71–80 of 227 posts
You know that a coin has 2 faces (known=2) and that if you toss it, 1 face will be up (unknown which=1).
The ratio is more about something in the mind than intrinsic to the objects.
That helped me understand why in the Monty Hall problem, when you switch doors, the probability of getting the prize increases.
> The scientific method only works because the rules of the universe happen to be simple, while the set of observations it offers is vast. Kolmogorov complexity captures this defining characteristic of our reality. I've never seen this spelled out so beautifully!
We could state that our simple rules works, but what does it mean "to work"? For example, a spider sees reality not like us, it feels vibrations of it's web, runs to a source of vibrations and start to bite, to wrap intruding object with web. It would do it to a tuning fork, if you pressed it to spider's web. His simple rules of reality works though. Despite the fact that sometimes spider bites steel of a tuning fork without any benefits for the spider.
How could we know that our theories not just extended version of spider's? With the same issues, like they make us to do something absolutely pointless. How could we evaluate this fact? To ask our theories? But our theories already predicted that this pointless thing we do would be a good thing. We might ask our theories again and we'd get the same answer.
This statement seems as a tautology for me. Our rules are simple, because they are simple. Our theories work because they tell us, that they work.
Earlier quoted context omitted.
> the amount of error a perfect Bayesian makes as they observe event and bet on the next one, ad infinitum, is finite. There's a little assumption that you're leaving out, namely that the Kolmogorov complexity of the data generating process is finite . From Wikipedia: > expected cumulative errors made by the predictions based on Solomonoff's induction are upper-bounded by the Kolmogorov complexity of the (stochastic)…
If quantum is really random, then our universe has infinite Kolmogorov complexity.
So even if quantum is really random, I'd bet an unbiased coin will still land on heads with 50% probability every time.
Nice article, on the whole, and usefully provocative. But I have misgivings about making these close connections between information theory and scientific theory-making. As everyone knows, information theory leaves out any notion of semantics, as it should. But the important thing about our theories of the world is that they have meaning to us. The scientist searches for something that makes sense of the world, not a…
Tow things. First, probability is real: it's a construct in one's mind, and minds are just as real as dice or coins. https://www.lesswrong.com/posts/f6ZLxEWaankRZ2Crv/probabilit... Second, (and the author may be leading up to this), there's Solomonoff's theory of inductive inference, which he has proven complete : when we apply Occam's razor (where the prior probability of each possible theory drops exponentially wit…
> the amount of error a perfect Bayesian makes as they observe event and bet on the next one, ad infinitum, is finite. There's a little assumption that you're leaving out, namely that the Kolmogorov complexity of the data generating process is finite . From Wikipedia: > expected cumulative errors made by the predictions based on Solomonoff's induction are upper-bounded by the Kolmogorov complexity of the (stochastic)…
Every time there was a significant advance in physics, it tended to go towards simplification and unification. Geocentrism required epicycles. Then Keppler came up with his ellipses. Then Newton unified celestial and terrestrial laws. Maxwell & Einstein allowed us to view time as less special dimension than we thought it was…
I won't presume about the initial state of the universe, to the extent such a notion is even meaningful. But the fact that it is governed by mathematics, and relatively simple maths at that, sounds likelier and likelier every quarter-century.
And I'm not even talking about everyday life, where we can observe in practice that the simplest theories about who ate the last cookie (little Mike, who lives in the house) are more often true than the more outlandish ones (magical imps, which we never witnessed).
Maybe, as philosopher Robert Pirsig theorized in "Zen and the Art of Motorcycle Maintenance", you must rely on Quality!
Earlier quoted context omitted.
I don't think it's demonstrably false: If you don't know that the coin is weighted, the probability is 50%. Probabilities are predictions and estimates, not fundamentally about the thing itself, but about what we know about the thing.
> If you don't know that the coin is weighted, the probability is 50%. No, if it's weighted it's not 50%. Your prior probability is 50%, but neither a Bayesian nor a frequentist would claim the true probability is known before testing.
Earlier quoted context omitted.
> the amount of error a perfect Bayesian makes as they observe event and bet on the next one, ad infinitum, is finite. There's a little assumption that you're leaving out, namely that the Kolmogorov complexity of the data generating process is finite . From Wikipedia: > expected cumulative errors made by the predictions based on Solomonoff's induction are upper-bounded by the Kolmogorov complexity of the (stochastic)…
If quantum is really random, then our universe has infinite Kolmogorov complexity.
Under the many-world interpretation, when you send a photon through a half sieved mirror, the universe splits in one version where the photon goes through, and one universe where the photon doesn't. This is all very deterministic.
What the researcher subjectively observe however is another matter. If the universe splits, so does the researcher. The problem of observing outcomes turns into an anthropy problem: if I split myself in two copies, in which copy am I likeliest to find myself into? I'm not sure making bets about that even makes sense: which copy I find myself into has no bearing in the final state of the universe.
The fluid nature of probability is the center of the insurance universe. Probability is always a moving target in the insurance world. Indeed, if it weren't, there wouldn't be much of a need for actuaries. Much of actuarial training revolves around the idea of credibility -- how credible is your sample set, what alterations should you make to old data to make it relevant to today, and what data should you add to it as a complement in order to relieve the model of the biases inherent in your sample size. This is inherently Bayesian in it's approach.
Where it truly gets interesting is that insurance companies are very cognizant of tail risk -- the 1-in-100, 1-in-250, 1-in-500 events that can cause insurer insolvency if not properly accounted for. You can survive a miscalculated loss trend within reasonable bounds, but if you haven't thought about the potential Cat 5 hurricane that hits Miami-Dade then you are going to have some very unhappy investors. When it comes to these types of events, you mostly need to be in the right ballpark. The order of magnitude matters more than the exact number -- albeit the exact number matters quite a bit for regulatory reasons. This type of calculation for property lines has largely been outsourced to the stochastic models developed by companies such as AIR and RMS. A sudden change in their models, which I think is likely after this record breaking hurricane season, can inflict capital pressure on the industry almost instantly.[1]
There are some actuarial papers from around 50 years ago that discuss information entropy as another way to approach the issue of constructing probability models, but they never really caught on. It seems that is likely due to the lack of widespread computing power. I'm hoping these ideas can gain some steam now that we can construct some of these distributions from Python and R.
[1] There is a fantastic article by Michael Lewis that describes this issue at great detail: https://www.nytimes.com/2007/08/26/magazine/26neworleans-t.h...