Re: The danger of classical hypothesis and significance tests [was Re: MADLY AMUSED]

illywhacker wrote:
On May 29, 7:26 pm, "S. F. Thomas" <thomas7...@xxxxxxxxxxxxx> wrote:

paradigm

You might want to look here:

http://math.ucr.edu/home/baez/crackpot.html

bearing in mind your post here:

http://www.math.yorku.ca/Who/Faculty/Monette/Ed-stat/0150.html

paying particular attention to points 8, 18, 19, 22, 26, and 37.

I am afraid I will not be discussing further with you unless you can
make a substantive point.

It appears I may have misjudged you. I thought after your intervention in the deplorable series of exchanges between John Smith/Jack Tomsky and Luis Afonso, that you were a person of some maturity and judgment. From our brief exchange I see, sadly, that you are not what at first you seemed. Your mode of argumentation here is no different from what Galileo suffered under the Catholic Church so many centuries ago -- argumentum ab auctoritate. Given the pained and heartfelt way in which Jaynes (see your citation earlier in the thread) likened himself to Galileo going up against the classical orthodoxy, I would have thought a Bayesian such as you would refrain from taking up the sort of position you here do, namely that of self-appointed Defender of the Faith, taking aim at what you too quickly deem to be misguided crackpots. Surely, it is sufficient to let cracked pots to leak all on their own. If they are cracked, surely they will in the end carry no water; if they are not, then they give us something to think about, don't they? I, at least, have less patience with those who commit the twin fallacies of argumentum ab auctoritate and argumentum ad hominem, especially those doing so from behind a cloak of anonymity.

Let me however address the following few points, such as they are, made in your earlier post, which is effectively a parallel one, as I am seeing them both at the same time:

1) First let me get the unkind ad hominem attack out of the way. I cannot speak for my publisher. My book speaks for itself. You are welcome to read it, or not, according to your taste or inclination.

2) The problem, which you pretend not to see or understand, is that which has beset the Bayesian approach from the very beginning -- namely the justification, or lack, for the notion of prior. After all the axiomatizations, claims of rationality, consistency, coherence and all the rest, the problem, alas, remains.

Nevertheless, it is clear, and Fisher would agree, that the likelihood function "supplies a natural order for the possibilities under consideration." Therefore, the only question at issue is what calculus should be applied to the likelihood function for purposes of marginalization (to eliminate nuisance parameters) and of change of variable (to characterize loss functions in decision problems). I understand clearly that Bayesians proffer:

Posterior = Likelihood x Prior

with necessary re-scaling accomplished per Bayes theorem. From this, it is clear that where the prior is taken as flat (whether "properly" or not), the effect is to give

Posterior = Likelihood

appropriately re-scaled to conform with the requirements of probability theory. Therefore, what in effect Bayes accomplishes is to take the likelihood function, and turn it into probability density.

That actually works well, and I quite agree with Jaynes that the proof of the inferential pudding is (ultimately) in the eating. Nevertheless, the justification for taking Likelihood, which is manifestly not a probability distribution or density function in the way it is derived from the data under the model, and turning it into Posterior pdf is a matter of some philosophical delicacy and was so from the beginning, that is, starting with the Rev. Thomas Bayes himself.

I do not fault the neo-Bayesians for having returned to this question, because the classical approaches, which rest on a considered rejection of the early Bayesian methods, turn unwieldy and complex very quickly. And as Jaynes has pointed out, they also fall into positive error on occasion. Nevertheless, it is my contention that the neo-Bayesians have been barking up the wrong tree. The efforts at justification have in my opinion simply not succeeded, and moreover are doomed to fail. This contention is admittedly a bold assertion, but I may justify it very simply by making the observation, without fear of contradiction, that *all* of what the data say about the parameter under the maintained model is summed up in the likelihood function. Therefore the likelihood can, and I agree with those who so argue, should, stand alone as the fundamental inferential object from which all tasks of inference should proceed. Therefore, the Prior must, in principle, be irrelevant to the task of inference.

Whether you agree with that last statement or not, what is indisputable is that there is at least one other tree up which inferential dogs may bark. And that tree amounts simply to taking the likelihood function as a representation of uncertainty in its own right, and deriving from foundational first principles what is the calculus that should apply to it. It is with that background that I would again suggest, if you believe Cox to be persuasive, why not simply rescale the likelihood directly to conform to the Coxian conclusion, and treat it as probability density in its own right, reconstrued as the the "only consistent calculus of plausibility" as it pertains to uncertainty in the (constant) parameter, as described by the likelihood. That remains a serious suggestion for those disposed to accept the Coxian conclusion. It is not however the way I would go.

My intuition leads me in a different direction. The essential insight derives from fuzzy set theory, which is to recognize (i) that the (absolute) likelihood function minimally satisfies the conditions of the membership function of a fuzzy set, and more fundamentally (ii) that uncertainty in a model parameter is of a different (fuzzy) sort than uncertainty (probabilistic sort) in the next occurrence of a random variable. The whole point of my book was to sort out the differences between the two kinds of uncertainty, and in the process to develop a calculus of the likelihood or fuzzy or possibilistic sort. In practice, we then find that the inferential conclusions then agree pretty well, though not exactly, with those obtained using a Bayesian analysis with flat prior. The rule for change of variable (extension principle to the fuzzicists) is, though, different. Thus in a decision analysis, the expected loss may be explicitly, and irreducibly, fuzzy, given the data and the model, not always and inevitably reduced to a single point as under the Bayesian analysis. Be all that as it may, the fact remains that we have, at least since Zadeh's fuzzy insights, an alternative tree up which to bark for those so inclined. If you shake the tree hard enough, there are even some delicious fuzzy fruit that seem to fall, for example the very notion of fuzzy probability, the notion of regression when the data are fuzzy, etc. You are of course welcome to keep barking up the Prior tree. I won't hold it against you, and I ask in return that you don't hold it against me if I choose at this point to bark up the Fuzzy tree.

I say all that to explain where it is "I'm coming from", to use the vernacular. My entry into the discussion as earlier explained was triggered by your use of the term "dangerous" in relation to the classical approach. I thought perhaps I had missed something in my earlier in-principle review of this approach. Yes they are sometimes grossly in error -- I have at least one example in my book taken from Godambe and Sprott. However, I have not been able to fault the paradigm per se, because whenever such error is discovered, it is usually possible, as we see also with Jaynes' examples, to rectify. Like the Bayesians, I lost patience with that a long time ago, but in the interest of strict fairness, I must and do concede that at least in principle they have the possibility of rectifying. I though am most certainly not clever enough to see how in every instance. And I see that the great Jaynes also throws his hands up on occasion, and the great Wald likewise.

3) Now as to the question of randomness, the trap of circularity is obvious to anyone who has done any foundational work -- it is the challenge of defining probability and randomness together, not in an uninterpreted mathematical way, but in an empirical-axiomatic way that anyone can understand, and where the potential for circularity is avoided by grounding the discussion in an appropriate foundational context. The approach I have taken in my book requires first a scene-setting but informal discussion of the notion of the phenomenon, as that which we are concerned with modeling, and a more formal discussion of the notion of model of phenomenon, both in terms of morphology (intension), meaning the objects, attributes, and relationships which we bring into focus for purposes of explaining a phenomenon, and in terms of extension, meaning an elaboration of what constitutes an instance, and implicitly, all instances of a phenomenon, within the morphology we choose for the moment to consider. With this background, the notion of proportion, with respect to the extension set, is easy to explain, by means merely of counting and classifying. The notion of probability then follows by equation between a statistical (counting and classifying) model over the extension set of some (model) of a phenomenon, and "random" selection of instances from the extension set, with randomness carrying the essential meaning that selection of the instance is blind, that is, uninformed by prior observation of any attribute in any way correlated with the attributes within the morphology. This is as brief as I can make it to convey the essential idea. For the full formal treatment, there is of course the book. The main thing though is that there is nothing "meaningless" about the concept of randomness, quantum-theoretic or otherwise.

I will stop here as this post is quite long enough, so I will leave aside the non-central issues. As I see now that you are not a good arguer -- your resort to argumentum ab auctoritate, argumentum ad hominem, and uncalled-for generalized ex cathedra smear, are sufficient proof of that -- I will not engage you further. If you come out from behind your cloak of anonymity, and withdraw your generalized and uncalled-for smears, I may change my mind. I have taken the time to make the points I do above because this is a public forum, after all, and one never knows who is watching. Certainly the reader is free to make up their own mind as to who is the cracked pot, since those who resort to ex cathedra posturing and pointing of fingers are likely as not to be crackpot themselves, as we saw with Galileo.


illywhacker;

Regards,
S. F. Thomas



illywhacker wrote:
> On May 29, 7:26 pm, "S. F. Thomas" <thomas7...@xxxxxxxxxxxxx> wrote:
>> illywhacker wrote:
>>> On May 29, 4:25 am, "S. F. Thomas" <thomas7...@xxxxxxxxxxxxx> wrote:
>>>> illywhacker wrote:
>>>>> On May 28, 8:52 pm, "S. F. Thomas" <thomas7...@xxxxxxxxxxxxx> wrote:
>>>>>> illywhacker wrote:
>
> [cut]
>
>> I have to concede, so far, that some sufficiently clever classicist may
>> rectify whenever a Jaynes comes along to point out clear error.
>
> How can a classicist possibly reproduce a Bayesian calculation with a
> non-binary prior? This is silly.
>
>>>> But I'm more concerned with the issue of principle.
> [cut]
>>> There is a failure of principle, indeed many. One is called Cox's
>>> theorem.
> [cut]
>> How does this defeat the classical approach? I see this as a device to
>> rationalize a Bayesian treatment of uncertainty as it applies to
>> characterizing what an unknown (constant) parameter might be.
>
> On the contrary: it means that alternative treatments of uncertainty
> are either equivalent to a Bayesian treatment, or irrational.
> Classical statistics is not equivalent to Bayesian approaches. In
> addition, the Bayesian calculations are simpler than the classical
> ones; this too is an argument against classical approaches.
>
> [cut - treated later]
>
>>> Indeed the very use of the meaningless word
>>> 'randomness' in anything other than a strictly mathematical context
>>> ('random variable') is another failure of principle.
>> Again, at best an argument *for* your particular paradigm, not *against*
>> any other.
>
> The fact that a 'paradigm', as you put it, is based on meaningless
> notions is not an argument against it?
>
>> I don't agree that the concept of randomness is meaningless,
>> but I do agree that its definition requires some care if one is not to
>> get ensnared in a trap of circularity.
>
> So let's hear it. What is your definition? You are full of bold
> assertions but no actual arguments.
>
> [cut]
>
>> I have not per se addressed Cox's
>> theorem, but I see no reason to. Any theorem is only as good as the
>> starting axioms, and I can see right from the get-go that "real numbers"
>> are introduced in a way that precludes fuzziness as a separate category
>> of uncertainty.
>
> You have understood neither the background to Cox's theorem nor the
> status of fuzziness.
>
> 1) Once a total ordering of the truth values is assumed, they might as
> well be real numbers.
>
> 2) The truth values of fuzzy *are* real numbers. The proposition to
> which the truth values apply are of course not necessarily real
> numbers, but neither are they in probability theory.
>
> 3) In fact, the truth values of fuzzy propositions are not even at the
> same semantic level as probabilities: they cannot be, as Cox's theorem
> shows. Rather, they are just propositions like any others; they are a
> low-resolution picture of the world. In other words, the truth values
> of fuzzy propositions are *arguments* to probability distributions.
>
> [cut]
>> I think the classicists have a point that there is a problem.
>
> You still have not said what the problem is! Do you have any idea what
> constitutes a mathematical or scientific argument? By the way, how do
> you explain this quote form the ACG Press web page: 'Welcome. We are
> proud to offer you our first book: "Time and money. Your guide to
> economic freedom
> the path to financial independence"'?
>
> [cut]
>>> I can never understand what people mean when they imagine that they
>>> can get by without priors.
> [cut]
>> I do not disagree that "prior"
>> information is sometimes relevant to a contemplated decision problem. In
>> that case, such prior information could in principle be included with
>> the data, and we are back to the problem of characterizing, alone, what
>> the data say about the parameters of interest.
>
> Of course! This is what Bayes theorem does for you. It tells you how
> to incorporate new data into your prior so that you are ready for even
> newer data. Were we to start from the dawn of time and take into
> account all the data that has gone into the evolution of human
> physiology, psychology, and knowledge, then maybe we would need only a
> uniform (or more correctly, an ignorance) prior. But if we want to
> summarize this knowledge in a way relevant for applications, then we
> need a non-trivial prior. Without a prior, you do not even know of the
> existence of space or time or measurement or air or the density of
> water.
>
>> Btw, if Cox is used as justification, why do you need Bayes? Just
>> rescale the likelihood function so it conforms to Cox, and off you go.
>
> What? Evidently you have not read and understood Cox's theorem yet.
>
>> Suppose you want to estimate a function. If
>>> you do not place constraints on this function, the task is impossible,
>>> since a finite number of data points can never determine a function.
>>> So you place constraints. But this is a prior!
>> No, this is a model.
>
> What do you think a prior is? It is a model!
>
> You seem to lack the basic knowledge and competences necessary to
> understand this subject.
>
> illywhacker;
.

Relevant Pages

  • Re: different priors (flat, uniform, etc)
    ... Now that we have the Bayesian recipe resolved about the problem ... the beta distribution is a member of the ... RF> conjugate prior family -- meaning both the prior AND posterior ... likelihood function as the posterior distribution. ...
    (sci.stat.math)
  • Re: Highest Posterior Density
    ... You need to be able to supply the Bayesian ingredients of a prior ... obtain your posterior distribution of your PARAMETER ... you do not need to perform the integration as the ...
    (sci.stat.math)
  • Re: Bayesian continued and shuffling
    ... From a Bayesian point of view did I not use prior knowledge? ... An observation about hand shuffling: ... For the record I try not to adjust and keep using pure probability ...
    (rec.gambling.poker)
  • Re: Where it all began: Illywhackers lack of understanding about Bayesian inference
    ... RF> maximum of the posterior distribution, ... RF> bother with Bayesian inference? ... RF> Never even heard of the acronym MAP in Bayesisian inference. ... It is called a "diffuse prior" not a uniform prior. ...
    (sci.stat.math)
  • Re: Reference for Savage/Berger
    ... >>specifying one's prior distributionrealistically, ... >>computational problems of obtaining the posterior distributions from ... > This is overused as a reason for not being a Bayesian. ... That is a seldom used statement of reality by a true Bayesian. ...
    (sci.stat.edu)