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Comment by tbabb | Hacker News Reader

Each additional unchecked assumption that is added to a theory is another opportunity to be wrong. These probabilities of failure, however small, are compounded with multiplication, so the probability of failure rises exponentially with the number of unchecked assumptions. (10 assumptions each with 95% chance of correctness have only a collective 59% chance of being right— .95^10 = .59).

To maximize the chance of choosing a model which corresponds to reality, we must involve as few unchecked assumptions as possible when deciding between theories that agree with the data; and this matters more than any particular assumption having a relatively high chance of being right, because of this exponential sensitivity.

So if we want theories that are correct, yes, we should value simplicity.

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I don't understand how we can reason with "probabilities of failure" about such fundamental things such as laws of physics.

Under the frequentist interpretation of probability, 95% of having a "correct assumption" means that as you observe iid experiments of this assumption, the long-run average of them will have 95/100 of such experiments exhibit the assumption as true. But "experiment" here isn't a physics experiment, it's a _universe_ in which this assumption could be true. We're in some _fixed_ universe. This assumption holds in ours with probability 0 or 1 (we don't know which).

So, I think the 95% you're referring to here is a Bayesian belief level. This is coherent in that it doesn't require some ambient set of multiple universes to meaningfully describe what is meant by "probability", but the prior required for this interpretation is a bit of weird beast. That is, we have some prior over the space of all possible assumptions for models of physics and update our beliefs based on evidence (we can collect evidence from our single universe multiple times and update our beliefs in a coherent way). In the limit of evidence, this prior matters less, but constants matter here! How much evidence do we need before we can be confident the "belief levels" we're throwing around aren't that subjective anymore? We don't really have a good sense for what the structure over this space of "assumptions of physical models" is, so we can't really answer this question.

Within specific settings for statistical learning theory, we find that simple models generalize well. But that's an implication: _if_ you have a class of simple models and it fits the data well, _then_ you'll generalize well. When it comes to answering the question, "Which theories generalize well?", such analysis is incomplete.

> I don't understand how we can reason with "probabilities of failure" about such fundamental things such as laws of physics.

A sketch: Suppose we have two theories, one ("A") with assumptions (a, b); the other ("B") with assumptions (a, p, q, r, s, t); and our evidence 'e'.

Pr(A) ~= Pr(a|e) * Pr(b|e)

That is, the probability of each theory being right is the probability that each of its assumptions are simultaneously true. It might be difficult to come up with a specific number for each of those component terms, but we'd do well to estimate the total probability of each theory simply by counting the terms, since in the limit that will matter more than the probability of each term (assuming we think none of them are obviously low).

Also note that when counting, we can trivially factor out the common assumptions "a" (a.k.a. the things we don't wish to doubt or vary between theories, e.g. QM in "ordinary" regimes, GR, the Newtonian approximation to GR, etc.)

Yes, there is a "ground truth" theory which is absolutely true or false, but we don't have access to it. And I don't see how it's more problematic to use probability here than on any other classical event on which we have imperfect information, like a specific dice roll, or a baseball game, or the outcome of an election— One specific thing will happen, and no other outcome was possible, but we can still use probability to model our incomplete knowledge. How is physics different?

Example stress tests of this idea:

e = the motion of the planets; A := "G = m1*m2/r^2"; B := [long list of epicycle parameters] --> pick "A".

e = the varied appearance/adaptations of animals/species; A := [reproduction, inheritance, variation, selection]; B := foreach animal x {"God zotted $x into existence like that because just_so_story($x)" --> pick "A".

e = my empty garage; A := "there's nothing in it"; B := "there's a dragon in it; the dragon is invisible; the dragon dodges your touch; the dragon has no heat signature; the dragon floats and leaves no footprints; ..." --> pick "A".

e = the behavior of the universe; A := [the standard model]; B := [the standard model; also it's a simulation; there are intelligent beings who set up the simulation; there is an external universe in which the simulation is occurring; the number of simulations happening in this universe is large; ...] --> pick "A".

I think that you may have missed my point.

First off, why are assumptions independent? Why are you allowed to factor p(a|e) * p(b|e) = p(a,b|e)? Assumptions aren't just independent binary variables -- it's not necessarily true that you can have some product measure over the set of assumptions (a, b, p, q, r, s, t) simultaneously.

More importantly, you dived write into some notation (Pr) without telling me what it _means_, which is what my OP was about.

> And I don't see how it's more problematic to use probability here than on any other classical event on which we have imperfect information, like a specific dice roll, or a baseball game, or the outcome of an election— One specific thing will happen, and no other outcome was possible, but we can still use probability to model our incomplete knowledge. How is physics different?

Here's a crucial difference. We can roll a die 100 times, 1000 times, 10K times, and the frequency of a six landing as we increase the number of trials will tend to 1/6. That's what we mean (if we're frequentists) when we say the probability of a six landing is 1/6. We can't "roll" universes with physics models.

> First off, why are assumptions independent?

Because I've defined them that way. I mean them to be independent choices you could make when designing your model that could be varied to fit the data. If two aspects of the model are not independent; i.e. they are covariant in some way, then there is some common parameter that explains them both, and that parameter is the one that should be seen as an input to the model.

> We can roll a die 100 times, 1000 times, 10K times [...] That's what we mean (if we're frequentists)

We're not frequentists.

You can't "re-roll" the 2016 election 10K times, either. There was only one, and there was only one way it could come out; we just didn't know enough to say what it would be before it happened. All the particles in all the voters were obeying the laws of physics at every moment; never was there any freedom for a different outcome. Nonetheless, even though there was/is only one "ground truth" that could ever be, we assigned probabilities to each possible outcome, given our incomplete knowledge.

This is a pretty standard application of probability. State estimators (e.g. the Kalman filter) are doing the same thing— you have some noisy readings of reality, and you use Bayesian logic on some assumed probability distributions to pick the estimate from the space of possible "ground truths" that has the highest probability of being the right one.

Concretely: I'm measuring roll rate, local acceleration, compass heading, barometric pressure, and GPS, all with significant error, and I want to know where my quadcopter is most likely to be at the current moment. There is only one true answer to that question, the quadcopter is in one place, not 10,000 places (or 10,000 flights), there is a single ground truth. But Bayes will give me a probability, given my readings, that any given estimate is the true ground truth (and some math will help me solve for the highest one).

In this case, instead of assigning probabilities to possible election outcomes or system state "ground truths", the "configuration space" is models of reality. But all we've changed is the domain of our probability distribution; the math doesn't care what kind of thing our "ground truth" represents. And it doesn't matter if reality contains only one "ground truth" or many; the fact is that we are choosing between many options (and we are ranking them by likelihood).

Re independence: identifying whether or not these physical assumptions covary is not that easy. That's my point: assumptions a, b, c, d could easily have some mutual incompatibility that makes them non-independent. It's an active area of research.

Re probability, I'm glad you committed to the Bayesian interpretation. Bayes gives you a _degree of belief_, based on your priors.

It's quite fortuitous that you mention the 2016 election. As you say, there's only one instance here. Which is why the prior matters a lot. We can incorporate (partial) evidence from past elections, but it's going to be very sensitive to the priors that we place, since the net amount of evidence we're working with is very small.

As we found out in 2016, that means these beliefs aren't worth much in such low data scenarios, since the prior has a large impact! https://projects.fivethirtyeight.com/2016-election-forecast/

This brings me to my original point:

> In the limit of evidence, this prior matters less, but constants matter here! How much evidence do we need before we can be confident the "belief levels" we're throwing around aren't that subjective anymore? We don't really have a good sense for what the structure over this space of "assumptions of physical models" is, so we can't really answer this question.

> identifying whether or not these physical assumptions covary is not that easy

But still tractable, I'd say. My core claim is that counting independently-variable assumptions will be a highly performant way to select between theories which agree with the data. Or put another way, it's the best Fermi approximation calculation for measuring "how good is your theory". How you do that for any given theory, while important to do correctly, is an implementation detail which I think is secondary to the discussion of whether doing it at all is a good idea. :) (seems like we might agree that it is?)

> We can incorporate (partial) evidence from past elections, but it's going to be very sensitive to the priors

To the extent that election forecasts are unreliable, I think that's because they are forced to involve a lot of assumptions (e.g. similarity to past elections) that turn out not to correspond well to reality. Models which make fewer such assumptions will do likely do better! (and IMO fivethirtyeight's forecasts did the best job of this out of any; most of the rest put Hillary at around 97%).

Unfortunately with elections, there is a comparatively high lower bound on the number of assumptions we must make, thanks to the complexity of their dynamics and sparsity of data/knowledge we have about each. I think this is much less the case with physics, where we are varying comparatively small physical assumptions to explain mountains of data. But in either case, I contend that the most performant models will make fewer (unmeasured) independent assumptions.

> How much evidence do we need before we can be confident the "belief levels" we're throwing around aren't that subjective anymore?

The point I'm trying to make is bigger-picture than the above level of detail: Counting independent assumptions, in the limit, matters more than the specific constants of each assumption (assuming they're not low/zero), precisely because it's so hard to come up with "accurate" numbers for each.

That is to say, the probability is not sensitive to those belief levels: We could choose widely varying distributions for the probabilities of our assumptions, including choosing probabilities very close to 1, and it will hardly ever matter to the total probability as much as the absolute number of independent assumptions we make.

> So if we want theories that are correct, yes, we should value simplicity.

I'm reminded of a problem I was given on a math exam: a hand-drawn polynomial on a 5x5 grid, where the curve intersected the grid in 5 points. "Find the equation of this cubic polynomial."

The given integral points weren't possible for any cubic; if the function was a polynomial, it was a quartic at a minimum.

Sadly, my teacher was not trying to trick us. Simplicity, itself, was an unchecked assumption which rendered correctness impossible.

>> So if we want theories that are correct, yes, we should value simplicity.

There is no a priori reason why simple theories should be correct. The suggestion that any model can be correct is presumptuous - you might call it an assumption ;-) To borrow from George Box, all models are wrong, but some are useful...

Sure there is! The Universal Distribution is the ultimate prior, see for example, Solomonoff induction: https://en.m.wikipedia.org/wiki/Solomonoff%27s_theory_of_ind...

Assumption 1: Whathisface's universal star force lightning theory is true. Assumption 2: No other assumptions.

General relativity has a couple postulates, my theory has one. I guess I'm more likely to be right.

I suspect you're being willfully dense.

The theory you paint would be composite of other assumptions (involving lighting, forces, or stars, e.g.?) which in turn are either in contradiction with observation (implying Pr=0%), or rely on further (very low probability) assumptions in order to avoid contradiction.

If that's not true, then you've just given a strange name to the empty set of assumptions, which is the null hypothesis; and the null hypothesis DOES get epistemological privilege.

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