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nhaliday : fixed-point   9

Is there a common method for detecting the convergence of the Gibbs sampler and the expectation-maximization algorithm? - Quora
In practice and theory it is much easier to diagnose convergence in EM (vanilla or variational) than in any MCMC algorithm (including Gibbs sampling).

https://www.quora.com/How-can-you-determine-if-your-Gibbs-sampler-has-converged
There is a special case when you can actually obtain the stationary distribution, and be sure that you did! If your markov chain consists of a discrete state space, then take the first time that a state repeats in your chain: if you randomly sample an element between the repeating states (but only including one of the endpoints) you will have a sample from your true distribution.

One can achieve this 'exact MCMC sampling' more generally by using the coupling from the past algorithm (Coupling from the past).

Otherwise, there is no rigorous statistical test for convergence. It may be possible to obtain a theoretical bound for the convergence rates: but these are quite difficult to obtain, and quite often too large to be of practical use. For example, even for the simple case of using the Metropolis algorithm for sampling from a two-dimensional uniform distribution, the best convergence rate upper bound achieved, by Persi Diaconis, was something with an astronomical constant factor like 10^300.

In fact, it is fair to say that for most high dimensional problems, we have really no idea whether Gibbs sampling ever comes close to converging, but the best we can do is use some simple diagnostics to detect the most obvious failures.
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october 2019 by nhaliday
gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.

...

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
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october 2017 by nhaliday
Controversial New Theory Suggests Life Wasn't a Fluke of Biology—It Was Physics | WIRED
First Support for a Physics Theory of Life: https://www.quantamagazine.org/first-support-for-a-physics-theory-of-life-20170726/
Take chemistry, add energy, get life. The first tests of Jeremy England’s provocative origin-of-life hypothesis are in, and they appear to show how order can arise from nothing.
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august 2017 by nhaliday
Lecture 6: Nash Equilibrum Existence
pf:
- For mixed strategy profile p ∈ Δ(A), let g_ij(p) = gain for player i to switch to pure strategy j.
- Define y: Δ(A) -> Δ(A) by y_ij(p) ∝ p_ij + g_ij(p) (normalizing constant = 1 + ∑_k g_ik(p)).
- Look at fixed point of y.
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june 2017 by nhaliday

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