tag:blogger.com,1999:blog-15005476.post8882676617290052424..comments2018-08-25T21:24:44.954-04:00Comments on Rationally Speaking: Why we should use odds, all the timeUnknownnoreply@blogger.comBlogger40125tag:blogger.com,1999:blog-15005476.post-58714124135695748192013-09-12T15:21:36.177-04:002013-09-12T15:21:36.177-04:00Love your practicality. I'm new to the world ...Love your practicality. I'm new to the world of stats and Bayesians and have struggled with the "conceputal opaqueness" you have referred to other places. With "Odds", I get the real world nature of it as opposed to just something theoretical. I'm giving 10 to 1 that learned something important here.<br />The Big Easyhttps://www.blogger.com/profile/03976700610847399797noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-58131049818162931832011-09-01T21:19:00.970-04:002011-09-01T21:19:00.970-04:00@Jeremy: You are definitely right about expected u...@Jeremy: You are definitely right about expected utility calculations, as in a lottery. That counterexample did not occur to me, but should have, and it does make odds less advantageous for such purposes.<br /><br />However, the very thing that makes odds bad for expected utility calculations seems like a feature when one moves to the realm of propositional beliefs.<br /><br />I think a lot of this business depends on how one mentally pictures odds vs probabilities. I imagine a probability as slice on a pie chart, which conduces to expected utility calcs for sure. But I imagine odds as the quantities in two sides of a weigh scale, which does a nice job of capturing the importance of the diminishing marginal impact of extra information. In terms of expected utility, 99% probability of winning may be only 1.1 times better than 90%. But in terms of <i>collecting evidence</i>, getting 99% probability really is something like 11 times harder than getting 90% probability, not 1.1 times.<br /><br />Also, and this will come into play later, Bayes theorem looks WAY nicer in odds form than the ugly probability form.ianpollockhttps://www.blogger.com/profile/15579140807988796286noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-6506703852514041832011-08-27T00:53:50.693-04:002011-08-27T00:53:50.693-04:00Sorry, I didn't make it explicit that the priz...Sorry, I didn't make it explicit that the prizes were all equal. And yes, lotteries are all losing propositions. My example was to demonstrate the flaws of odds, not comment on lotteries. Point is, we aren't wired to compare odds, we're wired to compare probabilities.Jeremyhttps://www.blogger.com/profile/06393351226439600979noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-2625760800380539952011-08-26T23:11:31.589-04:002011-08-26T23:11:31.589-04:00Jeremy wrote:
> If the first lottery's odds...Jeremy wrote:<br />> If the first lottery's odds of a winning are 9 to 1 (pr=0.1) and the second's 99 to 1 (pr=0.01) you should spend ten times more. But if the first lottery's odds are 1 to 99 (pr=0.99) and second's 1 to 9 (pr=0.9) you should only spend 1.1 times more. <<br /><br />I think you need to rework your example. First, you didn't specify the value of the prize in each of the lotteries. Usually lotteries with higher chances of winning have smaller prizes. Second, lotteries are generally losing propositions, so it's wisest not to spend anything! Finally, and related to the last point, it's unusual to find a lottery where the odds of winning is greater than 1 (i.e. greater than 50% chance of winning). What would the value of the prize be in such a case?Nick Barrowmanhttps://www.blogger.com/profile/11224940659269649220noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-79693184683726205162011-08-26T20:29:06.275-04:002011-08-26T20:29:06.275-04:00Here's why we should never use odds in making ...Here's why we should never use odds in making decisions: if a lottery has ten times higher odds of winning compared to another, how much more money should you spend on the first lottery? <br /><br />The answer is...it depends. If the first lottery's odds of a winning are 9 to 1 (pr=0.1) and the second's 99 to 1 (pr=0.01) you should spend ten times more. But if the first lottery's odds are 1 to 99 (pr=0.99) and second's 1 to 9 (pr=0.9) you should only spend 1.1 times more. Comparing the odds of two choices doesn't actually tell you anything.<br /><br />If I tell you the chances of winning one lottery over another is ten times better, you know right away you should spend ten times more. The human brain can compare probabilities but can't compare odds. If you're going to be making decisions, and what good is a probability if you have no other to compare it to, you're better off sticking with probabilities.<br /><br />This is a problem in epidemiology where we use a lot of ratios. People's intuition is to interpret ratios of odds as ratios of probabilities because ratios of odds are entirely foreign to us. This is fine when dealing with tiny odds but when dealing with high odds (such as your example) the difference can be huge (10 compared to 1.1) and can lead to radically different interpretations. In fact, some people preferentially report odds because their effect sizes just look bigger.<br /><br />Also, I think the benefit you're attributing to odds can actually be attributed to expressing either odds or probabilities as fractions. In fact, totally without evidence to back this up, I'd say most people would say 1 chance in 3 is more intuitive than saying an odds of 2 to 1.Jeremyhttps://www.blogger.com/profile/06393351226439600979noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-31363927637396841422011-08-26T15:00:13.057-04:002011-08-26T15:00:13.057-04:00I've been thinking about it still more, and I ...I've been thinking about it still more, and I think I would attach a degree of belief of 1 to propositions like "existence exists," which would be true no matter how deceived you are.Timothyhttps://www.blogger.com/profile/04338789669131796827noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-4344520857102087052011-08-21T20:05:50.389-04:002011-08-21T20:05:50.389-04:00Rarely do we have enough quantitative information ...Rarely do we have enough quantitative information to state any probabilities (or odds, if you prefer) that have any connection to reality. Rather than try and fail miserably, why not simply use the all natural *qualitative* probabilistic reasoning (something seems "more likely" than something else now, but would seem "less likely" if we were to learn that something else is the case; we are "very sure" or "indifferent" or "very uncertain" etc.).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15005476.post-57154457247819403742011-08-20T18:12:41.407-04:002011-08-20T18:12:41.407-04:00Well, other people have already commented, but I w...Well, other people have already commented, but I will go anyway.<br /><br />I would say that statistics goes about what we ignore. Maybe I am a little bit skeptic and/or negative here. Or maybe a little bit of semantics.<br /><br />And about the example of Peter Singer's lunch, I see it like wishful thinking. There are a lot of assumptions going on, which are not checked self-consistently. Or at least, it seems to me so. I agree however it is a lot of fun, but to me it is just fun and nothing else. <br /><br />Moreover, if we can construct all the possible cases, it would be fine, but in most of the cases this is not possible and the interpretation becomes difficult, not to mention that we are making the problem bigger.<br /><br />Again, maybe I am too strict, but still, if all of us start to make such """subjective"" computations, will we be able to share the results? <br /><br />And in the end, we have to decide, and this is not probabilistic at all.<br /><br />Finally, sorry if this seems too random. I hope that adds something to the other contributions.Oscarhttps://www.blogger.com/profile/04360507492938258763noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-60234715839498735572011-08-20T16:08:47.818-04:002011-08-20T16:08:47.818-04:00I know of a few people for whom that would be a si...<i>I know of a few people for whom that would be a significant improvement! ;)</i><br /><br />I don't doubt it, but I'd say the odds of them adopting that practice are 500 to 1 (give or take).Thameronhttps://www.blogger.com/profile/05056803143951310082noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-5150432394377966662011-08-20T14:11:38.946-04:002011-08-20T14:11:38.946-04:00Part (most) of my preference for lurking is just a...Part (most) of my preference for lurking is just a self-discipline issue in that I find it hard to resist spending excessive amounts of time reviewing the matter in my head. By way of evidence, and at the risk of rambling or making myself look even more foolish, here's this post.<br /><br />But, I was puzzling what you (Eamon) might be referring to when you mentioned a contradiction, and speculating, it might be something like: <br /><br />-If there are no propositions that have a probability of 1, then (by the possible-worlds interpretation) there does not exist any proposition that is true in all possible worlds. Then the proposition, "there does not exist any proposition that is true in all possible worlds" is not true-in-all-possible-worlds, so there is a world where it is false. Then there is a world where there exists a proposition that is true in all possible worlds, which provides us a contradiction.<br /><br />The problem I see is that such derivation requires logical inference, and doubt that attaches to logics' axioms would mean that there is some (however infinitesimal) chance that the contradiction does not obtain. So within the formalism, as it were, you'd say there are propositions true in all possible worlds, but when assessing your degrees of belief, you wouldn't as aliens might be utterly deceiving you about what logic is. (Which might be a convoluted way of saying I don't see why it would be of concern that he might deny the laws of logic (as we know them) are true in all possible worlds, since the propositions that logic assumes obviously wouldn't escape his no-prob-1 view.)Timothyhttps://www.blogger.com/profile/04338789669131796827noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-5258641481622104042011-08-20T10:16:44.866-04:002011-08-20T10:16:44.866-04:00> when Eve mentions that she is 100% certain of...> when Eve mentions that she is 100% certain of the defendant’s guilt, a quick conversion shows that she gives odds of 100:0, aka “infinity.” ... The fact that odds explode as mathematical objects when they try to map absolute certainty is a nice feature probabilities don’t have. <<br /><br />But odds don't explode when you go the other direction. When the probability of an event is zero, the odds is zero. When the odds of event A is infinite, the odds of event ~A is 0. <br /><br />I recently examined somewhat similar issues from a <a href="http://logbase2.blogspot.com/2011/08/landscape-of-probability.html" rel="nofollow">slightly different angle</a>.Nick Barrowmanhttps://www.blogger.com/profile/11224940659269649220noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-76254100306167077922011-08-19T23:20:29.153-04:002011-08-19T23:20:29.153-04:00>I offered an argument as to why one ought to a...>I offered an argument as to why one ought to assign 1 to logical and mathematical propositions in the face of external-world skepticism.<<br /><br />Just briefly, and sorry to reiterate, but - I still don't see how so; Ian's comment drew a distinction between mathematical probability and an epistemic degrees-of-belief, which we'd model using probability. In particular, your comment "Ian grants that 1 + 1 = 2 is a mathematical truth, and ipso facto a necessary truth (i.e. a statement true in all possible worlds)" seems to be what's at issue. If we suspend skepticism to explore an axiom system, then we pronounce things like "this proposition has prob=1" under the axioms, but when we use such a system to model actual beliefs, it doesn't seem we can ascribe a probability of 1 to them. Perhaps the issue is that where I'm reading Ian to be discussing epistemic skepticism (as he says epistemology), you're not? I'd infer Ian is willing to assign probability of 1 to many propositions whilst working in formal contexts ("100% correct as mathematics"), but he's saying we shouldn't when discussing our degrees of belief.Timothyhttps://www.blogger.com/profile/04338789669131796827noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-85736488759303270552011-08-19T19:49:29.439-04:002011-08-19T19:49:29.439-04:001+1=2 is at best a tautology. It's a descript...1+1=2 is at best a tautology. It's a descriptive truth, but only if we have agreed to accept it as such. <br />Except that in reality it can only describe itself with complete accuracy and otherwise is always to some degree an approximation of whatever it's being used to describe outside of the descriptive system we've agreed to use.Baron Phttps://www.blogger.com/profile/04138430918331887648noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-69657296177026203612011-08-19T17:30:34.964-04:002011-08-19T17:30:34.964-04:00Timothy,
Thank you for your reply. Don't fret...Timothy,<br /><br />Thank you for your reply. Don't fret, I really do not care to carry on an extended exchange either. I will simply offer the following and leave matters alone. <br /><br />Re: 'But how amazing it would be if someone managed to rebut the millennia-old problem of epistemic skepticism here in the comments section! Even epistemologists aren't so confident of their work.'<br /><br />I did not offer a refutation of external-world skepticism. I offered an argument as to why one ought to assign 1 to logical and mathematical propositions in the face of external-world skepticism. I could have provided more detail in the argument, sure, but as it is it is sufficient. (Though I did not, I could have gone further and showed that insofar as Ian refuses to assign logical and mathematical truths 1 [and logical and mathematical falsehoods 0], he finds himself employing a probability calculus which is a straightforward extension of a deductive logic whilst denying that the laws of that deductive logic are true in all possible worlds.) <br /><br />Re: Possible worlds <br /><br />Logicians employ possible worlds semantics ubiquitously in order to explicate, amongst other modal notions, a desired concept of logical necessity. Moreover, Ian initiated possible worlds talk when he mentioned skeptical alternatives to our being certain that 1 + 1 = 2. Anyway, the issue is simple: Ian grants that 1 + 1 = 2 is a mathematical truth, and ipso facto a necessary truth, but yet advises not to grant 1 to P (1 + 1 = 2) because 1 + 1 = 2 *might* not be true in a possible world in which aliens massively deceive us. <br /><br />In a nutshell, he grants that 1 + 1 = 2 is a necessary truth (i.e. a statement true in all possible worlds) but then admits a possible world in which 1 + 1 = 2 *may not be true*. <br /><br />As for the rest of your comment, it is completely off the mark, particularly the bit about epistemic agents. <br /><br />P.S. Ian, I suspect your 'hundredth part stake' amendment would not work. Perhaps you should offer up something on this matter for consideration in future?Cian Eamon Marleyhttps://www.blogger.com/profile/09070168038290681070noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-23067301367921517612011-08-19T07:27:45.199-04:002011-08-19T07:27:45.199-04:00What are the odds that, if you pick a natural numb...What are the odds that, if you pick a natural number randomly, it is a multiple of 3? One third? But there are the SAME amount of natural numbers and of multiples of 3. Your guess about the odds depends on your HYPOTHESIS about how are the numbers ordered and what are the statistical process by which you pick one instead of other. (For example, if you assume that the natural numbers are given in their 'natural' order -1, 2, 3...-, then the odds seem to be 1/3; but if you assume that ALL the numbers are mixed in an infinite lottery drum and THEN you pick one, then the odds are 1/2). <br />The question, hence, is that in many cases we don't have ANY sensible idea about what are the 'underlying statistical process' of our pickings, so we don't have ANY reason to opt for some odds instead of others.Jesús P. Zamora Bonillahttps://www.blogger.com/profile/07054631110263426886noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-16610695934569997172011-08-19T05:06:50.055-04:002011-08-19T05:06:50.055-04:00>But if 1 + 1 = 2 is a mathematical truth (and ...>But if 1 + 1 = 2 is a mathematical truth (and it is), then it is necessarily true. If it is necessarily true, then it is true in all possible worlds. If the world in which aliens massively deceive us is a possible world, then in that world it is true that 1 + 1 = 2. The world in which we are massively deceived by aliens is logically possible. Therefore, it follows that in that world 1 + 1 = 2 is true. So, even if *we are being massively deceived by aliens*, we must assign necessary truths a 1.<<br /><br />I am not looking to engage in another of those lengthy exchanges I've had recently, but I'll overcome my better judgment to offer my thoughts. Take them for what they're worth; perhaps Ian might expound on the issue better.<br /><br />Eamon, I do not see an argument in your posts that a proposition that has a likelihood of 1 under a set of axioms, or even multiple sets of axioms, implies the axiom set, or at least the set of axiom sets, itself is "necessarily true," which (it seems to me) is what you would need to show to substantiate the first sentence I've quoted, since otherwise epistemic doubt that attaches to the axioms would then attach to propositions that the axioms attribute a probability of 1 to. <br /><br />Additionally, I do not see how your implicit citation of the modal logics helps your argument. Possible-worlds semantics is just an interpretation of the formal operations the logics perform. It doesn't tell us that propositions it assigns the "necessarily true" value to are necessarily true independently of the logic's own epistemological status. <br /><br />But how amazing it would be if someone managed to rebut the millennia-old problem of epistemic skepticism here in the comments section! Even epistemologists aren't so confident of their work. Perhaps it might help to persuade me if you were to show more expressly what the contradiction is that your first comment's last paragraph mentions, and discuss why the contradiction is of epistemological rather than just logical significance?<br /><br />>Betting odds directly measure the ratio at which money changes hands in a bet and, at best, only indirectly measure degrees of belief. Many factors associated with betting odds restrict their utility in indirectly measuring uncertainty.<<br /><br />"Betting" is just an interpretation; Ian's other interpretation, "'odds of 2 to 1 in favour of rain tomorrow' means something like 'days like this are followed by twice as many rainy days as non-rainy days, to the best of my knowledge'" seems less behavioral. Also I'm not sure why you cast your discussion in terms of the epistemic agent. It's been quite some time since I've had this material at all, but as I recall the modal epistemic agent knows all the logical consequences of her or his belief, and in any case there are many idealized assumptions for epistemic agents. Although Ian's reply here casts the issue more in terms of utility, so perhaps behavioral considerations are important for Ian's position.Timothyhttps://www.blogger.com/profile/04338789669131796827noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-19590434538059108562011-08-19T03:07:31.597-04:002011-08-19T03:07:31.597-04:00"@Baron: What do you mean by an algorithmic s..."@Baron: What do you mean by an algorithmic scale here?"<br /><br />That would be the strategic process that set up the the optional choices available for whatever behavioral responses were to be "probably" involved.Baron Phttps://www.blogger.com/profile/04138430918331887648noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-53123652929081232562011-08-19T01:46:49.735-04:002011-08-19T01:46:49.735-04:00Thanks to all for the comments! FYI, sometimes wor...Thanks to all for the comments! FYI, sometimes work & other commitments leave me less time to answer them than I would like, so things will sometimes be a bit... laggy.<br /><br />@Hector: Alas, I think you're right about bias appearing in people's assessments of subjective probabilities. However, I like the alternatives even less.<br />As to the Bayesian/Frequentist jihad, I'm planning to write something about that in the fullness of time, so I will refrain (with some difficulty) from arguing the point now. :)<br /><br />@Eamon: I had trouble following your reasoning on possible worlds and necessary truths, but I will take another run at it tomorrow. In practice I do think it's a good idea to have an implied ceteris paribus clause ("given gross model accuracy" or something) and work from there so that one does not have to mention the aliens too much. Saying 100% still gives me the willies, though.<br /><br />As to your reasons for not using odds as a measure of degrees of belief, I originally had a paragraph on risk aversion, but decided the post was already too long. My approach is this: in doing a thought experiment involving betting (or, for that matter, actual betting), consider what amount of money would ruin you if you lost it, then make the "stake" maybe a hundredth part of that amount. Something significant but not ruinous.<br /><br />@t-b: I definitely would never advocate assigning a probability to a statement without knowing at least roughly what the statement means and what its referents are. As for pulling a number out of thin air, it can happen that we have essentially zero knowledge on a question (ignorance prior). The point of my Peter Singer example is that this is extremely rare, however. Usually, if you even know what a proposition means, you've already got enough background knowledge to start seeing lumps in the distribution.<br /><br />@Thameron: I know of a few people for whom that would be a significant improvement! ;)<br /><br />@Baron: What do you mean by an algorithmic scale here?<br /><br />@AlexSL: You are probably right - my trouble is that when I try to pin down exactly where odds are inapplicable (beyond the caveats above), I fail to do so convincingly. However, take all of this with a grain of salt. It is my new hammer, and boy does everything look like a nail.ianpollockhttps://www.blogger.com/profile/15579140807988796286noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-30879157645633979372011-08-19T00:53:50.000-04:002011-08-19T00:53:50.000-04:00Well, sometimes 1 + 1 = 10.
You know, there are 1...Well, sometimes 1 + 1 = 10.<br /><br />You know, there are 10 types of people in the world: those who know binary and those who don't. :-)<br /><br />Anyway, reinforce one thing Hector said above. I'm Brazilian, and while I do know, intellectually, what odds are, it is never very intuitive. I got better with time, though.J. Marcelo Alveshttps://www.blogger.com/profile/09967299561849915314noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-23800132143926698402011-08-18T20:34:28.725-04:002011-08-18T20:34:28.725-04:00Ian,
Betting odds directly measure the ratio at w...Ian,<br /><br />Betting odds directly measure the ratio at which money changes hands in a bet and, at best, only indirectly measure degrees of belief. Many factors associated with betting odds restrict their utility in indirectly measuring uncertainty. <br /><br />Epistemic agents may or may not have moral or religious qualms about betting. Epistemic agents are likely to possess differing attitudes toward risk. Given diminishing and increasing marginal utility of money, wealthier agents are apt to have a low aversion to risk and less wealthy agents are apt to have a high aversion to risk, resulting in the tendency of both groups to accept odds which overvalue and undervalue their credences in the events or propositions under consideration, respectively. <br /><br />Considerations of risk are inextricably tied to matters of expected utility which lead to difficulties like the St. Petersburg paradox. In light of such difficulties, I would advise to drop attempts to cast talk of measuring uncertainty in terms of betting odds entirely.Cian Eamon Marleyhttps://www.blogger.com/profile/09070168038290681070noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-65829285643934102412011-08-18T20:08:43.158-04:002011-08-18T20:08:43.158-04:00I submit this snark I recently read on the blog of...I submit this snark I recently read on the blog of an investment adviser by the name of John Hussman:<br /><br /><i>A Bayesian is someone who, vaguely expecting a horse, and glimpsing the tail of a donkey, concludes he has probably seen a mule.</i><br /><br />More seriously, I deeply distrust any argumentation that essentially goes "we should always use this approach, and base all our reasoning on it". That never works out, not for Objectivists, not for Positivists, not for Solipsists, and thus I conclude Bayesianally that it is most probable to not work out for the idea of attaching odds to everything either. (What are the odds that it is useful to attach odds to everything?)Alex SLhttps://www.blogger.com/profile/00801894164903608204noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-28050396872377256542011-08-18T19:56:39.875-04:002011-08-18T19:56:39.875-04:00"...of European ancestry, which ceteris parib..."...of European ancestry, which ceteris paribus makes various other world cuisines (e.g., Mexican, Finnish) somewhat less likely than not."<br />My background knowledge says that Finland is part of Europe.<br />However, as an example it otherwise holds true -- they do not usually consume British- or French- type breakfasts.Margaret K. Westfallhttps://www.blogger.com/profile/15920706327571834856noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-61665631985905007852011-08-18T19:01:44.549-04:002011-08-18T19:01:44.549-04:00Jesus,
Tough question. If one rejects the law of ...Jesus,<br /><br />Tough question. If one rejects the law of excluded middle when it comes to mathematical propositions, I suspect one cannot provide a cogent answer. <br /><br />However, if the question were rephrased a bit<br />('What exactly are the odds that a constructive proof for Goldbach's conjecture will be given within a specified time period?'), then perhaps we can say something like 20:1 against. I am willing to bet $20 on a return of $1 that no proof will be on offer for Goldbach's conjecture within the next five years.Cian Eamon Marleyhttps://www.blogger.com/profile/09070168038290681070noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-36572839592761235472011-08-18T18:22:38.222-04:002011-08-18T18:22:38.222-04:00What are exactly your odds that the Goldbach conje...What are exactly your odds that the Goldbach conjecture is true? (I admit up to four decimals).Jesús P. Zamora Bonillahttps://www.blogger.com/profile/07054631110263426886noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-83047424834298672292011-08-18T16:21:24.880-04:002011-08-18T16:21:24.880-04:00If, as I believe, we do most of our thinking on th...If, as I believe, we do most of our thinking on the less than fully conscious level, and do so by use of our own biological forms of predictive logic, we can't help but make predictions on an algorithmic scale of possible to probable rather than on an exponential scale of odds that go from simply possible to more and much more possible. We might make accurate yes or no decisions on a long term basis using odds, but for the short term, we need a better way to decide when possible becomes actionably probable.Baron Phttps://www.blogger.com/profile/04138430918331887648noreply@blogger.com