read from a fixed sequence and emit a string of outputs. For a plausible improvement, get the user to log the full sequence of writes, so that you can see on which instruc...
To my knowledge, it's not discussed explicitly in the wider literature. I'm not a statistician by training though, so my knowledge of the literature is not brilliant.
On the other hand, talking to working Bayesian statisticians about "what do you do if we don't know what the model should be" seems to reliably return answers of broad form "throw that uncertainty into a two-level model, run the update, and let the data tell you which model is correct". Which is the less formal version of what Jaynes is doing here.
This seems to be a reasona...
Thank you for calling out a potential failure mode. I observe that my style of inquisition can come across as argumentative, in that I do not consistently note when I have shifted my view (instead querying other points of confusion). This is unfortunate.
To make my object level opinion changes more explicit:
I have had a weak shift in opinion towards the value of attempting to quantify and utilise weak arguments in internal epistemology, after our in person conversation and the clarification of what you meant.
I have had a much lesser shift in opinion of
Fermat considered the sequence of functions f(n,x) = x^n for n = 0, 1, 2, 3, ....
Only very kind of. Fermat didn't have a notion of function in the sense meant later, and showed geometrically that the area under certain curves could be computed by something akin to Archimedes' method of exhaustion, if you dropped the geometric rigour and worked algebraically. He wasn't looking at a limit of functions in any sense; he showed that the integral could be computed in general.
The counterexample is only "very simple" in the context of knowing that the...
It possible that "were known in general to lead to paradoxes" would be a more historically accurate phrasing than "without firm foundation".
For east to cite examples, there's "The Analyst" (1734, Berkeley). The basic issue was that infinitesimals needed to be 0 at some points in a calculation and non-0 at others. For a general overview, this seems reasonable. Grandi noticed in 1703 that infinite series did not need to give determinate answers; this was widely known in by the 1730's. Reading the texts, it's fairly clear that th...
That it worked in every instance of continuous functions that had been considered up to that point, seemed natural, and extended many existing demonstrations that a specific sequence of continuous functions had a continuous limit.
A need for lemmas of the latter form are endemic, for a concrete class of examples, any argument via a Taylor series on an interval implicitly requires such a lemma, to transfer continuity, integrals and derivatives over. In just this class, you get numerical evidence came from the success of perturbative solutions to Newtonian mechanics, and theoretical evidence in the existence of well behaved Taylor series for most functions.
Observationally, the vast majority of mathematical papers do not make claims that are non-rigorous but as well supported as the Basel problem. They split into rigorous proofs (potentially conditional on known additional hypotheses eg. Riemann), or they offer purely heuristic arguments with substantially less support.
It should also be noted that Euler was working at a time when it was widely known that the behaviour of infinite sums, products and infinitesimal analysis (following Newton or Leibnitz) was without any firm foundation. So analysis of these obje...
I do not think that they are "making it up"; that phrase to me seems to attach all sorts of deliberate malfeasance that I do not wish to suggest. I think that to an outside observer the estimate is optimistic to the point of being incredible, and reflecting poorly on CEA for that.
These 291 people haven't pledged dollar values. They've pledged percentage incomes. To turn that into a dollar value you need to estimate whole-life incomes. Reverse engineering an estimate of income (assuming that most people pledge 10%, and a linear drop off in pledgers with 50% donating for 40 years), yields mean lifetime earnings of ~£100K. That's about the 98th centile for earnings in the UK.
Hi Will,
I'm glad to hear that a general response is being collated; if there are things where CEA can improve it would seem like a good idea to do them, and if I'm wrong I would like to know that. Turning to the listed points:
I went into that conversation with a number of questions I sought answers to, and either asked them or saw the data coming up from other questions. I knew your time was valuable and mostly targeted at other people there.
Adam explicitly signed off on my comment to Luke. He saw the draft post, commented on it, recommended it be put
The primary source of the post was an extensive email exchange with Adam Casey (currently working full time at CEA). Since we are friends, this was not primarily in an official capacity. I also asked Adam to cross check the numbers whilst wearing a more official hat.
I was encouraged by him and Alexey Morgunov (Cambridge LWer) to make the substance of this public immediately after Will Crouch came up to Cambridge.
Can I just make clear my role here. 1) I've had general conversation with Jonathan about CEA and MIRI, in which several of these criticisms were raised. 2) I checked over the numbers for the GWWC impact assessment. 3) I've also said that criticism in public is often productive, and that public scrutiny of CEA on LW would be helpful for people choosing between it and MIRI. 4) I saw a draft of the post before it was published.
I want to make it very clear that: 1) I do not endorse this post. 2) I did not do detailed fact-checking for it. 3) I do not want t...
Whose status ordering are you using? Getting someone who is not a mathematician to TMS is harder; within the Natural Sciences it is possible, and there are O(1) Computer Scientists, philosophers or others. For the historians, classicists or other subjects, mathmos are not high status. In terms of EtG, these groups are valuable - most traders are not quants.
In that case, having a claim on every page of the GWWC site claiming that £112.8M have been pledged seems deceptive. 291 people have pledged, and [by a black box that doesn't trivially correspond to reality] that's become £112.8M. I know that at least 3 people in Cambridge have seen that statistic and promptly laughed at GWWC. The numbers are implausible enough that <5s Fermi estimates seem to refute it, and then the status of GWWC as somewhat effective rational meta-charity is destroyed. Why would someone trust GWWC's assessment of charities or likely impact over, say, GiveWell, if the numbers GWWC display are so weird and lacking in justification?
Talking about effective altruism is a constraint, as is talking about mathematics. Being a subject society makes it easier to get people from that subject to attend; it also makes it harder to convince people from outside that subject to even consider coming.
TMS pulls 80+ people to most of its talks, which are not generally from especially famous mathematicians. TCSS got 600 people for a Pensrose-Rees event. Both TCSS and TMS have grown rapidly in 18-24 months, having existed for far longer. This seems to indicate that randomly selected student societies h...
This holds for graduates who earn less than average as well. Is there data showing that the predominant source of career changes are people who would otherwise earn substantially less than mean? Is there data suggesting that the career changes are increasing incomes substantially?
We might mean many things by "2 + 2 = 4". In PA: "PA |- SS0 + SS0 = SSSS0", and so by soundness "PA |= SS0 + SS0 = SSSS0" In that sense, it is a logical truism independent of people counting apples. Of course, this is clearly not what most people mean by "2+2=4", if for no other reason than people did number theory before Peano.
When applied to apples, "2 + 2 = 4" probably is meant as: "apples + the world |= 2 apples + 2 apples = 4 apples". the truth of which depends on the nature of "the worl...
I want to note that I may be confused: I have multiple hypotheses fitting some fraction of the data presented.
The first is what seems to be in the example. The second is what the strategy handles. The third is what I get when I try to interpret:
...This technique is about f
So, it seems to me that what you describe here is not moving up a hierarchy of goals, unless there are serious issues with the mechanisms used to generate subgoals. It seems like slogans more appropriate to avoiding the demonstrated failure mode are:
"Beware affective death spirals on far-mode (sub)goals" or "Taboo specific terms in your goals to make them operationally useful" or possibly even "Check that your stated goals are not semantic stop-signs"
As presented, you are claiming that:
...I wanted to be a perfectly honed instru
Thanks. Definite typo, Fixed.
Better directions to the JCR (with images) are here.
ETA: Also fixed the list of meetups to link there.
The foundational problem in your thesis is that you have grounded "rationality" as a normative "ought" on beliefs or actions. I dispute that assertion.
Rationality is more reasonably grounded as selecting actions so as to satisfy your explicit or implicit desires. There is no normative force to statements of the form "action X is not rational", unpacked as "If your values fall into {large set of human-like values}, then action X is not optimal, choosing for all similar situations where the algorithm you use is run".
Th...
From your own summary:
I think that trolley problems contain perfect information about outcomes in advance of them happening, ignore secondary effects, ignore human nature, and give artificially false constraints.
Which is to say they are idealised problems; they are trued dilemmas. Your remaining argument is fully general against any idealisation or truing of a problem that can also be used rhetorically. This is (I think) what Tordmor's summary is getting at; mine is doing the same.
Now, I think that's bad. Agree/disagree there?
So, I clearly disagree...
The thrust of your argument appears to be that: 1) Trolley problems are idealised 2) Idealisation can be a dark art rhetorical technique in discussion of the real world. 3) Boo trolley problems!
There are a number of issues.
First and foremost, reversed stupidity is not intelligence. Even if you are granted the substance of your criticisms of the activists position, this does not argue per se against trolley problems as dilemmas. The fact that they share features with a "Bad Thing" does not inherently make them bad.
Secondly, the whole point of cons...
In the wider sense, MML still works on the dataset {stock prices, newspapers, market fear}. Regardless of what work has presently been done to compress newspapers and market fear, if your hypothesis is efficient then you can produce the stock price data for a very low marginal message length cost.
You'd write up the hypothesis as a compressor-of-data; the simplest way being to produce a distribution over stock prices and apply arithmetic coding, though in practice you'd tweak whatever state of the art compressors for stock prices exist.
Of course the side ef...
You and I both agree on Bayes implying 1/21 in the single constant case. Considering the 2 constant game as 2 single constant games in series, with uncertainty over which one (k1 and k2 the mutually exclusive "this is the k1/k2 game")
P(H | W) = P(H ∩ k1|W) + P(H ∩ k2|W) = P(H | k1 ∩ W)P(k1|W) + P(H|k2 ∩ W)P(k2|W) = 1/21 . 1/2 + 1/21 . 1/2 = 1/21
This is the logic that to me drives PSB to SB and the 1/3 solution. I worked it through in SB by conditioning on the day (slightly different but not substantially).
I have had a realisation. You work dir...
Continuity problem is that the 1/2 answer is independent of the ratio of expected number of wakings in the two branches of the experiment
Why is this a problem?
The next clause of the sentence is the problem
unless the ratio is 0 (or infinite) at which point special case logic is invoked to prevent the trivially absurd claim that credence of Heads is 1/2 when you are never woken under Heads.
The problem is special casing out the absurdity, and thus getting credences that are discontinuous in the ratio. On the other hand, you seem to take 1/21in PSB (...
No; P(H|W) = 1/21
Multiple ways to see this: 1) Under heads, I expect to be woken 1/10 of the time Under tails, I expect to be woken twice. Hence on the average for every waking after a head I am woken 20 times after a tail. Ergo 1/21.
2) Internally split the game into 2 single constant games, one for k1 and one for k2. We can simply play them sequentially (with the same die roll). When I am woken I do not know which of the two games I am playing. We both agree that in the single constant game P(H|W) = 1/21.
It's reasonably clear that playing two single const...
Continuity problem is that the 1/2 answer is independent of the ratio of expected number of wakings in the two branches of the experiment, unless the ratio is 0 (or infinite) at which point special case logic is invoked to prevent the trivially absurd claim that credence of Heads is 1/2 when you are never woken under Heads.
If you are put through multiple sub-experiments in series, or probabilistically through some element of a set of sub-experiments, then the Expected number of times you are woken is linearly dependent on the distribution of sub-experiment...
You're woken with a big sign in front of you saying "the experiment is over now", or however else you wish to allow sleeping beauty to distinguish the experimental wakings from being allowed to go about her normal life.
Failing that, you are never woken; it shouldn't make any difference, as long as waking to leave is clearly distinguished from being woken for the experiment.
No. I assert P(H|W) = 1/21 in this case.
Two ways of seeing this: Either calculate the expected number of wakings conditional on the coin flip (m/20 and m for H and T). [As in SB]
Alternatively consider this as m copies of the single constant game, with uncertainty on each waking as to which one you're playing. All m single constant games are equally likely, and all have P(H|W) = 1/21. [The hoped for PSB intuition-pump]
Before I am woken up, my prior belief is that I spend 24 hours on Monday and 24 on Tuesday regardless of the coin flip. Hence before I condition on waking, my probabilities are 1/4 in each cell.
When I wake, one cell is driven to 0, and the is no information to distinguish the remaining 3. This is the point that the sleeping twins problem was intended to illuminate.
Given awakenings that I know to be on Monday, there are two histories with the same measure. They are equally likely. If I run the experiment and count the number of events Monday ∩ H and Monday ...
As I see it, initially (as a prior, before considering that I've been woken up), both Heads and Tails are equally likely, and it is equally likely to be either day. Since I've been woken up, I know that it's not (Tuesday ∩ Heads), but I gain no further information.
Hence the 3 remaining probabilities are renormalised to 1/3.
Alternatively: I wake up; I know from the setup that I will be in this subjective state once under Heads and twice under Tails, and they are a priori equally likely. I have no data that can distinguish between the three states of identi...
The reason it corresponds to Sleeping Beauty is that in the limit of a large number of trials, we can consider blocks of 20 trials where heads was the flip and all values of the die roll occurred, and similar blocks for tails, and have some epsilon proportion left over. (WLLN)
Each of those blocks corresponds to Sleeping Beauty under heads/tails.
No; between sedation and amnesia you know nothing but the fact that you've been woken up, and that 20 runs of this experiment are to be performed.
Why would an earlier independent trial have any impact on you or your credences, when you can neither remember it nor be influenced by it?
It isn't a probability; the only use of it was to note the method leading to a 1/2 solution and where I consider it to fail, specifically because the number of times you are woken is not bound in [0,1] and thus "P(W)" as used in the 1/2 conditioning is malformed, as it doesn't keep track of when you're actually woken up. In as much as it is anything, using the 1/2 argumentation, "P(W)" is the expected number of wakings.
...No. You will wake on Monday with probability one. But, on a randomly selected awakening, it is more likely that it's
Of course P(W) isn't bound within [0,1]; W is one of any number of events, in this case 2: P(You will be woken for the first time) = 1; P(You will be woken a second time) = 1/2. The fact that natural language and the phrasing of the problem attempts to hide this as "you wake up" is not important. That is why P(W) is apparently broken; it double counts some futures, it is the expected number of wakings. This is why I split into conditioning on waking on Monday or Tuesday.
(Tuesday, tails) is not the same event as (Monday, tails). They are distinct ...
The point of the PSB problem is that the approach you've just outlined is indefensible.
You agree that for each single constant k_i P(H|W) = 1/21. Uncertainty over which constant k_i is used does not alter this.
So if I run PSB 20 times, you would assert in each run that P(H|W) = 1/21. So now I simply keep you sedated between experiments. Statistically, 20 runs yields you SB, and each time you answered with 1/21 as your credence. Does this not faze you at all?
You have a scenario A where you assert foo with credence P, and scenario B where you also assert foo with credence P, yet if I put you in scenario A and then scenario B, keeping you sedated in the meantime, you do not assert foo with credence P...
The claim is implied by your logic; the fact that you don't engage with it does not prevent it from being a consequence that you need to deal with. Furthermore it appears to be the intuition by which you are constructing your models of Sleeping Beauty.
Imagine we repeat the sleeping beauty experiment many times. On half of the experiments, she'd be on the heads path. On half of the experiments, she'd be on the tails path.
Granted; no contest
If she is on the tails path, it could be either monday or tuesday.
And assuredly she will be woken on both days...
P(Monday ∩ H | W) = P(Monday ∩ T | W). Regardless of whether the coin came up heads or tails you will be woken on Monday precisely once.
P(Monday ∩ T | W) = P(Tuesday ∩ T | W), because if tails comes up you are surely woken on both Monday and Tuesday.
You still seem to be holding on to the claim that there are as many observations after a head as after a tail; this is clearly false. There isn't a half measure of observation to spread across the tails branch of the experiment; this is made clearer in Sleeping Twins and the Probabilistic Sleeping Beauty probl...
It seems there are a few meta-positions you have to hold before taking Bayesianism as talked about here; you need the concept of Winning first. Bayes is not sufficient for sanity, if you have, say, an anti-Occamian or anti-Laplacian prior.
What this site is for is to help us be good rationalists; to win. Bayesianism is the best candidate methodology for dealing with uncertainty. We even have theorems that show that in it's domain it's uniquely good. My understanding of what we mean by Bayesianism is updating in the light of new evidence, and updating correctly within the constraints of sanity (cf Dutch books).
The same one that you're currently seeing; for all values of E there is a value of F such that this is consistent, ie that D has actually predicted you in the scenario you currently find yourself in.
The game is to pick a box numbered from 0 to 2; there is a hidden logical computation E yielding another value 0 to 2. Omega has a perfect predictor D of you. You choose C.
The payout is 10^((E+C)mod 3), and there is a display showing the value of F = (E-D)mod 3.
If F = 0, then:
And similarly for F = 1, F = 2 play C = F+1 as the only stable solution (which nets ...
This ad-hoc fix breaks as soon as Omega makes a slightly messier game, wherein you receive a physical clue as to a computation output, and this computation and your decision determine your reward.
Suppose that for any output of the computation there is a a unique best decision, and that furthermore this set of (computation output, predicted decision) pairs are mapped to distinct physical clues. Then given the clue you can infer what decision to make and the logical computation, but this requires that you infer from a logical fact (the predictor of you) to the physical state to the clue to the logical fact of the computation.
The underlying issue is what we take the purpose of debate or discussion to be. Here we consider discourse to be prior to justified belief; the intent is to reveal the reasonable views to hold, and then update our beliefs.
If there is a desire to justify some specific belief as an end in itself, then the rules of logical politeness are null; they have no meaning to you as you're not looking to find truth, per se, but to defend an existing position. You have to admit that you could in principle be wrong, and that is a step that, by observation, most people d...
Even if it's the case that the statistics are as suggested, it would seem that a highly effective strategy is to ensure that there are multiple adults around all the time. I'll accept your numbers ad arguendo (though I think they're relevantly wrong).
If there's a 4% chance that one adult is an abuser, there's a 1/625 chance that two independent ones are, and one might reasonably assume that the other 96% of adults are unlikely to let abuse slide if they see any evidence of it. The failure modes are then things like abusers being able to greenbeard well eno... (read more)