A mathematical proof of Ockham’s razor?

$ i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$

 

Ockham’s razor is a principle often used to dismiss out of hand alleged phenomena deemed to be too complex. In the philosophy of religion, it is often invoked for arguing that God’s existence is extremely unlikely to begin with owing to his alleged incredible complexity. Bild A geeky brain is desperately needed before entering this sinister realm.

In a earlier post I dealt with some of the most popular justifications for the razor and made the following distinction:

Methodological Razor: if theory A and theory B do the same job of describing all known facts C, it is preferable to use the simplest theory for the next investigations.

Epistemological Razor: if theory A and theory B do the same job of describing all known facts C, the simplest theory is ALWAYS more likely.”

Like the last time, I won’t address the validity of the Methodological Razor (MR) which might be an useful tool in many situations.

My attention will be focused on the epistemological glade and its alleged mathematical grounding.

{\frac{x+y}{xy}}

Example: prior probabilities of models having discrete variables

Presentation of the problem

We consider five functions that predicts an output Y (e.g. the velocity of a particle in an agitated test tube) which depends on an input X (e.g. the rotation speed).

Those five functions themselves depend on a given number of unknown parameters $latex a_i $.

$latex f1(a1)[X] $
$f2(a1,a2)[X] $
$latex f3(a1,a2,a3)[X] $
$latex f4(a1,a2,a3,a4)[X] $
$latex f5(a1,a2,a3,a4,a5)[X] $

To make the discussion somewhat more accessible to lay people, we shall suppose that the $latex a_i$ can only take on five discrete values: {1,2,3,4,5}
Let us suppose that an experiment was performed.
For x = 200 rpm (rotation per minute), the measured velocity of the particle was y = 0.123 m/s.

Suppose now that there is only one set of precise values that allows the function fi to predict the measurement E.
For example
f1(2)[200 rpm]= f2(1,3)[200 rpm]= f3(5,2,1)[200 rpm]=f4(2,1,4,5)[200 rpm]=f5(3,5,1,3,2)[200 rpm]= 0.123 m/s.

Now we want to evaluate the strength of the different models.
How are we to proceed?

Many scientists (including myself) would say that the five functions fit perfectly the data and that we would need further experiments to discriminate between them.

$latex your-latex-code-here$

The objective Bayesian approach

Objective Bayesians would have a radically different approach.
They believe that all propositions (“The grass is greener in England than in Switzerland”, “Within twenty years, healthcare in Britain will no longer be free”, “The general theory of relativity is true”…) is associated with a unique precise degree of belief every rational agent knowing the same facts should have.

They further assert that degrees of belief ought to obey the laws of probability using diverse “proofs” such as the Dutch Book Argument (but see my critical analysis of it here).

Consequently, if at time t0, we believe that model M has a probability p(M) of being true, and if at t2 we get new measurement E, the probability of M should be updated according to Bayes’ theorem:

$latex p(M|E) = \frac{p(M)*p(E|M)}{(p(E|M)+p(E|\overline{M})}$.

p(M|E) is called the posterior, p(M) is the prior, p(E|M) is the likelihood of the experimental values given the truth of model M and p(E|M)+p(E|non M) is the total probability of E.
A Bayesian framework can be extremely fruitful if the prior p(M) is itself based on other experiments.

But at the very beginning of the probability calculation chain, p(M) we are in a situation of “complete ignorance”, to use the phrase of philosopher of science John Norton.

Now back to our problem.

An objective Bayesian would apply Bayes’ theorem and conclude that the probability of a model fi is given by:

p(fi|E) = p(fi)*p(E|fi)/(p(E|fi)+p(E|non fi))

Objective Bayesians apply the principle of indifference, according to which in utterly unknown situations every rational agent assigns the same probability to each possibility.

As a consequence, we get p(f1)=p(f2)=…=p(f5)=0.2

p(E|fi) is more tricky to compute. It is the probability that E would be produced if fi is true.

 

 

 

 

 

For this reason O(i,j) is usually referred to as an Ockham’s factor, because it penalizes the likelihood of complex models. If you are interested in the case of models with continuous real parameters, you can take a look at this publication. The sticking point of the whole demonstration is its heavy reliance on the principle of indifference.

The trouble with the principle of indifference

I already argued against the principle of indifference in an older post. Here I will repeat and reformulate my criticism.

Turning ignorance into knowledge

The principle of indifference is not only unproven but also often leads to absurd consequences. Let us suppose that I want to know the probability of certain coins to land odd. After having carried out 10000 trials, I find that the relative frequency tends to converge towards a given value which was 0.35, 0.43, 0.72 and 0.93 for the four last coins I investigated. Let us now suppose that I find a new coin I’ll never have the opportunity to test more than one time. According to the principle of indifference, before having ever started the trial, I should think something like that:

Since I know absolutely nothing about this coin, I know (or consider here extremely plausible) it is as likely to land odd as even.

I think this is magical thinking in its purest form. I am not alone in that assessment.

The great philosopher of science Wesley Salmon (who was himself a Bayesian) wrote what follows. “Knowledge of probabilities is concrete knowledge about occurrences; otherwise it is uselfess for prediction and action. According to the principle of indifference, this kind of knowledge can result immediately from our ignorance of reasons to regard one occurrence as more probable as another. This is epistemological magic. Of course, there are ways of transforming ignorance into knowledge – by further investigation and the accumulation of more information. It is the same with all “magic”: to get the rabbit out of the hat you first have to put him in. The principle of indifference tries to perform “real magic”. “

Objective Bayesians often use the following syllogism for grounding the principle of indifference.

1)If we have no reason for favoring one outcomes, we should assign the same probability to each of them

2) In an utterly unknown situation, we have no reason for favoring one of the outcomes

3) Thus all of them have the same probability.

The problem is that (in a situation of utter ignorance) we have not only no reason for favoring one of the outcomes, but also no grounds for thinking that they are equally probable.

The necessary condition in proposition 1) is obviously not sufficient.

This absurdity (and other paradoxes) led philosopher of mathematics John Norton to conclude:

“The epistemic state of complete ignorance is not a probability distribution.”

The Dempter Shafer theory of evidence offers us an elegant way to express indifference while avoiding absurdities and self-contradictions. According to it, a conviction is not represented by a probability (real value between 0 and 1) but by an uncertainty interval [ belief(h) ; 1 – belief(non h) ] , belief(h) and belief(non h) being the degree of trust one has in the hypothesis h and its negation.

For an unknown coin, indifference according to this epistemology would entail  belief(odd) = belief(even) = 0, leading to the probability interval [0 ; 1].

Non-existing prior probabilities

Philosophically speaking, it is controversial to speak of the probability of a theory before any observation has been taken into account. The great philosopher of evolutionary biology Elliot Sober has a nice way to put it: ““Newton’s universal law of gravitation, when suitably supplemented with plausible background assumptions, can be said to confer probabilities on observations. But what does it mean to say that the law has a probability in the light of those observations? More puzzling still is the idea that it has a probability before any observations are taken into account. If God chose the laws of nature by drawing slips of paper from an urn, it would make sense to say that Newton’s law has an objective prior. But no one believes this process model, and nothing similar seems remotely plausible.”

It is hard to see how prior probabilities of theories can be something more than just subjective brain states.

Conclusion

The alleged mathematical demonstration of Ockham’s razor lies on extremely shaky ground because:

1) it relies on the principle of indifference which is not only unproven but leads to absurd and unreliable results as well

2) it assumes that a model has already a probability before any observation.

Philosophically this is very questionable. Now if you are aware of other justifications for Ockham’s razor, I would be very glad if you were to mention them.

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3 thoughts on “A mathematical proof of Ockham’s razor?

  1. Now if you are aware of other justifications for Ockham’s razor, I would be very glad if you were to mention them.

    If there are two competing explanations with equal explanatory power, and the assumptions of the simpler explanations are a subset of the more complex explanation, then the simpler can explanation logically can only be more likely or equally likely than the more complex one, but cannot possibly be less likely.
    Example: if explanation 1 relies on assumptions A and B, and explanation 2 relies on A, B, and C – then the two explanations are equally likely to be correct if the probability of C being true is 1, and explanation 1 is more likely in every other case.
    If the simpler explanation relies on assumptions that are NOT a subset of the assumptions for a more complex explanation, then I doubt that there can be anything like a formal proof of Occam´s Razor being valid.

    • Hello Andy, thanks for your answer.

      To get the epistemological razor (see my definitions above) you need to prove that the probability of the more complex explanation is strictly inferior to that of the simpler one. The probability of C might very well be unknown, so that we don’t know the extent of the penalty.

      I think there is a simpler way to go: if C does not add anything to the explanation of the phenomenon at hand, C is not a part of the explanation .

      Therefore it is a separate problem/claim/phenomenon which should be studied in an entirely different context.

      I generally doubt, however, that propositions which are not events possess probabilities.
      I believe that events which have not occurred (such as technological singularity) or historical happenings have a physical probability but the same cannot be said about the existence of numbers, the soul, God or the truth of string theory.

      In such situations, I resort to likelihoodism and consider the quantity p(E|theory), that is to say the probability of the ensemble of our evidence given the truth of the theory .
      In this way it is possible to compare theory1, theory2 and theory3 with respect to their agreement with reality.
      The theories leading to the worse predictions (in terms of the probabilities of the predictions of the data in the real world) should no longer be pursued.

      Concrete example: p(E|Young Earth Creationism) < p(E| old earth creationism) < p(E | Behe's intelligent design) < p(E| evolution).

      Doing this is perfectly sufficient for all my concerns as a scientist (and that of all people I know), there is no need to introduce subjective degrees of belief into the picture.

  2. This absurdity (and other paradoxes) led philosopher of mathematics John Norton to conclude: “The epistemic state of complete ignorance is not a probability distribution.”

    The Dempter Shafer theory of evidence offers us an elegant way to express indifference while avoiding absurdities and self-contradictions.

    This is really neat! Bayesian probabilities convolve two sets which ought to be kept separate: evidence-for, and evidence-against. Of course we often combine these sets when we need to act, but we still keep them separate after the action.

    After being instigated by a friend, I’ve thought a lot about how ‘unknown’ is a different type or category than ‘true’ or ‘false’. I think you’re really onto something, Lotharson!

    To what extent have you examined the criticism of Dempter Shafer theory? I doubt we will ever find the perfect way to evaluate evidence; instead, I predict we’ll just find better and better models, or at least models that work in more and more situations. 🙂

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