Probabilistic Proof of Occam's Razor

Occam's Razor is an abductive heuristic used in scientific hypothesis construction and in other situations where we're looking for an explanation for something. The bottomline of Occam's razor consists of the following rule:

[quote=Wikipedia]Entia non-sunt multiplicanda praeter necessitatem (Entities must not be multiplied beyond necessity)[/quote]

The wikipedia article, if you wish to pursue the topic further, is detailed enough and provides a variety of views on the logic behind Occam's Razor, its impact on science, its limitations and so forth.

That said, in this post, my aim is to provide a simple proof of Occam's Razor and I would like you all to give your views on whether my proof is any good or not.

My proof will require an imaginary scenario but it will, in all likelihood, be familiar enough to you all. Mutatis mutandis, the scenario is applicable to science and all other occasions where an explanation is being sought.

Imagine a doctor who's called to a house where young girl has fallen ill. He picks up the necessary medical equipment and arrives at the house.

Upon arriving he notices that the family has also consulted the local shaman and he's also at the house, looking into the matter of the girl's illness.

The doctor greets the family and ask to be taken to the patient. After inquiring into the illness and its circumstances the doctor comes to the conclusion that the likely cause is food the girl ate at a neighbor's house the other day.

The shaman too carries out his own investigation and comes to the same conclusion - it's the food the girl ate at her neighbor's - but then the shaman doesn't stop there; he claims that the neighbor is a witch and that there's an evil spirit that possessed the girl through the food and is making her sick.

Let's now compare the explanatory hypotheses of the doctor and the shaman, assigning made up probabilities to each component of the hypotheses:

Doctor's hypothesis:

1. The food the sick girl ate at her neighbor's is the cause. Probability of this being true is, say, 2/3 = 67%

The probability the doctor's hypothesis is true = 67%

Shaman's hypothesis:

1. The food the sick girl ate at her neighbor's is the cause. Probability of this being true is, say, 2/3 = 67%

AND

2. The neighbor's a witch. The probability of this being true is, say, 3/4 = 75%

AND

3. There are evil spirits. The probability of this being true is, say, 9/10 = 90%

The probability that the shaman's hypothesis is true = (67%)*(75%)*(90%) = 45%

As you can see, even when I assigned high probabilities to components 2 (neighbor's a witch) and 3 (there are evil spirits) for the shaman, the shaman's hypothesis is less probable than the doctor's hypothesis: shaman's hypothesis has a 45% chance of being true and doctor's hypothesis has a 67% chance of being true.

The only relevant difference between the doctor and the shaman is the number of components ("entities") required to explain the girl's sickness: the shaman's hypothesis has more components ("entities") than the doctor's. And we observe, very clearly, that multiplying entities reduces the overall probability of a hypothesis being true.

Ergo, since multiplying entities in a hypothesis reduces the probability of the hypothesis being true, Occam's Razor is a well-justified rule.

:chin:

N.B: An important part of Occam's Razor is what exactly do "entities" mean? Since Occam's Razor is about explanatory hypotheses, we need to review what they (explanatory hypotheses) are?

An explanatory hypothesis, for me, attempts to provide some reasons why some possibilities are observed as facts of this universe and some possibilities don't actualize. The way an explanatory hypothesis is built resembles the familiar axiomatic systems of math: some initial propositions are assumed as true and then the explanatory hypothesis is constructed around these initial propositions or "axioms". Given this is true about explanatory hypothesis, the only viable candidate I see as a referent for "entities" are the initial propositions of an explanatory hypothesis for there is no sense in which any other component of an explanatory hypothesis can multiply i.e. increase numerically.