Re: probability definition
This is a multi-part message in MIME format. --FF841A0334127EDA335D19E4 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit I'm glad to hear that somebody has his eye on the ball. Unfortunately, a designation of a region like "western Puerto Rico" means so many different things to so many different people, that I disbelieve its utility. With the definition you quote, we should have a 100% chance of precipitation almost every day. --FF841A0334127EDA335D19E4 Content-Type: text/x-vcard; charset=us-ascii; name="rabeldin.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Richard A. Beldin Content-Disposition: attachment; filename="rabeldin.vcf" begin:vcard n:Beldin;Richard tel;home:787-255-2142 x-mozilla-html:TRUE url:netdial.caribe.net/~rabeldin/Home.html org:BELDIN Consulting Services version:2.1 email;internet:[EMAIL PROTECTED] title:Professional Statistician (retired) adr;quoted-printable:;;PO Box 716=0D=0A;Boquerón;PR;00622; fn:Richard A. Beldin end:vcard --FF841A0334127EDA335D19E4-- = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: probability definition
- Original Message - From: "Alex Yu" [EMAIL PROTECTED] To: "Shareef Siddeek" [EMAIL PROTECTED] Cc: [EMAIL PROTECTED] Sent: Thursday, March 01, 2001 5:01 PM Subject: Re: probability definition 1. Very Interesting. Would it be possible to get a copy of your paper on Probability. I gave a review of "The Meanings of 'Probability'" several years ago for our ASA chapter, and would like to redo it. For a quick walk through of various prob. theories, you may consult "The Cambridge Dictionary of Philosophy." pp.649-651. Basically, propensity theory is to deal with the problem that frequentist prob. cannot be applied to a single case. Propensity theory defines prob. as the disposition of a given kind of physical situation to yield an outcome of a given type. The following is extracted from one of my papers. It brielfy talks about the history of classical theory, Reichenbach's frequentism and Fisherian school: Fisherian hypothesis testing is based upon relative frequency in long run. Since a version of the frequentist view of probability was developed by positivists Reichenbach (1938) and von Mises (1964), the two schools of thoughts seem to share a common thread. 2. Von Mises (1957) quotes Johannes von Kries and goes on the address the use as "I shall assume therefore a definite probability of the death of Caius, Sempronius or Titus in the course of the next year"., as support of his concept of "probability in a collective". He does include the "single event" as part of his "collective". He then states, "The term 'probability' will be reserved for the limiting value of the relative frequency in a true collective which satisfies the condition of randomness." With respect to Caius, Sempronius and Titus" he was considering the collective of aged rulers of Rome. However, it is not necessarily true. Both Fisherian and positivist's frequency theory were proposed as an opposition to the classical Laplacean theory of probability. 3. My reading of Fisher was that he opposed the Laplacian view because it had no mathematical basis, Bayes however did, and was fully accepted. In the Laplacean perspective, probability is deductive, theoretical, and subjective. To be specific, this probability is subjectively deduced from theoretical principles and assumptions in the absence of objective verification with empirical data. Assume that every member of a set has equal probability to occur (the principle of indifference), probability is treated as a ratio between the desired event and all possible events. This probability, derived from the fairness assumption, is made before any events occur. Positivists such as Reichenbach and von Mises maintained that a very large number of empirical outcomes should be observed to form a reference class. Probability is the ratio between the frequency of desired outcome and the reference class. Indeed, the empirical probability hardly concurs with the theoretical probability. For example, when a dice is thrown, in theory the probability of the occurrence of number "one" should be 1/6. But even in a million simulations, the actual probability of the occurrence of "one" is not exactly one out of six times. It appears that positivist's frequency theory is more valid than the classical one. However, the usefulness of this actual, finite, relative frequency theory is limited for it is difficult to tell how large the reference class is considered large enough. 4. The idea of a "limiting condition" is based on the same understanding of differential calculus and infinite series. That is a limit is reached, not on the value of N, only that some value converges to a limit as N increases. This does not depend on any arbitrary large value of N 5. von Mises also based his position on the laws of probability, which can be defined as a "natural" outcome from the frequentist view, where limiting occurs as a converging value of a ratio. He differentiated between the "actual occurances" and as a "thought experiment". It was the latter that he was using. Fisher (1930) criticized that Laplace's theory is subjective and incompatible with the inductive nature of science. However, unlike the positivists' empirical based theory, Fisher's is a hypothetical infinite relative frequency theory. In the Fisherian school, various theoretical sampling distributions are constructed as references for comparing the observed. 6. My reading of the historical record is that this was the K. Pearson school that did this. Fisher stuck to the Uniform, Normal and Poisson. Since Fisher did not mention Reichenbach or von Mises, it is reasonable to believe that Fisher developed his frequency theory independently. 7. Fisher was aware of many who challenged his views, but chose not to respond, except
Re: probability definition
This is a multi-part message in MIME format. --3C6331B8C260BF767681A8B3 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit The "definition" via axioms provides a mathematical structure that we interpret either as "relative frequency" or as "degrees of belief". Indeed, I think that any phenomenon which satisfies the axioms can serve as an "interpretation". As they say, "If it walks like a duck, ...". As far as the probability of rain tomorrow, I always explained to my students that the language is so imprecise that the numerical value has only rhetorical utility. We need to know: 1) How much rain in cm. ? 2) In which locations? 3) During what time span? Does 70% probability mean that it rains in 70% of the locations or 70% of the time or what? Your instincts are correct. That example is severely flawed because we have not made the experiment clear. Continue to question the simple examples. You will learn from it. --3C6331B8C260BF767681A8B3 Content-Type: text/x-vcard; charset=us-ascii; name="rabeldin.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Richard A. Beldin Content-Disposition: attachment; filename="rabeldin.vcf" begin:vcard n:Beldin;Richard tel;home:787-255-2142 x-mozilla-html:TRUE url:netdial.caribe.net/~rabeldin/Home.html org:BELDIN Consulting Services version:2.1 email;internet:[EMAIL PROTECTED] title:Professional Statistician (retired) adr;quoted-printable:;;PO Box 716=0D=0A;Boquerón;PR;00622; fn:Richard A. Beldin end:vcard --3C6331B8C260BF767681A8B3-- = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: probability definition
In article [EMAIL PROTECTED], "Richard A. Beldin" [EMAIL PROTECTED] wrote: This is a multi-part message in MIME format. --3C6331B8C260BF767681A8B3 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit The "definition" via axioms provides a mathematical structure that we interpret either as "relative frequency" or as "degrees of belief". Indeed, I think that any phenomenon which satisfies the axioms can serve as an "interpretation". As they say, "If it walks like a duck, ...". As far as the probability of rain tomorrow, I always explained to my students that the language is so imprecise that the numerical value has only rhetorical utility. We need to know: 1) How much rain in cm. ? 2) In which locations? 3) During what time span? Does 70% probability mean that it rains in 70% of the locations or 70% of the time or what? Your instincts are correct. That example is severely flawed because we have not made the experiment clear. Continue to question the simple examples. You will learn from it. Although in this case, in the US, the definition is quite precise for the National Weather Service. From their Operations Manual, it is "the likelihood of occurrence...of a precipitation event at any given point in the forecast area. The time period to which the PoP applies must be clearly stated (or unambiguously inferred from the forecast wording) since, without this, a numerical PoP value is meaningless." Elsewhere in the Ops Manual, a "precipitation event" is the occurrence of at least 0.01" of liquid equivalent precipitation (i.e., rain, melted snow). It has been shown that, given this definition, the PoP is equal to the expected areal coverage of the precipitation. Whether other groups issuing PoPs (media, other countries' weather services) follow the same definition, I don't know.At the locations where verification data are available, NWS PoP forecasts out through 48 hours are remarkably reliable. That is, if the PoP is N%, precipitation occurs very close to N% of the time. Harold -- [EMAIL PROTECTED] http://www.nssl.noaa.gov/~brooks/ Standard disclaimer Head, Mesoscale Applications Group, National Severe Storms Laboratory Norman, Oklahoma = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: probability definition
For a quick walk through of various prob. theories, you may consult "The Cambridge Dictionary of Philosophy." pp.649-651. Basically, propensity theory is to deal with the problem that frequentist prob. cannot be applied to a single case. Propensity theory defines prob. as the disposition of a given kind of physical situation to yield an outcome of a given type. The following is extracted from one of my papers. It brielfy talks about the history of classical theory, Reichenbach's frequentism and Fisherian school: Fisherian hypothesis testing is based upon relative frequency in long run. Since a version of the frequentist view of probability was developed by positivists Reichenbach (1938) and von Mises (1964), the two schools of thoughts seem to share a common thread. However, it is not necessarily true. Both Fisherian and positivist's frequency theory were proposed as an opposition to the classical Laplacean theory of probability. In the Laplacean perspective, probability is deductive, theoretical, and subjective. To be specific, this probability is subjectively deduced from theoretical principles and assumptions in the absence of objective verification with empirical data. Assume that every member of a set has equal probability to occur (the principle of indifference), probability is treated as a ratio between the desired event and all possible events. This probability, derived from the fairness assumption, is made before any events occur. Positivists such as Reichenbach and von Mises maintained that a very large number of empirical outcomes should be observed to form a reference class. Probability is the ratio between the frequency of desired outcome and the reference class. Indeed, the empirical probability hardly concurs with the theoretical probability. For example, when a dice is thrown, in theory the probability of the occurrence of number "one" should be 1/6. But even in a million simulations, the actual probability of the occurrence of "one" is not exactly one out of six times. It appears that positivist's frequency theory is more valid than the classical one. However, the usefulness of this actual, finite, relative frequency theory is limited for it is difficult to tell how large the reference class is considered large enough. Fisher (1930) criticized that Laplace's theory is subjective and incompatible with the inductive nature of science. However, unlike the positivists' empirical based theory, Fisher's is a hypothetical infinite relative frequency theory. In the Fisherian school, various theoretical sampling distributions are constructed as references for comparing the observed. Since Fisher did not mention Reichenbach or von Mises, it is reasonable to believe that Fisher developed his frequency theory independently. Backed by a thorough historical research, Hacking (1990) asserted that "to identify frequency theories with the rise of positivism (and thereby badmouth frequencies, since "positivism" has become distasteful) is to forget why frequentism arose when it did, namely when there are a lot of known frequencies." (p.452) In a similar vein, Jones (1999) maintained that "while a positivist may have to be a frequentist, a frequentist does not have to be a positivist." Chong-ho (Alex) Yu, Ph.D., MCSE, CNE Academic Research Professional/Manager Educational Data Communication, Assessment, Research and Evaluation Farmer 418 Arizona State University Tempe AZ 85287-0611 Email: [EMAIL PROTECTED] URL:http://seamonkey.ed.asu.edu/~alex/ = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
probability definition
Hello, I have a question regarding the definition of probability. If I understand correctly, probability may be defined using just axioms. However, my textbook also uses a relative frequency definition, in which a probability is defined as being the proportion of times an outcome occurs in repeated trials of an experiment. This makes sense when one flip of the coin is one trial, and in repeated trials, the proportion of heads is 1/2. But what about a situation (an ex. in my textbook) where the probability of rain tomorrow is 0.70. How do you define this experiment? Perhaps you measure rainfall, temperature, pressure, etc. for each day over a long time period. Then the probability of rain tomorrow is the proportion of times that rain occurred on days with similar values for temp., humidity, etc.? This seems a bit awkard to me. Also, how many trials must one perform an experiment, before you know that the proportion converges to a particular fraction? Any help on interpretation of relative frequency probabilities would be greatly appreciated. In many cases, it seems difficult, at least for textbook examples, to define what the actual experiment is. ___ Send a cool gift with your E-Card http://www.bluemountain.com/giftcenter/ = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: probability definition
Hi Alex, Can you provide the definition of probability under each way? In other words, can you explain a little more on each way of defining probability? As it is, some of them are clear (e.g., Frequentist theory) and others are not clear to me. Thanks. Siddeek Alex Yu wrote: Probability can be defined in at least five different ways: 1. Classical Laplacean theory of probability: The prob.is derived from the fairness assumption e.g. a fair coin. It is also called equiproability. 2. Frequentist theory: It is developed by von Mises and Reichenbach. Prob. is the relative frequency in the long run by limiting observations. 3. Propensity: It is based upon the physical or the objective property of the events. 4. Logical: developed by Carnap. Prob. is defined like Y logically entails X. 5. Subjective or Bayesian: degree of belief There is no easy answer to your question. It depends on which point of view you chose. Chong-ho (Alex) Yu, Ph.D., MCSE, CNE Academic Research Professional/Manager Educational Data Communication, Assessment, Research and Evaluation Farmer 418 Arizona State University Tempe AZ 85287-0611 Email: [EMAIL PROTECTED] URL:http://seamonkey.ed.asu.edu/~alex/ = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ = = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: probability definition
In article [EMAIL PROTECTED], James Ankeny [EMAIL PROTECTED] wrote: Hello, I have a question regarding the definition of probability. If I understand correctly, probability may be defined using just axioms. However, my textbook also uses a relative frequency definition, in which a probability is defined as being the proportion of times an outcome occurs in repeated trials of an experiment. This makes sense when one flip of the coin is one trial, and in repeated trials, the proportion of heads is 1/2. But what about a situation (an ex. in my textbook) where the probability of rain tomorrow is 0.70. How do you define this experiment? Perhaps you measure rainfall, temperature, pressure, etc. for each day over a long time period. Then the probability of rain tomorrow is the proportion of times that rain occurred on days with similar values for temp., humidity, etc.? This seems a bit awkard to me. Also, how many trials must one perform an experiment, before you know that the proportion converges to a particular fraction? Any help on interpretation of relative frequency probabilities would be greatly appreciated. In many cases, it seems difficult, at least for textbook examples, to define what the actual experiment is. I think it is dangerous, and even useless, to ATTEMPT to define probability. In physics, one no longer even tries to define length or mass, just specify their properties. It is the same with probability. A quantum mechanical model has a joint probability distribution for observations, but is worse between them. Just as we use the postulated properties for length and mass, we should use those for probabilities. We do have the nasty problem that there is no way we can accurately calculate probabilities, unless very strong additional assumptions are made. -- 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/ =
