Comment by canjobear | Hacker News Reader

Comment by canjobear | Hacker News Reader

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What's often lost in the discussions about whether entropy is subjective or objective is that, if you dig a little deeper, information theory gives you powerful tools for relating the objective and the subjective.

Consider cross entropy of two distributions H[p, q] = -Σ p_i log q_i. For example maybe p is the real frequency distribution over outcomes from rolling some dice, and q is your belief distribution. You can see the p_i as representing the objective probabilities (sampled by actually rolling the dice) and the q_i as your subjective probabilities. The cross entropy is measuring something like how surprised you are on average when you observe an outcome.

The interesting thing is that H[p, p] <= H[p, q], which means that if your belief distribution is wrong, your cross entropy will be higher than it would be if you had the right beliefs, q=p. This is guaranteed by the concavity of the logarithm. This gives you a way to compare beliefs: whichever q gets the lowest H[p,q] is closer to the truth.

You can even break cross entropy into two parts, corresponding to two kinds of uncertainty: H[p, q] = H[p] + D[q||p]. The first term is the entropy of p and it is the aleatoric uncertainty, the inherent randomness in the phenomenon you are trying to model. The second term is KL divergence and it tells you how much additional uncertainty you have as the result of having wrong beliefs, which you could call epistemic uncertainty.

Thanks, that's an interesting perspective. It also highlights one of the weak points in the concept, I think, which is that this is only a tool for updating beliefs to the extent that the underlying probability space ("ontology" in this analogy) can actually "model" the phenomenon correctly!

It doesn't seem to shed much light on when or how you could update the underlying probability space itself (or when to change your ontology in the belief setting).

This kind of thinking will lead you to ideas like algorithmic probability, where distributions are defined using universal Turing machines that could model anything.

Amazing! I had actually heard about solomonoff induction before but my brain didn't make the connection. Thanks for the shortcut =)

You can sort of do this over a suitably large (or infinite) family of models all mixed, but from an epistemological POV that’s pretty unsatisfying.

From a practical POV it’s pretty useful and common (if you allow it to describe non- and semi-parametric models too).

Couldn't you just add a control (PID/Kalman filter/etc) to coverage on a stability of some local "most" truth?

Could you elaborate? To be honest I have no idea what that means.

I think what you're getting at is the construction of the sample space - the space of outcomes over which we define the probability measure (e.g. {H,T} for a coin, or {1,2,3,4,5,6} for a die).

Let's consider two possibilities:

1. Our sample space is "incomplete"

2. Our sample space is too "coarse"

Let's discuss 1 first. Imagine I have a special die that has a hidden binary state which I can control, which forces the die to come up either even or odd. If your sample space is only which side faces up, and I randomize the hidden state appropriately, it appears like a normal die. If your sample space is enlarged to include the hidden state, the entropy of each roll is reduced by one bit. You will not be able to distinguish between a truly random coin and a coin with a hidden state if your sample space is incomplete. Is this the point you were making?

On 2: Now let's imagine I can only observe whether the die comes up even or odd. This is a coarse-graining of the sample space (we get strictly less information - or, we only get some "macro" information). Of course, a coarse-grained sample space is necessarily an incomplete one! We can imagine comparing the outcomes from a normal die, to one which with equal probability rolls an even or odd number, except it cycles through the microstates deterministically e.g. equal chance of {odd, even}, but given that outcome, always goes to next in sequence {(1->3->5), (2->4->6)}.

Incomplete or coarse sample spaces can indeed prevent us from inferring the underlying dynamics. Many processes can have the same apparent entropy on our sample space from radically different underlying processes.

Right, this is exactly what I'm getting at - learning a distribution over a fixed sample space can be done with Bayesian methods, or entropy-based methods like the OP suggested, but I'm wondering if there are methods that can automatically adjust the sample space as well.

For well-defined mathematical problems like dice rolling and fixed classical mechanics scenarios and such, you don't need this I guess, but for any real-world problem I imagine half the problem is figuring out a good sample space to begin with. This kind of thing must have been studied already, I just don't know what to look for!

There are some analogies to algorithms like NEAT, which automatically evolves a neural network architecture while training. But that's obviously a very different context.

We could discuss completeness of the sample space, and we can also discuss completeness of the hypothesis space.

In Solomonoff Induction, which purports to be a theory of universal inductive inference, the "complete hypothesis space" consists of all computable programs (note that all current physical theories are computable, so this hypothesis space is very general). Then induction is performed by keeping all programs consistent with the observations, weighted by 2 terms: the programs prior likelihood, and the probability that program assigns to the observations (the programs can be deterministic and assign probability 1).

The "prior likelihood" in Solomonoff Induction is the program's complexity (well, 2^(-Complexity), where the complexity is the length of the shortest representation of that program.

Altogether, the procedure looks like: maintain a belief which is a mixture of all programs consistent with the observations, weighted by their complexity and the likelihood they assign to the data. Of course, this procedure is still limited by the sample/observation space!

That's our best formal theory of induction in a nutshell.

Someone else pointed me to Solomonoff induction too, which looks really cool as an "idealised" theory of induction and it definitely solves my question in abstract. But there are obvious difficulties with that in practice, like the fact that it's probably uncomputable, right?

I mean I think even the "Complexity" coefficient should be uncomputable in general, since you could probably use a program which computes it to upper bound "Complexity", and if there was such an upper bound you could use it to solve the halting problem etc. Haven't worked out the details though!

Would be interesting if there are practical algorithms for this. Either direct approximations to SI or maybe something else entirely that approaches SI in the limit, like a recursive neural-net training scheme? I'll do some digging, thanks!

Correct anything thats wrong here. Cross entropy is the comparison of two distributions right? Is the objectivity sussed out in relation to the overlap cross section. And is the subjectivity sussed out not on average but deviations on average? Just trying to understand it in my framework which might be wholly off the mark.

Cross entropy lets you compare two probability distributions. One way you can apply it is to let the distribution p represent "reality" (from which you can draw many samples, but whose numerical value you might not know) and to let q represent "beliefs" (whose numerical value is given by a model). Then by finding q to minimize cross-entropy H[p, q] you can move q closer to reality.

You can apply it other ways. There are lots of interpretations and uses for these concepts. Here's a cool blog post if you want to find out more: https://blog.alexalemi.com/kl-is-all-you-need.html

I'm not sure what you mean by objectivity and subjectivity in this case.

With the example of beliefs, you can think of cross entropy as the negative expected value of the log probability you assigned to an outcome, weighted by the true probability of each outcome. If you assign larger log probabilities to more likely outcomes, the cross entropy will be lower.

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