tag:blogger.com,1999:blog-15005476.post639974934874784255..comments2023-10-10T08:02:18.073-04:00Comments on Rationally Speaking: Odds again: Bayes made usableUnknownnoreply@blogger.comBlogger42125tag:blogger.com,1999:blog-15005476.post-30946652414616205372012-12-11T02:32:52.529-05:002012-12-11T02:32:52.529-05:00Ian, thanks, I am looking forward to it!Ian, thanks, I am looking forward to it!Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-71409149026947064842012-12-10T18:44:45.013-05:002012-12-10T18:44:45.013-05:00Richard, I think I am going to write a post on the...Richard, I think I am going to write a post on the DA in which I try to make sense of whether its logic works; stay tuned.ianpollockhttps://www.blogger.com/profile/15579140807988796286noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-86375069060065340122012-12-09T01:55:54.143-05:002012-12-09T01:55:54.143-05:00Unfortunately, if any linkage exists between the c...Unfortunately, if any linkage exists between the coin toss and the weather, you will never be able to extricate any statistical pattern from all of the noise. More to the point, you will never be able to do a controlled experiment to verify your original assertion, which was that existence of gods and the creation of machines by man are somehow linked. You are operating on conjecture not evidence.Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-71510351801781761122012-12-08T08:36:45.398-05:002012-12-08T08:36:45.398-05:00Richard - 100 people tossing coins confuses the is...Richard - 100 people tossing coins confuses the issue. I am talking about 'you' tossing a coin, and 'you' getting rain. Given limitations of time, cannot go into what does 'you' even means, but am saying that other objects in the local environment may have an interest in the results, and game them. You can call this magic if you like, it's just stuff we don't understand yet well enough to communicate to others in ways we can all agree. Some call this philosophy - others call it science, but both are established communication methods. <br /><br />The idea behind an 'informational' approach is simple:<br /><br />Given two objects (a sender, a receiver), we have some sort of constrained but fluid process behind it.DaveShttps://www.blogger.com/profile/15840516954793215700noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-520682664857387152012-12-07T16:30:12.410-05:002012-12-07T16:30:12.410-05:00Great question. I have to think about it.Great question. I have to think about it.ianpollockhttps://www.blogger.com/profile/15579140807988796286noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-62400055808162798442012-12-06T01:03:44.864-05:002012-12-06T01:03:44.864-05:00To get the ball rolling (pun intended), here are t...To get the ball rolling (pun intended), here are two thought experiments which at first glance seem to be equivalent but are actually quite different:<br /><br />1. Suppose I have two urns, labeled A and B. Urn A contains 10 balls, numbered from 1 through 10. Urn B contains 100 balls, numbered from 1 through 100. I pick an urn at random (you can't see which), and draw a ball from it at random. The number on the ball is 7. What are the odds that it was drawn from Urn A?<br /><br /> O(UrnA|Number7) = O(UrnA) * (P(Number7|UrnA)/P(Number7|UrnB))<br /> = 1 * (1/10) / (1/100) = 10 to 1 odds<br /><br />2. Suppose I have 110 sealed envelopes. 10 cards (forming group A) are numbered uniquely with a number from 1 to 10 on the outside, with a card inside that has the letter A printed on it. 100 cards (group B) are numbered uniquely with a number from 1 to 100 on the outside, with a card inside that has the letter B printed on it. 110 people enter the room, and each is handed one of the cards on entering. Your envelope has the number 7 printed on it. What are the chances that the card inside your envelope has the letter A? Are the odds<br /><br /> O(GroupA|Number7) = O(GroupA) * (P(Number7|GroupA)/P(Number7|GroupB))<br /> = 1 * (1/10) / (1/100) = 10 to 1 odds?<br /><br />This answer is wrong, because the prior odds for me giving you a card from group A are not 1 to 1, they are 1 to 10 against. So the correct answer is that (after factoring in the conditionals) your envelope has an equal chance of being in group A or group B. As confirmation, consider that there is just one other person in the room whose envelope has "7" on it. One of you has the A envelope, and the other has the B envelope.<br /><br />In the first experiment, I chose one of the two urns randomly (giving 1 to 1 odds for my prior) before I drew a ball. So there is a selection bias at work that makes the two experiments different.<br /><br />I think the Doomsday Argument proponents are trying to solve a problem of the second kind with a solution of the first kind. But they will reject this refutation. In Bostrum's words, ""It can be showed (see e.g. http://www.anthropic-principle.com/preprints/alive.html) that this greater prior probability that you are in the bigger race would exactly counterbalance and cancel the probability shift that the DA says you should make when you discover that you were born early (i.e. that you have a birth rank that is compatible with you being in the small race). This would annul the DA, but it only works if we know that there are both long-lasting and short-lasting races out there, and an anthropic argument can be made against that assumption -- if there were so many long-lasting races, how come we are not in one of them and having a great birth rank; for most observers would then be in such races and with great birth ranks." (http://www.anthropic-principle.com/?q=anthropic_principle/faqs)<br /><br />What gives? I think that what Bostrum suggests is, in effect, first removing the selection bias, then sneaking it back into the prior because, according to him, the evidence of your low birth number supports it. But if you do that, you can't multiply this new prior by the ratio of conditionals, since that ratio is already factored into the prior. In reality, this new "prior" is not really a prior at all.<br /><br />Have I missed something, or is Bostrum off his rocker?Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-62624989908467760512012-12-04T23:23:16.062-05:002012-12-04T23:23:16.062-05:00Ian, I'd love to hear your thoughts on the Doo...Ian, I'd love to hear your thoughts on the Doomsday Arguments of Gott, Carter, Bostrum, et al. I think there is a perfectly good refutation of these arguments, but from what I've read, the proponents are aware of the refutation and don't accept it. There seems to be a specific misapplication of Bayesian inference at work. I can go into more detail later...<br />Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-28030485747060819442012-12-04T18:59:10.413-05:002012-12-04T18:59:10.413-05:00Good points, Ian. I'd also point out that when...Good points, Ian. I'd also point out that whenever new evidence raises the probability that someone from class X did it, it lowers the probability for persons outside of class X. Once you determine the weapon was a pistol and not a shotgun, not only must you raise the probability that Yosemite Sam (a member of the pistol-totin' class) did it; you have to lower the odds on Elmer Fudd (habitually a shotgun carrier). Doing one while failing to do the other will lead to inconsistency. Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-65982699641954153142012-12-04T15:30:23.687-05:002012-12-04T15:30:23.687-05:00>…but "stating your priors" seems lik...>…but "stating your priors" seems like it could, in practice, be the same as "pull a probability out of your ass.”<br /><br />Ok, I think I see where you’re going with this. The first thing to note is that there are two schools of thought on priors. Subjective Bayesians (e.g., de Finetti) hold that there are no constraints on priors besides the laws of probability themselves. According to them, as long as your probability assignments obey the axioms of probability (usually cashed out with Cox’s theorem), they can take whatever values you please. So a subjective Bayesian could legitimately have a prior probability of 99% that Obama is a Cthulhu cultist, as long as they don’t become logically inconsistent (for example, by simultaneously claiming a 50% probability that he is not a Cthulhu cultist).<br /><br />Objective Bayesians (e.g., Jaynes, myself) think that there are almost always some kind of rational constraints on priors – as a trivial example, if I am about to flip a biased coin, & that is all the information I have, it makes no sense for me to assign a prior of 70% to heads, since I have zero evidence that could distinguish heads from tails at this stage. The only rational prior in this situation is 50%, even though no evidence has been collected and we *know* the coin is biased.<br /><br />However, I would wish to stress that even in situations where there is no obvious quantitative constraint on priors, it is still vital to use whatever qualitative, vague knowledge you have – even if it seems like “pulling something out of your ass.” The reason why that’s so vital is that *failing* to do so does, in fact, leave the barn door open to nonsense – <a href="http://xkcd.com/1132/" rel="nofollow">this comic</a> nicely shows why.<br /><br />To sum up, extraordinary claims require extraordinary evidence, BUT without using priors (explicitly or implicitly), you can’t actually make a *distinction* between an extraordinary claim and an ordinary one. Unable to make that distinction, the probability that the sun has exploded is treated exactly the same as any other hypothesis - we “let the data speak for themselves” and end up believing in lunacy, as long as the lunacy has p<0.05. By contrast, a Bayesian is going to have a prior on the sun exploding – potentially just a wild guess “pulled out of their ass” (hey, I don’t know whether it should be 1 in a billion or 1 in a trillion) – that keeps their beliefs in line with common sense when they see the result of that experiment.<br /><br />Now, it is certainly possible to quote stupid priors, and many people have & will. But doing without them completely is a recipe for epistemic insanity.<br /><br />> I've heard that people on LessWrong like to argue about the intentions of the post human super intelligence that is running our simulation. They wonder how one might court its favor and avoid pissing it off.<br /><br />I’m not really interested in becoming the local Stout Defender of Less Wrong, but keep in mind that it is basically a group philosophy blog (despite Yudkowsky’s dislike of philosophy), and as you know, philosophers like to play with thought experiments and wild speculative hypotheticals as a way to clarify concepts and test theories. Without a link to this discussion you reference, it’s hard for me to comment, but don’t jump to the conclusion that anybody on LW is seriously claiming such things are actually the case. Like Newcomb’s problem (also discussed on LW a lot), it may just be a way of testing one of the local decision theories, for example.ianpollockhttps://www.blogger.com/profile/15579140807988796286noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-13957282183476386252012-12-04T10:40:18.317-05:002012-12-04T10:40:18.317-05:00Yeah, Richard's approach is basically correct ...Yeah, Richard's approach is basically correct here.<br /><br />I would only add that one must be *very* careful about assigning probabilities of exactly 0 and 1 to anything. It's okay to do it as a shortcut to make the math simpler, but unless you know with *logical certainty* that nobody in the dining room could have committed the murder, AND that there is absolutely no possibility of an outsider to the cruise ship having snuck on, you've got to keep these possibilities, however unlikely, in the back of your mind as a longshot bet.<br /><br />>Something cautions me about putting too much weight on the spousal victimhood statistics as evidence. I get a feeling it is too generic; every crime is different.<br /><br />Absolutely. For one thing, there is an art to choosing reference classes correctly - for example, maybe spouses in the defendant's particular cultural group are *less* likely to kill each other than random strangers, even though the statistics for spouses in general have the opposite tendency.<br /><br />And unless the spousal stats are extreme (almost all murders are inter-spousal), you're going to need a lot more evidence in order to come near convicting Joe Doaks. At best, the slightly higher prior for a husband is a starting point for the police, directing inquiry to the likeliest of many unlikely suspects, in order that he may be ruled out.ianpollockhttps://www.blogger.com/profile/15579140807988796286noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-74654391015680803782012-12-04T01:40:43.525-05:002012-12-04T01:40:43.525-05:00No Max, if the conditional probability p(E|H) is g...No Max, if the conditional probability p(E|H) is greater than p(E|!H), then it is implied that evidence was sought. The ratio p(E|H)/p(E|!H) is a direct measure of the effectiveness of the search or experiment that *was* performed. We can't say, "Well, the ratio is 10 to 1, so we don't need to do the experiment, we'll just multiply the prior odds by 10". If we don't actually do the experiment, or search for the evidence, or we don't have the required tools to test the hypothesis, then the ratio is 1 to 1. A 1 to 1 ratio only represents ignorance of evidence, not absence of evidence.<br /><br />"Still, if I can't find my keys, that doesn't mean they don't exist." Well no, of course not...unless: your prior probability that the keys don't exist was zero (i.e., you never had keys in the first place). And if you are already certain that you have keys (p = 1, odds infinite), then any nonzero ratio of conditional probabilities leaves those odds at infinity. But let's say you have amnesia and really don't know if you had keys or not, and your prior odds are 1 to 1. Let's say you do a search that has a probability of 60% of not finding your keys if they exist, and a 100% probability of not finding them if they do not. Then not finding your keys reduces the odds that they exist to 3 to 5. Evidence of absence? I think so!Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-48423668047640013032012-12-04T01:13:25.325-05:002012-12-04T01:13:25.325-05:00"P(guilty)=1 in 7 billion"
You wouldn&#..."P(guilty)=1 in 7 billion"<br /><br />You wouldn't need to include the 6 billion people who were nowhere near North America at the time of the crime in your initial pool. In fact I would go much narrower. I would pick P(guilty) -- the probability of a randomly selected person being the perpetrator or a specific crime -- to be 1/N where N is the number of people who, a priori, had the opportunity to commit the crime, if the person is selected from that group (and 0 otherwise).<br /><br />Take a different example: A murder takes place on a cruise ship, which has total crew and passengers = 2000.<br />The prior odds O(PERP(X)) that random passenger or crew member X did it are 1 to 1999. Now 300 people were in the dining room at the time of the crime and could not have perpetrated it. D(X) = "in the dining room"; if X was not in the dining room (!D(X); I use the engineering notation "!" for not), then we have O(PERP(X)|!D(X)) = O(PERP(X)) * P(!D(X)|PERP(X))/P(!D(X)|!PERP(X)). The numerator P(!D(X)|PERP(X)) is 1 because we know the perp couldn't have been in the dining room. The denominator is 1699/1999, because 1699 of the 1999 people who aren't perps were not in the dining room. So the odds that X committed the crime, taking into account the 300 alibis, become 1 to 1699. And so on with each additional piece of evidence.<br /><br />Obviously you could just eliminate the 300 from the start, and assign odds 1 to 1699 without doing the Bayesian calculation, but I went through the exercise to show that the numbers work out as we would expect.<br /><br />You could get more sophisticated and divide the people into classes. The victim's spouse is in a class by him/herself, having a higher conditional probability (based on opportunity and possible motives) than the people in neighboring cabins; those residing on the same deck have higher conditional probabilities than those residing on other decks, those who have been seen interacting with the victim have higher conditional probabilities than those who have not, and so on. Each potential suspect could have odds calculated based on the classes to which he/she belongs or does not belong.<br /><br />Something cautions me about putting too much weight on the spousal victimhood statistics as evidence. I get a feeling it is too generic; every crime is different. Surely it is rational to include the information, but only if we know all other relevant facts are being presented. If the investigation was sloppy, or a dishonest prosecutor fails to disclose exculpatory evidence, then the statistical evidence could tilt the case against an innocent man the wrong way. And of course, a jury is only allowed to consider the evidence presented in the courtroom.<br />Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-29264752289559141732012-12-03T19:17:55.378-05:002012-12-03T19:17:55.378-05:00The math does seem to work out:
O(guilty|married t...The math does seem to work out:<br />O(guilty|married to victim) =<br />O(guilty)*P(married to victim|guilty)/P(married to victim|not guilty)<br /><br />Then, calculate P from O: P=O/(O+1)<br /><br />My simple solution was to directly calculate P(guilty|married to victim), which is the fraction of people married to a murder victim who were the killers.<br /><br />But I'm still confused about the precise meaning of P(guilty), maybe because there are two ways of looking at it.<br />First, there's Bayesian probability, where we start off with P(guilty)=prevalence of murderers in the population, and then update it based on cases with similar evidence.<br />But then there's the process of elimination, where we start off with P(guilty)=1 in 7 billion, and then narrow it down as we learn about the killer. Like, the fact that the victim was stabbed rules out anyone who couldn't have stabbed her.<br />How do you reconcile the two?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15005476.post-59837617250862234642012-12-03T15:58:02.794-05:002012-12-03T15:58:02.794-05:00Ian, thanks for your response. A couple of points ...Ian, thanks for your response. A couple of points of clarification. Bostrom doesn't explicitly make the claim. I reduce his Simulation Argument to that. Because, as Julia noted at the end of the SA podcast, the SA is tantamount to Intelligent Design. But boy do I agree with your skepticism about bizarre metaphysical claims, and that is exactly where I suspect that Bayesian Analysis might be "leaving the barn door open." I'm out of my depth here, because I don't really grock the math, but "stating your priors" seems like it could, in practice, be the same as "pull a probability out of your ass." And the further you get from that stated prior, either in a chain of a Bayesian calculations or just running too far with your conclusions, the more you are disconnected from reality. I've heard that people on LessWrong like to argue about the intentions of the post human super intelligence that is running our simulation. They wonder how one might court its favor and avoid pissing it off. That they feel comfortable doing this is proof to me that a barn door somewhere has been left open. And I don't think they are using frequentist probability to get there.Aaron Shurehttps://www.blogger.com/profile/00837439765332783167noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-9710632908518155772012-12-03T15:33:55.125-05:002012-12-03T15:33:55.125-05:00>But how would you use the likelihood ratio to ...>But how would you use the likelihood ratio to update the probability of guilt as you learn each of the above three pieces of evidence?<br />You'd have to do weird calculations like P(victim=wife | guilty) / P(victim=wife | not guilty)<br />...But the prior odds that a random person murdered this specific victim are not the prevalence of murderers, they're 1 in 7 billion.<br /><br />If you really want to rewind to the state of extreme ignorance where all of humanity from a Mongolian goatherd to Dick Cheney is equally suspect, then you'll have to include a *lot* of updates (based on location, age, physical ability, mobility, relationship...) in order to get even a manageable suspect list, let alone focus in on the husband in particular. Thankfully, our common sense does a lot of that work for us.<br /><br />But skipping ahead to the spousal relationship issue, I think the correct framing would be a prior on Joe Doaks' guilt as a "randomly chosen" citizen, followed by an update on the fact that Joe Doaks is married to the victim - it would look something like<br /><br />O(Joe Doaks guilty) = 1:100,000 (say)<br />P(married to victim|guilty)/P(married to victim|not guilty)=(fraction of murders in which victim is spouse of killer)/(fraction of innocent people married to a murder victim).ianpollockhttps://www.blogger.com/profile/15579140807988796286noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-56304335681683675922012-12-03T14:16:15.467-05:002012-12-03T14:16:15.467-05:00I want to start from knowing as little as possible...I want to start from knowing as little as possible about the defendant, and treat every piece of information as evidence. So, the first piece of evidence is that the defendant is a man. The second piece of evidence is that the victim is his wife. The third piece of evidence is that she was stabbed to death.<br />If we treat it all as background info, then the solution is simple. But how would you use the likelihood ratio to update the probability of guilt as you learn each of the above three pieces of evidence?<br />You'd have to do weird calculations like P(victim=wife | guilty) / P(victim=wife | not guilty)<br /><br />And in such calculations, does "guilty" refer to this specific crime, or to any similar crime?<br />Like, the prior odds that a random person has some cancer are the prevalence of this cancer in the population. The prior odds that a random husband murdered his wife are the prevalence of such murderers in the population. But the prior odds that a random person murdered this specific victim are not the prevalence of murderers, they're 1 in 7 billion.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15005476.post-31708162505005814002012-12-03T12:54:33.530-05:002012-12-03T12:54:33.530-05:00Nice article!Nice article!ianpollockhttps://www.blogger.com/profile/15579140807988796286noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-11358550667048623322012-12-03T12:34:05.286-05:002012-12-03T12:34:05.286-05:00>Knowing nothing else, we'd just take all t...>Knowing nothing else, we'd just take all the cases of wives who were stabbed to death, and see what fraction of them were stabbed by their husband.<br /><br />Sounds like a good start, although if the police know nothing else, it's not clear why he would have been arrested.<br /><br />>But how would you use the likelihood ratio here? What are the prior odds of guilt, knowing nothing about the guy? 2 in 7 billion?<br /><br />Well, if we treat the fact that he is the victim's husband as background information, we've got no evidence on which to update, so there is no need to use the likelihood ratio (or Bayes theorem). Just stick with your prior: prior odds =(# of murders carried out by spouses)/(# of murders NOT carried out by spouses).<br /><br />You would only use Bayes theorem if you found out something additional to your background information - for example, that the killer left AB+ blood on the scene and the husband has AB+ blood.<br /><br />>Is the probability that the victim is his wife greater if he killed her or if he didn't kill her? Weird question.<br /><br />Yeah, not sure I follow. ;)ianpollockhttps://www.blogger.com/profile/15579140807988796286noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-19088737223074563482012-12-03T05:31:17.936-05:002012-12-03T05:31:17.936-05:00Thank you, Ian, I find this version of the theorem...Thank you, Ian, I find this version of the theorem way clearer.<br /><br />I was also able to start an argument using the odds form of Bayes' theorem, and I credited your article: "<a href="http://assertiveatheism.blogspot.it/2012/12/betting-on-horses-and-resurrection-of.html" rel="nofollow">Betting on horses and the resurrection of Jesus (I)</a>".<br /><br />Thank you again!Il Censorehttps://www.blogger.com/profile/01584751016662084504noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-71723406601367542922012-12-03T04:53:43.131-05:002012-12-03T04:53:43.131-05:00Absence of evidence means you didn't look for ...Absence of evidence means you didn't look for evidence, or you didn't use the right instruments. If you looked everywhere you'd expect to find something if it exists, and didn't find it, that's evidence of absence. Still, if I can't find my keys, that doesn't mean they don't exist.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15005476.post-26247879073368998802012-12-03T04:36:29.674-05:002012-12-03T04:36:29.674-05:00How would you apply it to solve this problem?
A gu...How would you apply it to solve this problem?<br />A guy is accused of murdering someone.<br />Evidence: the victim is his wife, and she was stabbed to death.<br /><br />Knowing nothing else, we'd just take all the cases of wives who were stabbed to death, and see what fraction of them were stabbed by their husband.<br /><br />But how would you use the likelihood ratio here? What are the prior odds of guilt, knowing nothing about the guy? 2 in 7 billion? Is the probability that the victim is his wife greater if he killed her or if he didn't kill her? Weird question.<br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15005476.post-65146218046102409162012-12-01T23:59:41.231-05:002012-12-01T23:59:41.231-05:00All of your premises are axiomatic, not empirical....All of your premises are axiomatic, not empirical. The means we have for determining the chance of rain are empirical. We measure pressure, temperature, humidity, wind, and we study satellite photos. None of which are influenced by a coin toss, not even by a million coin tosses (unless we toss the million coins right into the clouds and that triggers precipitation by direct physical interaction). Even if your axiom happened to be true true (in an alternate universe where magic is real), you could never establish a causal link, because one week your neighbor who is performing the same experiment throws heads and cancels your tails, and the next week you both throw tails... Not to mention the ten people across town and the hundred in the neighboring city all trying the same thing. You will never be able to separate all the variables to get a meaningful result. In other words, the two things would remain statistically independent even if your axioms were true. Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-34630662059416889442012-12-01T21:18:43.762-05:002012-12-01T21:18:43.762-05:00Richard - re the independence stuff, sorry to drag...Richard - re the independence stuff, sorry to drag back to the gods stuff, which bugs Massimo, who I forgot to congratulate, seeing that A for A book in the Barnes and Noble today, but anyway if some god responsible for local weather also had some interest in your coin flip, I guess there could be some linkage. But if we permit an atheistic world (all gods are focused on some intergalactic event, having noticed that earth stuff runs with or without their intervention) we can still establish a linkage, but not as easy. With 60% chance of rain, and 50ish% chance afforded by the coin flip, every object that (a) stands to lose by the rain and (b) is aware of the upcoming flip may try to game the outcome of the flip. Besides you, there is every possible reason to think the coin is aware, and other elements in the 'local environment' whatever that means is aware. If any activity whatsoever in this universe takes place as a result of knowledge of both (a) and (b) then the chances have been affected. Maybe only two more drops fell near your front door as a result of what I can only weakly describe as a 'cosmic concurrence' but what more people would attribute to combined consciousnesses at work. All explainable from a naturalistic point of view, just not anytime in the next 20 years - I think.DaveShttps://www.blogger.com/profile/15840516954793215700noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-87342674681937793982012-12-01T20:54:40.080-05:002012-12-01T20:54:40.080-05:00Richard - if the number of believers in coffee cup...Richard - if the number of believers in coffee cups dropped from 100% to 58% I'd probably be more curious as to why that happened. Did 42% of them go off-grid and take up residence in places where utensils could not easily exist, say the sea? Did 42% of them start really thinking about these things and decided they could not exist without human observers? Did 42% of them start making judgments about the nature of all 'physical' stuff, then rendered it impossible to talk about it without assuming many other things, things more mathematical than physical? Not quite sure how your empiricism works. Are you saying that if you and I were both coffee beans who knew nothing about cups, once one of us sensed a coffee cup two aisles away in the grocery, there would be an easy way of advising the other about it? Empiricism is a useful tool for those who know how to use it and use agreed methods, and its use leads to consensus read belief. A 'proper' view of reality is one that views it as a social construct and little else.DaveShttps://www.blogger.com/profile/15840516954793215700noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-76156674054640671922012-12-01T20:40:06.449-05:002012-12-01T20:40:06.449-05:0080% agreed about better weather forecasts, the oth...80% agreed about better weather forecasts, the other 20% putting it to fear of more drastic weather change in the last decade than the previous 10, supported by both changes in the data and relative invariance of recent global warming predictions.DaveShttps://www.blogger.com/profile/15840516954793215700noreply@blogger.com