RE: About Probability
I would like to enter the arena. I see the original question as two questions, one about probability in a general sense, and the second about probability as used within Bayes Theorem. This is in line with the historical arguments. Most statisticians (from Fisher down to the present) recognize an objective probabilility and a subjective probability. In technical terms it is the ontologicaland epistemic viewpoints. The following probability classes have aspects of both ontological and epistemic viewpoints, which muddies the whole thing. There really is nothing very clear in any "definition" of probability. A frequentist has belief in that the replications "are random, identical draws, events, rolls, etc.". This is axiomatic to a frequentist definition of probability. It is an epistemic viewpoint. Von Mises clearlysawthat any frequentist definition of probability is circular. So we are back to the starting line. Lets look at a set of definitions: 1. The outcome of an independent replicate from a defined population. This is the basis of the frequentist definition, where probability comes from a large number of replicates and probability is the ratio of the number of desired events to the total number of replicates. The following are mathematical, and are highly ontological. 2. The greatest upper bound on a set of significance levels. 3. The degree that an observation (or experiment) supports a hypothesis. 4. A confidence, tolerance or prediction interval. 5. The weights (to and from) the hidden layers in a neural network. 6. The outcome from a mathematical expression/equation. The following are subjective and highly epistemic: 7. A degree of belief based on past experience. 8. The degree of membership in a fuzzy set. 9. A measure of truth (in the religious/judicial/legal sense). 10. A measure of what will happen based on a religious belief in cause and effect. The only criteria is that the outcome of any of the above from definitions 1 to 10, is that it can be represented by a real number from zero to one, where both zero and one represent states that (outside of mathematics) can never be reached. Definitions 2-4 have epistemic aspects in our belief in what is an appropriate distribution or test. Some of the experts further take the position that probability only can beinterpreted and used where "chance" is involved. Bayes entry in 1763 profoundly affected our viewpoints on probability, which even in 2000 has not settled down. In a mathematical sense, any mathematical expression that represents a probability (density function) can be used as a prior in the Bayesian approach. This varies from the non-informative to the informative. Most of the historical theoretical work on understating the Bayesian conceptwas based on the binomial distribution. To a Bayesian (Martz and many others), the cornerstone in Bayesian inference is subjective probability, to which a "degree of belief" can be attached. Therefore there is a lack of consistency among different investigators in the outcome or posterior conclusions. In addition decision theory can be applied to a Bayesian analysis, which gives a different viewpoint on making conclusions. In Bayesian statistics, anything goes as long as the fundamentals are followed: Posterior Distribution = (Prior Distribution) * (Likelihood function) / (Marginal distribution) The subjective aspect allows one to have great freedom in defining what probability is. If you can construct a prior on your belief in the actions of God, you can use it. DAH
Re: About Probability
In article [EMAIL PROTECTED], Alan Mclean [EMAIL PROTECTED] wrote: Herman Rubin wrote: In article [EMAIL PROTECTED], Alan McLean [EMAIL PROTECTED] wrote: I am sure there is a multitude of possible answers to this one. One way I would answer it is to say that probability is only applicable to *observable* events - that is, the occurrence of something which is in some way directly measurable. The existence of God is not observable in this sense, so probability is irrelevant to any discussion about the existence of God. This would exclude the application of probability to such things as nuclear physics. While we have to use observations to draw inferences, the probabilities of interest are not those about the observations, but about the underlying process. Not really. The probability models that constitute nuclear physics provide probabilities for events which can be measured - and thus provide a test of the models. (I know this is oversimplified.) The probability models do provide these, and this is how they are verified and the parameters estimated. But it is very often the case that these probabilities come from probabilities of events which cannot be observed, using a fairly complicated model. The basic theory does not deal directly with probabilities of the observations, but leads to this. In fact, in many cases, the events are not even observed, but there are enough to use the law of large numbers directly. While modern theory deals with the behavior of gas particles, these are translated into pressure and temperature, and the probabilistic models are tested by this macroscopic data. Another, related way to express this is to say that belief in the existence of God is a *model* for the universe. Within that model probability questions can be asked, but one cannot talk meaningfully of the existence of the model. (The same comment applies, for example, about general relativity as a theory which models the universe.) However, we use probability methods (actually statistical) to draw inferences not just within a model, but between models. One mistake, however, is to treat composite hypotheses or models as simple, as is the case here. A statistical model is a probability model. I actually goofed a bit here - I meant to say "...but one cannot talk meaningfully of **the probability of** the existence of the model." And whether the model is 'composite' or 'simple' does not change this. It makes a difference, and this is much misunderstood. If one has simple hypotheses, one can proceed with only the posterior probabilities of the hypotheses. On the other hand, for composite hypotheses, it is necessary to use also the distribution of the simple hypotheses within them. It is very easy to give examples where ignoring this leads to quite substantial errors. The difference between probability and statistics is that in probability one starts with the distribution, while in statistics, the major problem is that the distribution is not assumed. Repeatability is certainly (oops! - with high probability) not a prerequisite for probability to make sense. This is very definitely the case. Good! -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: About Probability
On 19 Sep 2000, Herman Rubin wrote: This would exclude the application of probability to such things as nuclear physics. While we have to use observations to draw inferences, the probabilities of interest are not those about the observations, but about the underlying process. No, the probabilities of interest are both those about the observations and those associated with the underlying process. One could take the view that predictions of probabilities of observable events are a means to an end: the continual development of better and better models of the underlying process. However, Hume's paradox guarantees that we will never be able to certify a given model as the "final one". Alternatively, one could take the view that the continual improvement of models is a means to an end: improved calculations of probabilities of observed events. Science can and has delivered such calculations which have proven tremendously useful, in the form of applications, even to non-scientists. A less partisan view is to respect the mutual feedback and interplay of improved models of underlying processes and improved calculations of observable events. In this view both types of probabilities are "of interest". = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: About Probability
In article [EMAIL PROTECTED], Christopher Tong [EMAIL PROTECTED] wrote: On 19 Sep 2000, Herman Rubin wrote: This would exclude the application of probability to such things as nuclear physics. While we have to use observations to draw inferences, the probabilities of interest are not those about the observations, but about the underlying process. No, the probabilities of interest are both those about the observations and those associated with the underlying process. I would disagree with this in most cases. The probabilities of the observations are not likely to give insight, at least in many cases. The process is what one makes assumptions about. One could take the view that predictions of probabilities of observable events are a means to an end: the continual development of better and better models of the underlying process. However, Hume's paradox guarantees that we will never be able to certify a given model as the "final one". I agree that we will never even have a "correct" model. That is one of the arguments against the use of p values. Alternatively, one could take the view that the continual improvement of models is a means to an end: improved calculations of probabilities of observed events. Science can and has delivered such calculations which have proven tremendously useful, in the form of applications, even to non-scientists. This is engineering, not science. Science is not much driven by it, and this is not adequately realized. A less partisan view is to respect the mutual feedback and interplay of improved models of underlying processes and improved calculations of observable events. In this view both types of probabilities are "of interest". But the probabilities of interest in applications are not likely to be those in the experiments. Those probabilities are mainly tools, not items desired per se. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: About Probability
On 19 Sep 2000, Herman Rubin wrote: No, the probabilities of interest are both those about the observations and those associated with the underlying process. I would disagree with this in most cases. The probabilities of the observations are not likely to give insight, at least in many cases. The process is what one makes assumptions about. Agreed. But there are also many cases where the probabilities of the observations do give insight. One cannot understand the double slit experiment in quantum physics without discussing the distribution of dots on the fluorescent screen which is the final observable record of the experiment. To understand the double slit experiment, which is the key to understanding the crucial concept of wave-particle duality, one requires discussion of both the underlying process (interference effects in the electron wavefunction, which is a "probability amplitude") as well as its observational manifestation (fringes in the distribution of the dots on the screen). To simply write down the interference term in the wavefunction gives no physical insight unless the observational consequences are linked to it. There are many other examples but I think the point has been made. Alternatively, one could take the view that the continual improvement of models is a means to an end: improved calculations of probabilities of observed events. Science can and has delivered such calculations which have proven tremendously useful, in the form of applications, even to non-scientists. This is engineering, not science. Science is not much driven by it, and this is not adequately realized. I stand corrected and will rephrase. In engineering, the probabilities "of interest" include calculations for observable events. But the probabilities of interest in applications are not likely to be those in the experiments. Those probabilities are mainly tools, not items desired per se. In an application, what is desired is a product with some target final observational state. Whoever designed the product will use the theoretical probability model of the underlying process as a means to accomplish the end. Every time the product is operated, an "experiment" is performed and the user wants the result of the experiment to match the prediction promised by the manufacturer. The user may be completely unaware of the underlying operating principles of the product. = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: About Probability
In article [EMAIL PROTECTED], Valar [EMAIL PROTECTED] wrote: Hello to everyone! I has a question for you that comes from a discussion that I had with a friend of mine. Due to the fact that with the Bayes Probability definition we can define a probability even for events that doesn't occur necessarily several times he say that is possible to associate a probability to the existence of God. But I think that in this case probability has non sense because I think that the Bayes definition is usable only with events that are a priori reapeatable (even if they occurred only one time or they never occurred) or that are composed by some sub-events reapeatable Attempts to reduce probability to anything else just lead to the type of confusion you seem to be having. About the only way to look at probability in the universe is to assume it exists, and satisfies the properties posited. For example he said me that we can associate a probability that with certain coditions it will rain, and we can do that even if these conditions occurr one time in our life, but I say that there is an important difference because the calculus of thi probability is made with physical consideration about several sub-events each with a probability that came from experience and physical models (that are based on experience too) What is the right opinion? When looked at carefully, one never gets models from experience; the models are created in the mind, and selected from experience. The appearance may be otherwise, when a simple mental model fits the observations quite well; physical scientists were lucky in that this happened. When it gets a little harder, this does not work very well, and even for simple situations, not good enough or too good data will make things difficult. With probability, it is essentially impossible to get good results empirically. Only modeling will provide understanding, and confirmation of the assumptions. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: About Probability
I am sure there is a multitude of possible answers to this one. One way I would answer it is to say that probability is only applicable to *observable* events - that is, the occurrence of something which is in some way directly measurable. The existence of God is not observable in this sense, so probability is irrelevant to any discussion about the existence of God. Another, related way to express this is to say that belief in the existence of God is a *model* for the universe. Within that model probability questions can be asked, but one cannot talk meaningfully of the existence of the model. (The same comment applies, for example, about general relativity as a theory which models the universe.) Repeatability is certainly (oops! - with high probability) not a prerequisite for probability to make sense. Have fun. Alan Valar wrote: Hello to everyone! I has a question for you that comes from a discussion that I had with a friend of mine. Due to the fact that with the Bayes Probability definition we can define a probability even for events that doesn't occur necessarily several times he say that is possible to associate a probability to the existence of God. But I think that in this case probability has non sense because I think that the Bayes definition is usable only with events that are a priori reapeatable (even if they occurred only one time or they never occurred) or that are composed by some sub-events reapeatable For example he said me that we can associate a probability that with certain coditions it will rain, and we can do that even if these conditions occurr one time in our life, but I say that there is an important difference because the calculus of thi probability is made with physical consideration about several sub-events each with a probability that came from experience and physical models (that are based on experience too) What is the right opinion? Thanx for the attention See you Valar PS I'm sorry for my english that isn't very good -- Posted from mailsrv.sa.infn.it [193.205.70.3] via Mailgate.ORG Server - http://www.Mailgate.ORG = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ = -- Alan McLean ([EMAIL PROTECTED]) Department of Econometrics and Business Statistics Monash University, Caulfield Campus, Melbourne Tel: +61 03 9903 2102Fax: +61 03 9903 2007 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: About Probability
In article [EMAIL PROTECTED], Alan McLean [EMAIL PROTECTED] wrote: I am sure there is a multitude of possible answers to this one. One way I would answer it is to say that probability is only applicable to *observable* events - that is, the occurrence of something which is in some way directly measurable. The existence of God is not observable in this sense, so probability is irrelevant to any discussion about the existence of God. This would exclude the application of probability to such things as nuclear physics. While we have to use observations to draw inferences, the probabilities of interest are not those about the observations, but about the underlying process. Another, related way to express this is to say that belief in the existence of God is a *model* for the universe. Within that model probability questions can be asked, but one cannot talk meaningfully of the existence of the model. (The same comment applies, for example, about general relativity as a theory which models the universe.) However, we use probability methods (actually statistical) to draw inferences not just within a model, but between models. One mistake, however, is to treat composite hypotheses or models as simple, as is the case here. Repeatability is certainly (oops! - with high probability) not a prerequisite for probability to make sense. This is very definitely the case. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
