Bayes strength

This article is mainly focused on Bayesianism, a popular interpretation of the concept of probability used to evaluate hypothesis and justify the extent to which we feel confident that they are true. It replaces in part the famous, or infamous, problem of induction. The problem of induction was first explicitly brought to light by David Hume, an interesting Scottish philosopher with a lovely writing-style. It has served as a wall to bang your philosophically-inclined head against for centuries. Bayesianism does not solve this problem, let this be clear, but it does in part propose a way of clarifying why induction is so persuasive. I don’t think it’ll pose any problem at all for someone to read this, whether you are normally interested in philosophy or not. In any case, it is an interesting topic that deserves your attention. The only question that remains is whether my manner of my writing about it does so too, something for you to judge.

Especially on philosophical things comments and criticism are much appreciated, so I would strongly invite you to do so. Also question about concepts or statements that are unclear to you are welcome, I will be more than happy to talk, discuss or explain related issues.

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… An essential feature of Neyman-Pearson statistics is the aim to deny a place to false hypotheses and to reserve a place for those that are true. As risk here, as with practically all demands in theories within the boundaries of the philosophy of science, is making a theory too cold, meaning you set standards too low and everything will become scientifically meaningful, or on the other hand making demands too hot, so as to exclude knowledge that definitely belongs to epistemological areas.

As we will see shortly, this is exactly what happens with Neyman-Pearson statistics. Still, it is interesting. It is especially interesting when we combine it with a view on just what sets Bayesianism apart from other statistical ways of justification. Both have their own shortcomings, since neither has a core that is solely objective. We’ll take a quick look at why the first doesn’t work, after which we’ll discuss the alternative offered to us by Bayesians.

The trick to keep in the good stuff, and jettison the bad stuff has been prominent since the positivists. Their war on metaphysics is a part of practically any curriculum in philosophy education and provides an easy way to start talking about verifying and falsifying hypotheses. Anyone busy in the area of epistemology knows to be true what was just said, that the difficult task is one of including and excluding; science and pseudoscience; empiricism and metaphysics. So many different accounts have been given, that identifying and solving problems is sure to give you work for a lifetime.

Neyman and Person tried it via a hypothesis test. A simplified way of expressing this talks of Rejection of H and A, where H is hypothesis one and A an alternative two that is incompatible with H. To derive, or not to derive, a rejection will depend on the height of the number to which the individual constant X refers, and on the ratio you will get after filling the number into the theory. The formulation itself is not important here, what is more important are its philosophical implications: In ideal circumstances, when A is true, H will come out false. They are aiming for this goal by going for small-size on the one hand, and for greater strength on the other. It is not sure that these two pillars can be combined: It is as much as saying that you want to minimize size, and maximize power within statistics.  This can also be found in what is known as ‘The Fundamental Lemma of Neyman and Pearson:

“In the case of simple dichotomy, there exists, for any possible size, a uniquely most powerful test of that size. It may be a mixed test, but it is unique.”[1]

And within statistics, much power is derived from large numbers, especially when your core is weak in qualitative measures. And that is something to which I cannot ascribe much strength (it has strong ties with so called ‘likelihood tests’). So too has the ‘optional stopping’, which brings along at least two problems. The first is the most essential: When is a proposition falsified or verified, in other words: When have we collected enough data in order for us to stop investigating? The answer is, with induction lurking around the corner, ‘never’. The other question strongly relates to this, only it is more specific by a focus on the formula’s goals (small, strength). When you for have the aforementioned two hypotheses that are mutually exclusive, you need to know both when one is granted more probability than the other, why, what is worthy of confirmation or disconfirmation and when does ‘the most likely outcome’ swap sides, i.e. when will enough con- or disconfirmation have taken place to speak of a different allegiance of belief?

Bayesianism deals with these issues in a very different manner. We can see this when we take a look at its most basic formula, according to A. Bird:

Bayesianism does not take off from grounds that are self evident. In fact Bayes theorem is accepted by almost all probabilistic minded-philosophers, what they do not accept are the conclusions Bayesians infer from them. How then do they themselves view the way in which hypotheses need confirming or disconfirming? Recall an alternative here, in order to make a clear contrast. Take Hempel, famous for the Raven-paradox. He struggled with a qualitative notion of confirmation by asking what it is for given evidence to confirm a hypothesis. How much confirmation is supported by evidence is the quantitative notion that should later be dealt with, after settling with the qualitative aspect. Bayesians think this is all wrong. For them, confirmation is something quantitative in nature right from the start. We can’t ask whether evidence confirms or disconfirms a hypothesis, unless we know how probable H was in the first place. This simply means that, for something to be able to be confirmed or disconfirmed, it needs a prior probability. The way in which results are obtained is quite straightforward really, at least in essence it is (things can get messy and complicated, though this author likes to keep things clear). After a hypothesis with a given probability is confronted with evidence, it is confirmed if the probability rises, and disconfirmed if the probably gets lower in light of the evidence.

Now this brings along some difficulties. How do you attach probability to something?  We can’t just rule out subjectivism, because what to is subjectivist to you might seem wholly reasonable to me. I’ll to illustrate this with a short and simplified historical example about the concept ‘gene’. After Darwin’s ‘On the origin of species’ biological knowledge was granted to enter a whole new paradigm, albeit reluctantly. Natural selection, especially after it was combined with G. Mendel’s heredity, got a boost that still reverberates today. Even so, genes were unknown entities at the beginning of the 20^th century. It took a long time after the term was invented before they were actually discovered. This did not prevent anyone from using the term within a referential framework, that is to say: it did not prevent biologists from assigning them certain characteristics and qualities, despite the fact no one had ever proven their existence. Does this fact make talking about or making predictions with ‘genes’ something unscientific at the beginning of the last century? I should think not. If anything, it was a clear and wonderful result of intellectual expectations, based on inferences from good, probable knowledge. Still if we would add up the probability numbers of such an entity existing, it is certainly not unimaginable that its result would be quite low given the fact that they are based on non-existing evidence, hence either their place in Bayes theorem is illegitimate, or the theorem casts aside genuine scientific innovation. Now don’t be disappointed, things are not that clear.

There are two ways to go here. The first is opting to build on prior knowledge, of which genes were no part. The second is including not just results of things we only grasp inductively, but also things we have not yet accounted for by evidence but which play a decent, functional role in our theories nevertheless. In many cases this would seem like cheating, but since we are talking about probability and (future) testing this is clearly not the case here. A difficulty that would certainly arise is of course how you would assign a certain probability number to genes, and this has strong ties with beliefs, however rational these in essence might be, for as we know, gene-adherents turned out to be right.

This brings us to the aforementioned subjectivism. There is no standard to set probability, at least not at the beginning of the process. This does not mean we should immediately bow our heads in despair. As I see it we have several good things that can lead to many good things, such as a demand for consistency with other beliefs in your web of knowledge and their mutual dependence of those beliefs (we don’t want ‘red wine is sacred liquid’ something probable, because it is atomic in its meaning and therefore quite useless indeed). Demands when it comes to falsifying, verifying, corroboration and repeatability of tests also spark the imagination. All of these have shortcoming however, which means Bayesianism will take at least a part of the blow by leaning on them.

For consistency makes it easy to stay dogmatic; think of the religious paradigm. Darwin’s evidence was inconsistent with it. Should we have laid evolution to rest as a theory, especially because many thought it, at first sight, so unlikely? Of course not. Assigning a higher probability to well known hypotheses rather than to the unknown ones is something intuitive. People ascribe beliefs to others and themselves every day, social interaction would not be able to function if this didn’t happen. In normal conversations this does not lead to problems, since we’re not typically occupied with justifying all of our beliefs. Some, like political or ethical points of view, might need elaboration but they seldom get into the depths where Bayesianism finds itself. So where do we start? An initial position will be necessary for any theory. And they can come up with nothing better than subjectivism.

That sounds like a negative judgment, although it isn’t. Bayesianism is our best alternative; the only thing it is not able to solve is the problem of induction. True, it gives us much explanatory power on why induction works and why it seems a good thing to put your faith in. In this article I have largely left aside the issue of objective versus subjective probability. And I had good reasons to do so, or at I least I thought I had. I don’t think objective probability is likely, not at all. Its ideal poses obstacles too readily in our path, even without impossible, infinite regression. In the end hardly anyone of us beliefs in rigid and objective standards, why would probability be any different? As far as subjectivism goes, we won’t be able to get it more objectivistic than this.