Posts Tagged With: Bayesianism

R.A. Fisher: ‘Two New Properties of Mathematical Likelihood’

17 February 1890–29 July 1962

Today is R.A. Fisher’s birthday. I’ll post some different Fisherian items this week in honor of it. This paper comes just before the conflicts with Neyman and Pearson erupted.  Fisher links his tests and sufficiency, to the Neyman and Pearson lemma in terms of power.  It’s as if we may see them as ending up in a similar place while starting from different origins. I quote just the most relevant portions…the full article is linked below. Happy Birthday Fisher!

Two New Properties of Mathematical Likelihood

by R.A. Fisher, F.R.S.

Proceedings of the Royal Society, Series A, 144: 285-307 (1934)

  The property that where a sufficient statistic exists, the likelihood, apart from a factor independent of the parameter to be estimated, is a function only of the parameter and the sufficient statistic, explains the principle result obtained by Neyman and Pearson in discussing the efficacy of tests of significance.  Neyman and Pearson introduce the notion that any chosen test of a hypothesis H0 is more powerful than any other equivalent test, with regard to an alternative hypothesis H1, when it rejects H0 in a set of samples having an assigned aggregate frequency ε when H0 is true, and the greatest possible aggregate frequency when H1 is true.

If any group of samples can be found within the region of rejection whose probability of occurrence on the hypothesis H1 is less than that of any other group of samples outside the region, but is not less on the hypothesis H0, then the test can evidently be made more powerful by substituting the one group for the other.

Consequently, for the most powerful test possible the ratio of the probabilities of occurrence on the hypothesis H0 to that on the hypothesis H1 is less in all samples in the region of rejection than in any sample outside it. For samples involving continuous variation the region of rejection will be bounded by contours for which this ratio is constant. The regions of rejection will then be required in which the likelihood of H0 bears to the likelihood of H1, a ratio less than some fixed value defining the contour. (295)…

It is evident, at once, that such a system is only possible when the class of hypotheses considered involves only a single parameter θ, or, what come to the same thing, when all the parameters entering into the specification of the population are definite functions of one of their number.  In this case, the regions defined by the uniformly most powerful test of significance are those defined by the estimate of maximum likelihood, T.  For the test to be uniformly most powerful, moreover, these regions must be independent of θ showing that the statistic must be of the special type distinguished as sufficient.  Such sufficient statistics have been shown to contain all the information which the sample provides relevant to the value of the appropriate parameter θ . It is inevitable therefore that if such a statistic exists it should uniquely define the contours best suited to discriminate among hypotheses differing only in respect of this parameter; and it is surprising that Neyman and Pearson should lay it down as a preliminary consideration that ‘the tesitng of statistical hypotheses cannot be treated as a problem in estimation.’ When tests are considered only in relation to sets of hypotheses specified by one or more variable parameters, the efficacy of the tests can be treated directly as the problem of estimation of these parameters.  Regard for what has been established in that theory, apart from the light it throws on the results already obtained by their own interesting line of approach, should also aid in treating the difficulties inherent in cases in which no sufficient statistics exists. (296)

Categories: Fisher, phil/history of stat, Statistics | Tags: , , , | 2 Comments

R.A. Fisher: ‘Two New Properties of Mathematical Likelihood’: Just before breaking up (with N-P)

17 February 1890–29 July 1962

In recognition of R.A. Fisher’s birthday tomorrow, I will post several entries on him. I find this (1934) paper to be intriguing –immediately before the conflicts with Neyman and Pearson erupted. It represents essentially the last time he could take their work at face value, without the professional animosities that almost entirely caused, rather than being caused by, the apparent philosophical disagreements and name-calling everyone focuses on. Fisher links his tests and sufficiency, to the Neyman and Pearson lemma in terms of power.  It’s as if we may see them as ending up in a very similar place (no pun intended) while starting from different origins. I quote just the most relevant portions…the full article is linked below. I’d blogged it earlier here.  You may find some gems in it.

‘Two new Properties of Mathematical Likelihood’

by R.A. Fisher, F.R.S.

Proceedings of the Royal Society, Series A, 144: 285-307 (1934)

  The property that where a sufficient statistic exists, the likelihood, apart from a factor independent of the parameter to be estimated, is a function only of the parameter and the sufficient statistic, explains the principle result obtained by Neyman and Pearson in discussing the efficacy of tests of significance.  Neyman and Pearson introduce the notion that any chosen test of a hypothesis H0 is more powerful than any other equivalent test, with regard to an alternative hypothesis H1, when it rejects H0 in a set of samples having an assigned aggregate frequency ε when H0 is true, and the greatest possible aggregate frequency when H1 is true.

If any group of samples can be found within the region of rejection whose probability of occurrence on the hypothesis H1 is less than that of any other group of samples outside the region, but is not less on the hypothesis H0, then the test can evidently be made more powerful by substituting the one group for the other. Continue reading

Categories: Fisher, phil/history of stat, Statistics | Tags: , , , | 3 Comments

R.A. Fisher: ‘Two New Properties of Mathematical Likelihood’

17 February 1890–29 July 1962

Exactly 1 year ago: I find this to be an intriguing discussion–before some of the conflicts with N and P erupted.  Fisher links his tests and sufficiency, to the Neyman and Pearson lemma in terms of power.  It’s as if we may see them as ending up in a similar place while starting from different origins. I quote just the most relevant portions…the full article is linked below.

by R.A. Fisher, F.R.S.

Proceedings of the Royal Society, Series A, 144: 285-307 (1934)

  The property that where a sufficient statistic exists, the likelihood, apart from a factor independent of the parameter to be estimated, is a function only of the parameter and the sufficient statistic, explains the principle result obtained by Neyman and Pearson in discussing the efficacy of tests of significance.  Neyman and Pearson introduce the notion that any chosen test of a hypothesis H0 is more powerful than any other equivalent test, with regard to an alternative hypothesis H1, when it rejects H0 in a set of samples having an assigned aggregate frequency ε when H0 is true, and the greatest possible aggregate frequency when H1 is true.

If any group of samples can be found within the region of rejection whose probability of occurrence on the hypothesis H1 is less than that of any other group of samples outside the region, but is not less on the hypothesis H0, then the test can evidently be made more powerful by substituting the one group for the other. Continue reading

Categories: Fisher, phil/history of stat, Statistics | Tags: , , , | 1 Comment

R.A. Fisher: ‘Two New Properties of Mathematical Likelihood’

17 February 1890–29 July 1962

I find this to be an intriguing discussion–before some of the conflicts with N and P erupted.  Fisher links his tests and sufficiency, to the Neyman and Pearson lemma in terms of power.  It’s as if we may see them as ending up in a similar place while starting from different origins. I quote just the most relevant portions…the full article is linked below.

by R.A. Fisher, F.R.S.

Proceedings of the Royal Society, Series A, 144: 285-307 (1934)

  The property that where a sufficient statistic exists, the likelihood, apart from a factor independent of the parameter to be estimated, is a function only of the parameter and the sufficient statistic, explains the principle result obtained by Neyman and Pearson in discussing the efficacy of tests of significance.  Neyman and Pearson introduce the notion that any chosen test of a hypothesis H0 is more powerful than any other equivalent test, with regard to an alternative hypothesis H1, when it rejects H0 in a set of samples having an assigned aggregate frequency ε when H0 is true, and the greatest possible aggregate frequency when H1 is true.

If any group of samples can be found within the region of rejection whose probability of occurrence on the hypothesis H1 is less than that of any other group of samples outside the region, but is not less on the hypothesis H0, then the test can evidently be made more powerful by substituting the one group for the other.

Consequently, for the most powerful test possible the ratio of the probabilities of occurrence on the hypothesis H0 to that on the hypothesis H1 is less in all samples in the region of rejection than in any sample outside it. For samples involving continuous variation the region of rejection will be bounded by contours for which this ratio is constant. The regions of rejection will then be required in which the likelihood of H0 bears to the likelihood of H1, a ratio less than some fixed value defining the contour. (295)…

It is evident, at once, that such a system is only possible when the class of hypotheses considered involves only a single parameter θ, or, what come to the same thing, when all the parameters entering into the specification of the population are definite functions of one of their number.  In this case, the regions defined by the uniformly most powerful test of significance are those defined by the estimate of maximum likelihood, T.  For the test to be uniformly most powerful, moreover, these regions must be independent of θ showing that the statistic must be of the special type distinguished as sufficient.  Such sufficient statistics have been shown to contain all the information which the sample provides relevant to the value of the appropriate parameter θ . It is inevitable therefore that if such a statistic exists it should uniquely define the contours best suited to discriminate among hypotheses differing only in respect of this parameter; and it is surprising that Neyman and Pearson should lay it down as a preliminary consideration that ‘the tesitng of statistical hypotheses cannot be treated as a problem in estimation.’ When tests are considered only in relation to sets of hypotheses specified by one or more variable parameters, the efficacy of the tests can be treated directly as the problem of estimation of these parameters.  Regard for what has been established in that theory, apart from the light it throws on the results already obtained by their own interesting line of approach, should also aid in treating the difficulties inherent in cases in which no sufficient statistics exists. (296)

Categories: phil/history of stat, Statistics | Tags: , , , | Leave a comment

Bad news bears: ‘Bayesian bear’ rejoinder- reblog

To my dismay, I’ve been sent, once again, that silly, snarky, adolescent, clip of those naughty “what the p-value” bears (see Aug 5 post),, who cannot seem to get a proper understanding of significance tests into their little bear brains. So apparently some people haven’t  seen my rejoinder which, as I said then, practically wrote itself. So since it’s Saturday night here at the Elbar Room, let’s listen in to a reblog of my rejoinder (replacing p-value bears with hypothetical Bayesian bears)–but you can’t get it without first watching the Aug 5 post, since I’m mimicking them.  [My idea for the rejoinder was never polished up for actually making a clip.  In fact the original post had 16 comments where several reader improvements were suggested. Maybe someone will want to follow through*.] I just noticed a funny cartoon on Bayesian intervals on Normal Deviate’s post from Nov. 9.

This continues yesterday’s post: I checked out the the” xtranormal” http://www.xtranormal.com/ website. Turns out there are other figures aside from the bears that one may hire out, but they pronounce “Bayesian” as an unrecognizable, foreign-sounding word with around five syllables. Anyway, before taking the plunge, here is my first attempt, just off the top of my head. Please send corrections and additions.

Bear #1: Do you have the results of the study?

Bear #2:Yes. The good news is there is a .996 probability of a positive difference in the main comparison.

Bear #1: Great. So I can be well assured that there is just a .004 probability that such positive results would occur if they were merely due to chance.

Bear #2: Not really, that would be an incorrect interpretation. Continue reading

Categories: Comedy, Metablog, significance tests, Statistics | Tags: , , | 42 Comments

Bad news bears: Bayesian rejoinder

This continues yesterday’s post: I checked out the the” xtranormal” http://www.xtranormal.com/ website. Turns out there are other figures aside from the bears that one may hire out, but they pronounce “Bayesian” as an unrecognizable, foreign-sounding word with around five syllables. Anyway, before taking the plunge, here is my first attempt, just off the top of my head. Please send corrections and additions.

Bear #1: Do you have the results of the study?

Bear #2:Yes. The good news is there is a .996 probability of a positive difference in the main comparison.

Bear #1: Great. So I can be well assured that there is just a .004 probability that such positive results would occur if they were merely due to chance.

Bear #2: Not really, that would be an incorrect interpretation.

Bear #1: Oh. I see. Then you must mean 99.6% of the time a smaller difference would have been observed if in fact the null hypothesis of “no effect” was true.

Bear #2: No, that would also be an incorrect interpretation.

Bear #1: Well then you must be saying it is rational to believe to degree .996 that there is a real difference?

Bear #2: It depends. That might be so if the prior probability distribution was a proper probabilistic distribution representing rational beliefs in the different possible parameter values independent of the data.

Bear #1: But I was assured that this would be a nonsubjective Bayesian analysis.

Bear #2: Yes, the prior would at most have had the more important parameters elicited from experts in the field, the remainder being a product of one of the default or conjugate priors.

Bear #1: Well which one was used in this study? Continue reading

Categories: Statistics | Tags: , , | 20 Comments

Stephen Senn: Randomization, ratios and rationality: rescuing the randomized clinical trial from its critics

Stephen Senn
Head of the Methodology and Statistics Group,
Competence Center for Methodology and Statistics (CCMS), Luxembourg

An issue sometimes raised about randomized clinical trials is the problem of indefinitely many confounders. This, for example is what John Worrall has to say:

Even if there is only a small probability that an individual factor is unbalanced, given that there are indefinitely many possible confounding factors, then it would seem to follow that the probability that there is some factor on which the two groups are unbalanced (when remember randomly constructed) might for all anyone knows be high. (Worrall J. What evidence is evidence based medicine. Philosophy of Science 2002; 69: S316-S330: see page S324 )

It seems to me, however, that this overlooks four matters. The first is that it is not indefinitely many variables we are interested in but only one, albeit one we can’t measure perfectly. This variable can be called ‘outcome’. We wish to see to what extent the difference observed in outcome between groups is compatible with the idea that chance alone explains it. The indefinitely many covariates can help us predict outcome but they are only of interest to the extent that they do so. However, although we can’t measure the difference we would have seen in outcome between groups in the absence of treatment, we can measure how much it varies within groups (where the variation cannot be due to differences between treatments). Thus we can say a great deal about random variation to the extent that group membership is indeed random.

The second point is that in the absence of a treatment effect, where randomization has taken place, the statistical theory predicts probabilistically how the variation in outcome between groups relates to the variation within. Continue reading

Categories: Statistics | Tags: , , , , , , | 28 Comments

Does the Bayesian Diet Call For Error-Statistical Supplements?

Some of the recent comments to my May 20 post leads me to point us back to my earlier (April 15) post  on dynamic dutch books, and continue where Howson left off:

“And where does this conclusion leave the Bayesian theory? ….I claim that nothing valuable is lost by abandoning updating rules.  The idea that the only updating policy sanctioned by the Bayesian theory is updating by conditionalization was untenable even on its own terms, since the learning of each conditioning proposition could not  itself have been by conditionalization.” (Howson 1997, 289).

So a Bayesian account requires a distinct account of empirical learning in order to learn “of each conditioning proposition” (propositions which may be statistical hypotheses).  This was my argument in EGEK (1996, 87)*. And this other account, I would go on to suggest, should ensure the claims (which I prefer to “propositions”) are reliably warranted or severely corroborated.

*Error and the Growth of Experimental Knowledge (Mayo 1997): http://www.phil.vt.edu/dmayo/personal_website/bibliography%20complete.htm. Scroll down to chapter 3.

Categories: Statistics | Tags: , , | 32 Comments

Objectivity (#5): Three Reactions to the Challenge of Objectivity (in inference):

(1) If discretionary judgments are thought to introduce subjectivity in inference, a classic strategy thought to achieve objectivity is to extricate such choices, replacing them with purely formal a priori computations or agreed-upon conventions (see March 14).  If leeway for discretion introduces subjectivity, then cutting off discretion must yield objectivity!  Or so some argue. Such strategies may be found, to varying degrees, across the different approaches to statistical inference.

The inductive logics of the type developed by Carnap promised to be an objective guide for measuring degrees of confirmation in hypotheses, despite much-discussed problems, paradoxes, and conflicting choices of confirmation logics.  In Carnapian inductive logics, initial assignments of probability are based on a choice of language and on intuitive, logical principles. The consequent logical probabilities can then be updated (given the statements of evidence) with Bayes’s Theorem. The fact that the resulting degrees of confirmation are at the same time analytical and a priori—giving them an air of objectivity–reveals the central weakness of such confirmation theories as “guides for life”, e.g., —as guides, say, for empirical frequencies or for finding things out in the real world. Something very similar  happens with the varieties of “objective’” Bayesian accounts, both in statistics and in formal Bayesian epistemology in philosophy (a topic to which I will return; if interested, see my RMM contribution).

A related way of trying to remove latitude for discretion might be to define objectivity in terms of the consensus of a specified group, perhaps of experts, or of agents with “diverse” backgrounds. Once again, such a convention may enable agreement yet fail to have the desired link-up with the real world.  It would be necessary to show why consensus reached by the particular choice of group (another area for discretion) achieves the learning goals of interest.

Continue reading

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Two New Properties of Mathematical Likelihood

17 February 1890--29 July 1962

Note: I find this to be an intriguing, if perhaps little-known, discussion, long before the conflicts reflected in the three articles (the “triad”) below,  Here Fisher links his tests to the Neyman and Pearson lemma in terms of power.  I invite your deconstructions/comments.

by R.A. Fisher, F.R.S.

Proceedings of the Royal Society, Series A, 144: 285-307 (1934)

      To Thomas Bayes must be given the credit of broaching the problem of using the concepts of mathematical probability in discussing problems of inductive inference, in which we argue from the particular to the general; or, in statistical phraselogy, argue from the sample to the population, from which, ex hypothesi, the sample was drawn.  Bayes put forward, with considerable caution, a method by which such problems could be reduced to the form of problems of probability.  His method of doing this depended essentially on postulating a priori knowledge, not of the particular population of which our observations form a sample, but of an imaginary population of populations from which this population was regarded as having been drawn at random.  Clearly, if we have possession of such a priori knowledge, our problem is not properly an inductive one at all, for the population under discussion is then regarded merely as a particular case of a general type, of which we already possess exact knowledge, and are therefore in a position to draw exact deductive inferences.

Continue reading

Categories: Likelihood Principle, Statistics | Tags: , , , , , | 2 Comments

U-PHIL (3): Stephen Senn on Stephen Senn!

I am grateful to Deborah Mayo for having highlighted my recent piece. I am not sure that it deserves the attention it is receiving.Deborah has spotted a flaw in my discussion of pragmatic Bayesianism. In praising the use of background knowledge I can neither be talking about automatic Bayesianism nor about subjective Bayesianism. It is clear that background knowledge ought not generally to lead to uninformative priors (whatever they might be) and so is not really what objective Bayesianism is about. On the other hand all subjective Bayesians care about is coherence and it is easy to produce examples where Bayesians quite logically will react differently to evidence, so what exactly is ‘background knowledge’?. Continue reading

Categories: Philosophy of Statistics, Statistics, U-Phil | Tags: , , | Leave a comment

“You May Believe You Are a Bayesian But You Are Probably Wrong”

The following is an extract (58-63) from the contribution by

Stephen Senn  (Full article)

Head of the Methodology and Statistics Group,

Competence Center for Methodology and Statistics (CCMS), Luxembourg

…..

I am not arguing that the subjective Bayesian approach is not a good one to use.  I am claiming instead that the argument is false that because some ideal form of this approach to reasoning seems excellent in theory it therefore follows that in practice using this and only this approach to reasoning is the right thing to do.  A very standard form of argument I do object to is the one frequently encountered in many applied Bayesian papers where the first paragraphs lauds the Bayesian approach on various grounds, in particular its ability to synthesize all sources of information, and in the rest of the paper the authors assume that because they have used the Bayesian machinery of prior distributions and Bayes theorem they have therefore done a good analysis. It is this sort of author who believes that he or she is Bayesian but in practice is wrong. (58) Continue reading

Categories: Philosophy of Statistics, Statistics | Tags: , | Leave a comment

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