Talk:Frequency probability

From Wikipedia, the free encyclopedia

Archives

Archive 1

[edit] Classical statistics redirect

It is my understanding that classical statistics encompasses more than just the frequency probability interpretation, but also focuses on parametric statistics and statistical hypothesis testing. Also, while the Bayesian perspective on probability is an alternative to the frequentist perspective, I think that there are more "alternatives" to classical statistics than just the Bayesian perspective or paradigm alone. For example, nonparametric statistics is decisively non-classical, but it is not necessarily non-frequentist.

On these grounds I think classical statistics deserves its own page. Cazort 19:06, 3 December 2007 (UTC)

Statisticians that are frequentists doesn't have to agree when it comes to statistical practice, and often they don't. So yes, I agree that it's not entirely correct to link classical statistic to this page. I think the main reason why we have this incorrect redirect is simply that no one has started a page under that name yet, so please go ahead! :-) iNic (talk) 19:04, 27 May 2008 (UTC)

[edit] Life on Mars

I dispute this "Life on Mars" example... It is not because this thought experiment is traditionally used by the subjectivists to explain the way they think, that they mean this to be a case where you would not be able to solve the problem thinking in other paradigms. A good argument for subjectivism is not necessarily a good counter-argument for frequencialism. This smeels to me as naïve maniqueism.

I believe there is a lot of urban legend and misunderstood statements around the different interpretations of probability. I would like to see a quote from a sound book about probabilities, written by an author of either of the said-to-be-conflicting sides, where he actually says that he believes frequentism cannot conceive to attribute a probability to the existence of life in Mars 1000000 years ago. I am afraid that what is written here is just an echo of uncertain discussions at university cafeterias!...

This statement is controversial only because life on Mars is such a large thing, difficult to grasp. Now, imagine for example the [Alexander_Oparin|Oparin experiment]. If a frequentist can assign a probability to "Life in Oparin's bottle 50 years ago", why wouldn't he assign to "Life on Mars 50 millions of years ago"?

I dislike making such arrogant claims inside Wikipedia, but I believe the use of this Mars thing as being the epitome of the (said) difference between frequencialism and subjectivism is plain wrong. It lacks basis as much as my claims might... Please, let's find something more specific, something de Finetti might have said when contrasting his theories to others, and not when explaining his ideas by their own. Or else we are building and artificial dissent. I believe we are trying hard to find dichotomies to make things look simpler to understand, while the subject is in fact much more difficult. Mixing controversial subjects as life, geology and chaos only make things worse.

In other words: please, let's be careful not to pick up an example of how subjectivism works, and try to transform it in an example of why the paradigms might be different!... I feel someone was feeling bad about the lack of a good example that would contrast the two paradigms, and tried to build one from an example created with other intention. I don't buy it... Please tell me where this use of the “Life on Mars” problem as a way to contrast the two paradigms came from. I bet $0.87 that this did not came from any books by an author with a Wikipedia page, while I would bet $0,95 that such an author might have used the example simply to demonstrate how subjectivism works (and not contrasting). (sorry for the large and confuse posting...) -- NIC1138 (talk) 22:15, 8 March 2008 (UTC)

Yes I think it's silly too and I have tried to delete it several times before. However, there are some quite strong opinions among some authors here that we need to talk about martians on this page, otherwise this page gets too biased toward frequentism, they claim. Well this page IS about frequentism, isn't it? So please go ahead and remove it and we will see what happens. I would also like to know the source for this example about martians. I have never seen it in a book. It is for sure not mentioned anywhere in the books that are listed as references here. iNic (talk) 19:18, 27 May 2008 (UTC)

[edit] What kind of experiments?

Spurred by a remark at Talk:Probability#Interpretations that it is doubtful that frequentists only deal with experiments – any repeatable random process will do, even a natural one – I replaced "experiment" by "trial", which I think is standard terminology for an independent observation of an outcome of a repeatable random process. However, this change was reverted (also at Probability interpretations) with edit summary: Experiment in the ordinary sense of the word is exactly what is intended here. I'm not sure what the "ordinary sense" of the word is that is intended here. Is it that of our Wikipedia article Experiment linked to? It defines the notion as follows: "In the scientific method, an experiment ... is a set of observations performed in the context of solving a particular problem or question, to retain or falsify a hypothesis or research concerning phenomena." But someone repeatedly flipping a coin or throwing a die and observing the outcomes does in general not do that to retain or falsify a hypothesis. Determining what fraction of children dies before the age of one year by inspecting statistics will also usually not be considered to be an experiment in the sense of the scientific method. The examples of use of the word given at dictionary.com – a chemical experiment; a teaching experiment; an experiment in living; a product that is the result of long experiment; to experiment with a new procedure[1] – also do not fit well with what I think is intended here.  --Lambiam 13:28, 22 May 2008 (UTC)

The requirement that we need to define an experiment to be able to talk about probabilies at all is very central for the so called frequency interpretation of probability. It is more central for the frequency interpretation than frequencies themselves (which in fact are not required). An experiment in this context doesn't really differ from random what we mean with a scientific experiment. An experiment is a set of complete instructions on how to perform some series of actions that will give the same result whoever performs them. In some cases a particular experiment always gives the same result, but in other cases different outcomes are possible. In the latter case, when the different outcomes doesn't depend on different initial conditions, we talk about random experiments. These random experiments can be understood and treated with the use of the probability concept. Not many disagree here. However, the Frequentists also reverse this implication by claiming that the probability concept can only be used in the context of random experiments.

A "trial" is just one actually performed random experiment and should not be confused with the experiment itself. Think of an experiment as the instructions for making a cake, and a trial is the actual cake you do when following the instructions, i.e., when performing the cake experiment. It is easily seen here that if you mix these concepts you might end up trying to eat the instructions, or read the cake.

Other central concepts in the frequency interpretation are design of experiments and hypothesis testing. It would, in fact, make more sense to call the frequency interpretation the experimental interpretation instead.

The Wikipedia article on what an experiment is is almost accurate, but not quite. An experiment is not a set of observations but rather a set of instructions that may or may not end up in some specific observations. If merely a set of claimed observations would count as a scientific experiment, well then observed UFO's with aliens onboard would be science, but it is not. The strict scientific approach is very central here. When determining what fraction of childern dies before the age of one year it is very important to design this experiment in the proper scientific way, for example by picking the children in a truly unbiased (randomized) way so that the result of the experiment will not be misleading. It is also very important to explicitly formulate a hypothesis before performing any experiment. To formulate a valid hypothesis after collecting the data is always possible and is therefore a flagrant example of violating the scientific method. So your comment about this example above is simply not true. Statisticians are commonly recognized as being scientists, in particular classical statisticians endorsing the so called frequency interpretation of probability. iNic (talk) 00:47, 23 May 2008 (UTC)

Let's take a concrete example. I'll be visiting Avignon, France in August, and want to know if I should pack an umbrella. If the chance of rain exceeds 25%, I'll pack one; otherwise, I'll risk it. So could I count what fraction of the same period in the years 1958 through 2007 were recorded for Avignon, France as having days with non-zero precipitation? What, in this case, is the experiment, and what is the hypothesis to be explicitly formulated before performing the experiment?  --Lambiam 15:09, 24 May 2008 (UTC)

Only pack your umbrella if you happen to have a very small one, otherwise I advice you to buy an umbrella in a store in France if you need to. Have a nice trip! Cheers iNic (talk) 21:36, 26 May 2008 (UTC)

Thanks for the advice. What about answering the questions?  --Lambiam 00:13, 27 May 2008 (UTC)

I did answer your questions above. If you use the links you can read more about the subjects you are interested in. However, as is the case with all science, if you really want to learn the craft you have to study the subject for some years at the university. In the process you will discover that science isn't religion, and can never be. If you want to find guidance for every step in life, like if you should bring an umbrella or not for a trip to France, in science you are looking in the wrong direction. Science can never give this kind of guidance to people in everyday situations. iNic (talk) 18:40, 27 May 2008 (UTC)

To me this is a condescending non-answer.  --Lambiam 19:18, 27 May 2008 (UTC)

OK it wasn't my intention to be perceived as condescending. I apologize in that case. I will try to explain what is wrong with your question about the umbrella. You indicate that metrological data for the city during the period 1958-2007 should give a good estimate for rain this year in that city. Sure, if you believe that go ahead and calculate this ratio and act accordingly. But is this frequentism? No. Is this science? No.

The problem is that I can claim that another ratio for rain is the correct one based on some other data (different time period, data from a larger area, or something completely different) and no one of us can really claim that one model is better than any other. And who knows, we might in fact measure exactly the same thing only that random effects make our result different.

In order to handle these issues and to be able to draw correct conclusions, different statistical tests have been developed. However, if all you know is one specific event (rain or not rain in a specific city this year) you will have way too little data (only one datum) to be able to test any hypotheses at all. This is why we need an experimental description so we can (at least in theory) reproduce the desired events as much as we desire. iNic (talk) 00:44, 28 May 2008 (UTC)

If you repeat "well-defined random experiments" you may also get different data. I want to estimate the probability of rain in Avignon during a specific week of August, and the best we can hope for is an approximation. If out of the 50 years 1958 through 2007 there was rain in Avignon in the period from 10 to 16 August in 14 of these years, it is eminently reasonable to use 14/50 = 28% as an estimate of the probability of rain in Avignon in the week from 10 to 16 August 2008. That is in no essential way different from estimating probabilities in "well-defined random experiments" by dividing the number of times an event occurred by the number of trials. Of course somebody can claim that one should instead use the meteorological data for Ouagadougou during March to give the "correct" result of 0%, but so what? Do you really believe that that claim is just as acceptable as the claim that 28% is a reasonable estimate?

If you think this is not "science", you agree with at least part of what I set out to change. But why is this not frequentist (other than your repeated statement that it isn't). Is there some frequentist manifesto that spells out the official frequentist creed?  --Lambiam 07:37, 28 May 2008 (UTC)

This is an old debate and you can read a lot about this controversy elsewhere. Your thoughts and arguments are clearly Bayesian in nature, not frequentist. The essential difference between them is that while Bayesians start with a question about a specific event and then gather information about that event in order to estimate a probability, frequentists go the opposite way by always starting with a class of events and a mathematical (probability) model for the class. This is an essential difference indeed. The frequentist manifesto you ask for is called the Kolmogorov axioms of probability. iNic (talk) 01:06, 1 June 2008 (UTC)

This is getting strange. There is nothing in the formulation of the Kolmogorov axioms that requires a notion of experiment, in the sense of the scientific method or otherwise. The essence of the pure frequentist view is that the probability of an uncertain event is by definition the frequency of that event based on previous observations, as the number of observations increases indefinitely. It appears that your interpretation of the frequentist interpretation as requiring to define an experiment to be able to talk about probabilities at all is more in your head than anywhere in the literature.  --Lambiam 18:07, 1 June 2008 (UTC)

Richard von Mises realized that we can't speak of probabilities unless some class of events is defined. He called this class the sample space, which is nothing more than all possible outcomes of an experiment. Kolmogorov, in turn, used von Mises concept of sample space as a necessary ingredient in his concept of a probability space. Probabilities as defined by Kolmogorov only exists on probability spaces. So in order to speak about probabilities we must have a probability space defined. And in order to have a probability space we must have a sample space, which are simply all possible outcomes of an ordinary experiment. As you can see this dependence on experiments is more fundamental than the Law of Large Numbers which is the canonical theorem within the theory connecting probabilities to frequencies. Very strange that you haven't seen anything of this in the literature you claim you have read. Can you please tell me which books you have read? iNic (talk) 23:42, 1 June 2008 (UTC)

For the Avignon case the sample space is a discrete two-element set, consisting of the well-defined mutually exclusive events "Precipitation was recorded in Avignon in the seven days from 10 to 16 August" and "No precipitation was recorded". The "experiment" of observing whether the precipitation event occurs is repeated each year, if one insists on calling this an experiment, but why should one do so? There is nothing in the concept of a sample space per se that ties the events to a notion of experiment as it is commonly understood. As in any axiomatic approach, the ingredients are essentially uninterpreted; whether the Avignon observations can be modelled by axiomatic probability theory is a matter of belief that can perhaps be bolstered by the success of similar assumptions in similar situations, but does not depend on whether the space is formed by the possible outcomes of an experiment in the sense of the scientific method, performed to test a hypothesis or such, or by the possible outcomes of a trial of any other well-defined and repeatable random process. It is incumbent upon you to come with citations from reliable sources that support the statement whose inclusion in the article you are defending. You could start with the references listed in any of our Frequency probability, Probability space, and Probability axioms articles.  --Lambiam 05:42, 3 June 2008 (UTC)

I'm not sure what you mean by "experiment as it is commonly understood." Anyway, if you want to delete the word "experiment" from the world of classical statistics you have to burn quite a few books. As I told you already some days ago the notions of design of experiments as well as hypothesis testing are very central to this subject, and consequently treated in a lot of books in this subject. But maybe Fisher was wrong when he coined his discipline "design of experiments"? Maybe he should have called it something completely different? But what? Any suggestions? When it comes to your umbrella example I totally agree with you that that isn't an example of proper scientific reasoning at all, and hence doesn't deserve to be called an experiment. At least we agree here. iNic (talk) 23:43, 3 June 2008 (UTC)

I mean experiment as the term is defined by the article it is hyperlinked to: "In the scientific method, an experiment (Latin: ex- periri, "of (or from) trying") is a set of observations performed in the context of solving a particular problem or question, to retain or falsify a hypothesis or research concerning phenomena. The experiment is a cornerstone in the empirical approach to acquiring deeper knowledge about the physical world." Why the rhetoric? There are many contexts in which scientists are trying to retain or falsify a hypothesis, and then it makes sense for them to design an experiment, and Fisher's methodology may well be helpful to them. There are many other contexts in which people try to obtain information that does not fit this description, and then it is not helpful to suggest that they should apply the scientific method, formulate a hypothesis, and design and perform an experiment, to retain or falsify it. The fact that it may often be meaningful to count how often an experimental event occurs, does not mean that everyone who wants to count occurrences of any random event must necessarily turn it into a scientific experiment. Pointing out that this is not a requirement does, conversely, not imply that everything having to do with experiments is pointless and should be scrapped. I'm still waiting for the references, but think I shouldn't hold my breath.  --Lambiam 03:21, 4 June 2008 (UTC)