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12th July 2009, 06:24 PM | #1 |
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Probabilities
I have a question regarding prior vs posterior probabilities -- can it be said that all prior probabilities are in fact 0.5, since any knowledge at all could be argued to contribute to only posteriors?
For instance, if I roll a typical die, is it valid to say the prior probability of rolling a 6 is actually 0.5, whereas only the posterior is 1/6, since knowing the die has 6 possible values is "further" knowledge? |
12th July 2009, 07:28 PM | #2 |
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No. Both "prior" and "posterior" refer to your knowledge at a particular point in time. As time advances, your "posterior" probability from one observation will become the "prior" probability for the next one.
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12th July 2009, 07:48 PM | #3 |
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I've read statistics textbooks that are critical of Bayesian Statistics and I have difficulty with it myself. It seems to confuse probability, which concerns random events that will take place in the future, such as the roll of a pair of dice, with confidence, which concerns personal belief about an existing state of nature, such as the identity of a playing card that is laying face down on a table.
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12th July 2009, 07:56 PM | #4 |
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An uninformative prior is one that is uniform in a space where the likelihood is data translated. Or something like that. I haven't really touched Bayesian stats outside of computational stats course in a long time...
/many people love Bayesian stats until they deal with nonparametric Bayesian |
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12th July 2009, 08:30 PM | #5 |
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How do you get 0.5? I'd be okay if you said it was "unknown" because you didn't know anything about the event until after you saw the die you rolled. But you could just as easily look at the die right before rolling and figure out the odds. In that case, I don't any other answer besides 1:6 is valid. That is, until you find the die has been tampered with, but I think that's beyond the scope of your question.
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12th July 2009, 08:36 PM | #6 |
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Prior and posterior probability (difference)
(First match in Google with "prior vs. posterior probability" ) |
12th July 2009, 09:53 PM | #7 |
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12th July 2009, 10:18 PM | #9 |
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13th July 2009, 04:18 AM | #10 |
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We Bayesians are quite aware that what the frequentist means by probability and what we mean by probability are not the same thing, but that they coincide in a broad range of circumstances and obey essentially the same mathematics, which is why we choose to call our level of belief in something a probability.
I don't think it's fair to criticise Bayesian methods for their choice of wording rather than for their actual value in decision making, assessing data and so on. As for 'prior probabilities' always being 0.5 - no. That reads like a somewhat incomplete formulation of a statement that in the lack of better founded priors you should choose a uniform prior. There's certainly no reason to use 0.5, it's quite inconsistent with the other questions you might ask (will I roll a 4? will I roll a 17?) - I'd argue much more that if you have no idea what to expect then you should wait till your first results are in before using any prior at all (arguably a uniform prior with limits tending to infinity), and then use those to generate some very broad initial prior. And work onwards from there carefully, taking past posteriors as future priors. |
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13th July 2009, 05:08 AM | #11 |
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That's because that's not what "probability" means.
Random events that will take place in the future are properly described using "frequency." If you believe that probability and frequency are identical, then you're a frequentist, which is the catch-all term for most non-Bayesians. If you believe that terms like "likelihood" apply to one-off events (I figure there's at least a 25% chance that Lance Armstrong will win this year's Tour de France) then you're not, properly speaking, a frequentist but a Bayesian. |
13th July 2009, 05:19 AM | #12 |
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I think he is getting .5 from the idea of binary statistics, either it rolls a 6, or it doesn't. The roll of a die, however, does not meet the criteria for using binary statistics, so the usage is false.
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13th July 2009, 08:43 AM | #13 |
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Yeah, that is what I was thinking.
As edd pointed out a uniform prior would be more rational, because it would occur to any rational human that additional numbers might be just as likely, but that just begs my question -- wouldn't the thought "well, it might also end up 17, or 25, etc" count as further knowledge? At any rate drkitten explained it in terms that make sense to me -- that the distinction isn't really static, it is dynamic and just happens to be related to what you know and don't know at a given point in time. So technically I can't go back and "remove" knowledge already acquired, I.E. the fact that a normal die is six sided will forever be "baked" into any prior I can come up with from this point onward. Is that about right, drkitten? |
13th July 2009, 09:26 AM | #14 |
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Let me illustrate what I mean about the problem that arises when discussing "probability" in connection with the identity of a playing card laying face down on a table, or any other existing state of nature for that matter.
The probability that a standard die will land with six spots up when rolled randomly is 1/6. That is a property of dice and is not subjective. If I ask three guys what that probability is, they should agree that it's 1/6, and if so, they'll all be correct. If one of them says otherwise he'll be wrong. Probability is an absolute objective state of nature that can often, although not always, be approximated by experiment or determined by analysis. Probability itself is not subject to opinion or differing points of view, although degree of confidence in an assessment of probability is. Probability is not frequency, it is only the expected value of a frequency. If a die is rolled several times, it's unlikely that the frequency that a six appears will be exactly 1 out of 6. It's more likely be either higher or lower. Confidence is different. Confidence is a property of a person and is affected by the knowledge that person has and his ability to properly analyze that knowledge, so different people can have different levels of confidence about an existing state of nature, based on what they know and how they analyze it. Imagine that you're sitting around a card table with Andy, Bill, and Charlie, who, incidentally, aren't aware of any difference between confidence and probability. You remove the four aces from a deck of cards and set the rest of the deck aside. You stipulate that the aces of Hearts and Diamonds are "red cards" and the aces of Spades and Clubs are "black cards". You ask what the probability is that a randomly drawn card from that set will be red. Everyone agrees that the probability is 1/2. You then shuffle the cards thoroughly so that it's impossible for anyone present to know which card is which, and you place one card face down in the center of the table. You then deal one card each to Andy, Bill, and Charlie, but warn them not to look at their cards yet. You ask them what the probability is that the card in the center of the table is red. They all impatiently repeat that the probability is 1/2. They feel that they've already answered that question. You then instruct Andy and Charlie, but not Bill, to each secretly peek at the card he's been dealt. You give each guy a pencil and a piece of paper, and instructions to privately write down his name and the probability that the card in the center of the table is red, then fold up his paper and give it to you. You then compare the papers. As it turns out, all three are different. Andy says 1/3, Bill says 1/2, and Charlie says 2/3. You then announce that not all answers are the same, and you ask how this could be. An argument breaks out because each person feels absolutely justified in his assessment of a property of a certain playing card, yet they can't all be correct. They begin to grasp at awkward phrases like "probability for you" and probability for me", but since they're talking about a property of a certain playing card, this sounds as weak and irrational as if they were speaking of "your truth" and "my truth". The bottom line is that there is no probability that the card in the center of the table is red. It's either red or it isn't. At that point in the exercise, it would only have made sense to ask each person to write down his own level of confidence that the card is red, and in that case, what each person wrote would have been correct and there would have been no conflict. Confidence is a measure of personal certainty about a state of nature. Probability is a state of nature. |
13th July 2009, 11:24 AM | #15 |
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Why can't one take the same point of view before the card is chosen at random from the four and placed on the table? Either it will be red or it won't be; we just don't know which. Why is there such a big difference between not knowing because it hasn't yet been chosen and not knowing because we haven't yet looked at it? Either way, it's the same card, and either way, we don't know which card it is.
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13th July 2009, 11:35 AM | #16 |
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14th July 2009, 12:38 AM | #17 |
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Hmm, I'm not sure I like the word 'confidence' to express this quantity. I'd prefer 'belief' or something like it.
If I get to have a peek at a card that is red, my belief that the card is black drops to zero. If I said my confidence was zero, it sounds too much like I've dropped back to a completely uninformed state rather than being in a completely informed state. We definitely need a nice new word for Bayesian probability/belief. And then we can go round telling everyone about this word and what it means and thereby teach the entire world the joys of Bayesian statistics Bayesability perhaps? |
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14th July 2009, 07:17 AM | #18 |
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14th July 2009, 07:50 AM | #19 |
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Yeah confidence is already used in statistics but doesn't it usually mean something somewhat different from probability? Certainly in the way I use it the two terms are not interchangeable.
What I meant more was that I would prefer confidence was restricted to the way it is currently used in frequentist statistics already, and wasn't taken to be a synonym for Bayesian probability. |
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14th July 2009, 07:58 AM | #20 |
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14th July 2009, 08:03 AM | #21 |
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Glad to be of help.
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But what a horrid waste of effort to re-learn what you already know. The only reason I can imagine wanting to do that is if I suspected that the die was NOT really random. But in this case, I can approximate the same thing by using as a prior the fact that the die is uniformly random and then generating a new distribution for the (possibly loaded) die as a posterior, which gives me the same results for less work.... |
14th July 2009, 08:24 AM | #22 |
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It appears to me that you just invalidated just about every branch of science that exists. The existence of randomness is certainly, at the very least, an elegant scientific theory that does an excellent job of describing observations and making predictions.
But even if you're right and randomness doesn't really exist, there's still a difference between probability and confidence, just as there's a difference between a horse and a unicorn. The central point is that the two words have different meanings. |
14th July 2009, 09:07 AM | #23 |
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I did!!!
I certainly didn't intend to...
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Same goes for pseudo-random number generators on a computer.
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14th July 2009, 10:00 AM | #24 |
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In a nutshell, probability is an objective concept that applies to the outcome of an event that may take place in the future. Confidence is a subjective property of an observer concerning a state of nature.
I have a similar complaint about the phrase "he looks suspicious", which I think has a confusing double meaning. To me, a person looks suspicious when he has a doubting expression on his face, suggesting that he suspects something. A person does not "look suspicious" when he's sneaking around and looking at other people's valuables as if he intends to steal them. We need another word for that. When you spot someone sneaking around like that, he's not suspicious, you are. It's another case of applying an attribute to an external object that is really an attribute of yourself. |
14th July 2009, 10:23 AM | #25 |
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In Spanish we have "suspicaz" as opposed to "sospechoso".
Someone that is "suspicaz" thinks that someone else might be guilty of something, or distrusts that other person. Someone that is "sospechoso" is the person being thought of as possibly guilty of something, or the person not being trusted. You look up either suspicaz or sospechoso in a Spanish-English dictionary, and for both you get "suspicious". Doesn't make much sense indeed to use the same word to describe both of them, but so are languages I guess. |
14th July 2009, 06:24 PM | #26 |
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If it helps, the prior and posterior probabilities when Bayesian analysis is used for diagnostic tests are measured frequencies.
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14th July 2009, 06:43 PM | #27 |
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Does this statement reflect the problem that Bayesians have with Frequentists, and the problem referred to in the OP - is it too easy to confuse the number of outcomes with the frequency of those outcomes?
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14th July 2009, 09:30 PM | #28 |
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I'll pass on that one. I have no idea what you're asking me.
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But that doesn't mean we actually expect that to happen. Indeed, unless the number of rolls is divisible by six, it's impossible. We only recognize that as the number of trials increases, the ratio of the actual outcome to the expected outcome tends to approach unity, the difference between the actual outcome and the expected outcome tends to increase without limit, and the probability of the actual outcome equaling the expected outcome (i.e. a die landing with each number up an equal number of times) asymptotically approaches zero.
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15th July 2009, 04:45 AM | #29 |
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It wasn't addressed to you specifically, but rather to anyone that may have some insight. I don't understand why there is any supposed conflict between Frequentism and Bayesianism.
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15th July 2009, 05:31 AM | #30 |
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Frequentists are just jealous because Bayesians have bigger computers.
http://mh1823.com/frequentists_and_bayesians.htm
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15th July 2009, 05:40 AM | #31 |
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I don't understand how the two statements you have highlighted are different. They seem to be saying the same thing - that sample means are used to estimate population means, and that an interval can be formed which is likely to contain the mean that would be obtained if the entire population were measured.
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15th July 2009, 05:44 AM | #32 |
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Confidence is a little different in statistics than what Towlie is describing. It is not subjective. When asked of a specific incident, three statisticians are going to give you the same confidence interval for any given confidence level. It is not how confident you are that the card is red, whether you looked at it or not, it is based on probability.
For example, if there was a class room and 100 students were going to walk into the room, at any time, based on the student population, you can give a probability of whether the next person that walks in is male or female, and the probability would be different based on the confidence level. You could say, I am 25% confident that the next person is male. Or you could say I am 75% confident that out of the next 10 people 5 of them will be male. Or you could say I am 95% confident that of the 100 people that are going to walk into the room, 50 or them will be male. |
15th July 2009, 06:53 AM | #33 |
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15th July 2009, 07:42 AM | #34 |
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Not quite. The Frequentist point of view excludes the possibility of saying there is a 95% chance the population value lies within the 95% confidence interval. But that's generally what we want to be able to say. The Bayesian credible interval does allow such statements to be made.
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15th July 2009, 08:55 AM | #35 |
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15th July 2009, 02:48 PM | #36 |
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According to frequentists:
The population mean is unknown, but fixed, so one cannot make any probability statements about it. A sample drawn from the population is the result of a random process (i.e., sampling randomly), so one can make probability statements about such samples. The endpoints of a confidence interval are computed from a random sample. The confidence interval is therefore random too, so one can make probability statements about it too. Before sampling, one can say, "the probability is 95% that a future random sample will be such as to yield a confidence interval that surrounds the fixed population mean." After sampling, the sample and the confidence interval computed from it are known, and the population mean is still fixed (though still unknown), so, strictly speaking, one cannot make any probability statement at all about their relation. Either the mean does lie within the computed interval or it doesn't; we just don't know which is the case. |
15th July 2009, 04:07 PM | #37 |
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The prior thread I was thinking of spoke to a peripherally related point, so it won't be of use.
The Bayesian is taking essentially the same information that the Frequentist uses, but modifying it based on prior information. What does this really mean when there is no prior information? I realize that this is of some value when the prior information is of a type that is unusable to a Frequentist (e.g. a guesstimate as to low, intermediate or high probability of pulmonary embolism based on clinical factors), but what if we are talking about the situation described in the OP, where there isn't really any prior information? Linda |
16th July 2009, 02:16 AM | #38 |
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16th July 2009, 04:20 AM | #39 |
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16th July 2009, 06:06 AM | #40 |
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