Probability theory implies Occam’s razor

Occam’s razor follows from the rules of probability theory. It need not be assumed, or taken as a prior. Looking at what it means to spread probability mass across a larger space, in the absence of any information on how to distribute that probability, shows why.

Spread-out distributions are less accurate distributions

Say I have some single-variable linear model , trying to predict some other single variable . Finding a good model means finding a good number that represents much a change in should affect the value we predict for . For concreteness, imagine the problem of predicting the weight of a dairy cow, using its height. Then is the length, and is how much a change in height changes the weight (and isn’t important in this post). We may know some things already about , before we have seen any data, like the smallest and largest values it could reasonably be, a vague guess to its average value, and some idea as to how much it tended to vary. From this information, we can get a prior probability distribution over . The probability distribution for a single parameter is a 2-dimensional curve. If we don’t know much about it, its distribution may be wide:

Unsure about ’s true value.

or narrow:

More confident about the value of .

Doing inference is the process of trying to get from the blue distribution to the pink distribution. When we have less information about , we’ll have more spread-out distributions, and when we have more information, narrower. In particular, the less information we have about to start with, the more spread out our probability will be, and the more additional information we’ll need to get to the same narrower pink curve. A worse situation than the blue would be the following:

Ignorant of the value of .

which represents us not even having any idea where to center the probability mass around, or knowing anything about how much it tends to vary.

So: getting to pink is desirable because less-spread-out distributions correspond to more accurate predictions. Blue to pink is easier than gray to pink, because blue is a less-spread-out distribution than gray is. And the more spread out your probability, the less you know, and the harder your inference task.

Adding a variable spreads out probability mass

Above, we only had one feature (also called parameter): . If instead we use the model , then our goal will be to accurately estimate both and - again, to make our probability volumes around them narrow. (Say is the cow’s age- then represents how important we think age is in predicting weight). If we set - the same as saying “age doesn’t matter at all”—we’re back at our original model with just . If b is not zero, then the new probability volume is a surface in 3 dimensions, instead of 2:

The joint probability distribution for and . (I call this the -surface later). For every value of , there is a probability distribution for , and vice-versa, giving many 2d surfaces. Many 2d surfaces make one 3d surface. The blue distribution is the one where b = 0.

That blue line is the same as our blue distribution for above. It’s the subspace of the -surface where b = 0. There is another such subspace for b = 1; another for b = 1.5123; and so on. If any of those other subspaces have non-zero probability mass, then that mass must have come from the b = 0 subspace. Mass that was all in the smaller subspace is now partly spread across the larger space. This is just the conservation of probability: in the previous plots, we were only considering the case , so 100% of our probability mass was being used on just the different values of . If b can be non-zero, but probabilities over the whole a-b surface still sum to 1 (as they must), then the subspace must now have less than 100% of the mass in it. This can improve predictions if is in fact informative about , but will damage predictions if is irrelevant to .

Another way of putting it: If you have no reason to think that age matters—if, for example, the problem at hand only concerns the weight of adult cows, and not the calves—then the probability mass you spread around onto different values of is being spread willy-nilly, with no link to reality. Since you’re spreading it willy-nilly, and sucking mass from the best-guess-as-far-as-we-know subspace to do so, your inferences are going to get worse. (Unless you get lucky—but if you have no reason to think matters, then there’s no reason to think you will get lucky, just like with those lottery boxes). This reasoning applies generally to situations of going from parameters to , and to models with all kinds of forms, not just polynomial equations. Of course, if you do have reason to think age matters, then the mass is no longer being spread randomly, and the redistributed mass in the higher dimension may improve predictions.

Statements of the razor

A restatement of Occam’s instruction is: “Entities are not to be multiplied without necessity.” In our example, is represented by a 2d probability surface. That one 2d surface is our “entity”. The geometric effect of including the additional parameter is that that single surface is copied—“multiplied”—many times (and each multiple can then vary). If this is done without any good reason to expect that actually matters, then we have done this “without necessity”. Occam himself wrote “It is futile to do with more what can be done with fewer”, and “plurality is never to be posited without necessity”. A restatement from Jaynes (chapter 20, which equipped me for this post): “Do not introduce details that do not contribute to the quality of your inferences.” In none of this is the word simplicity required, and thus philosophical disagreements over the meaning of “simplicity” are not relevant.

Yet another way of looking at it: Inference is occurring in a staggeringly huge probability space. It has a dimension for every possible variable you could think of, whether or not that variable is relevant in the least. (Like how every single polynomial is of the form , with arbitrarily big, but most polynomials used in practice have zillions of the ’s set to 0, to represent that the 3000th power of , or whatever, is not important.) Just like in the polynomials case, setting many parameters in the high-dimensional probability space to zero—saying they don’t matter—collapses the space to smaller dimension. The brain does this automatically, in amounts again staggering—if you ask me how much water I think is on Mars, I won’t for a second consider the current price of fish in Tasmania, or any of the incredible amount of possible things that might possibly matter for other problems (problems like the task of lunch planning in Africa). I instantly crumple hypothesis space into a pretty small subspace that involves the more tractable collection of things like “size of Mars”. That other stuff just doesn’t matter! Occam’s razor says “don’t think about things like fish prices unless you have a good reason to think they’re relevant”.

The alternative to Occam’s razor is poor inference

The alternative to using the correctly-understood version Occam’s razor is to try and guess at random what information is useful: “Ah, how much water on Mars? I’ll predict how much water is on it using the size of Mars, and… <Draws three cards from hat>… the price of fish in Tasmania, the number of characters ever written on, and the age of my best friend’s dog.” This model may give equally-good predictions as the model that only uses the size of Mars, or worse predictions, but it will not give better predictions except for by luck. The term “Occam prior” is sometimes used, but it’s strange to call a statement about how predictions cannot be improved by tossing in random stuff a “prior”, as if there were another choice. There is no prior that will cause the size-fish-characters-dog model of water on Mars to make better predictions than the size-only model, any more than there is a prior that will make it a good strategy to spread probability mass around at random.

It would be different if, instead of fish-characters-dog, we were considering using the variables “average distance from the sun” and “amount of gravity”. Using variables we have a reason to think matter does not violate the razor. Going back to the different phrasings at the beginning of this section, and considering these new variables: Yes, the entities are still being multiplied, but not without necessity; what is being done with more is better than what could be done with fewer; we reasonably expect a benefit to positing plurality; and details are being introduced, but they are ones that contribute to the quality of our inferences. Invoking Occam’s razor to mean “simpler is better” is not correct, because additional informative variables or interactions do not violate the razor. (In practice, one might have all kinds of good reasons for using a simpler model even where a complex one would give better inferences—maybe you want to run it on an Arduino and can’t afford the memory overhead of a complex model—but this kind of simplicity is not related to the razor.)

Horses and Zebras

The way I was originally taught Occam’s principle was with the example: “If you hear hoofbeats from the next street over, you should guess that it’s horses, not zebras”. But this is not a good example, because the situation is easy to analyze directly using Bayes’ theorem. Recall that the posterior probability is (proportional to) the prior times the likelihood . The odds ratio of horses over zebras is the ratio of the posteriors, , where is the prior on horses, is the likelihood of hearing hoofbeats from horses, and so on. Hoofbeats have the same likelihood for horses as they do zebras, so and they cancel, giving an odds ratio of just , the ratio of the priors. But I live in America! My prior on horses is a lot higher than my prior on zebras! . So the odds ratio is high above 1, and I’d guess horses. But all the action that causes me to guess horses is in the prior, and we never need to use the notion of simplicity—heck, for this, we don’t even need Occam’s razor.