(ML 18.3) Stationary distributions, Irreducibility, and Aperiodicity

[@kriti3011]

Fantastic video, clarified things i cudnt find anywhere else

[@joshuamhenrich]

These are seriously great. A next step would be making a PDF or something with your basic definitions you define here. Keep up the great work.

[@pelemanov]

For those who struggle with the same issue, check out video 18.6 starting at 06:25. Very clear intuition there and even explaining the name :-).

[@LeilaRmaths]

very valuable video.. Thank you so much

[@martinsdundurs9497]

Explanation of aperiodicity was interesting. Instructor, however, didn't make the final jump of conclusion there. It's interesting how periodicity arises in the context of 11:01

[@xxRAP13Rxx]

For your definition of an ergodic markov chain at the beginning, is it necessary that the markov chain is time-homogenous? Can a markov chain have its transition matrix change after each step and still be ergodic?

[@muxecoid]

I think if initially it is different than what it eventually settles into your MC is simply not time-homogenous.

[@pelemanov]

I mean, I understand it mathematically of course, but I don't understand it intuitively. What does this mean, why is this a nice property, a requisite for MCMC? What does it mean intuitively to multiply this row with T?

[@SuperGZK]

In MCMC it is standard practice allow the MC to run for some number of steps until it stabilizes. The initial behavior depends on the choice of starting point and is not representative of the long term behavior.

[@christosmichaelides1988]

Yes. There's typically a 'burn-in' period. This is a pre-determined amount of steps (e.g 2000) that we may choose to disregard completely. We are basically 'throwing away' some initial steps because we only care about the high probability region.

[@nonindividual]

(A very minor point) Your example following the definition of _aperiodic_ is slightly incorrect: because if k \in R_a, then M_k \subset R_a. (So that if 2 \in R_a, then surely 4 \in R_a.)

[@Jacob011]

It seems to me that the mechanics behind the PMF stationarity condition is finite dimensional analogue of the reproducing property in RKHS theory.

[@mbpm3]

what are you writing on?

[@Jacob011]

I strongly suspect he's using SmoothDraw with some graphics tablet, just like KhanAcademy.

[@Paivren]

what do the short terms 'pmf' or in one of the earlier videos 'pdf' mean?

[@dancetime4me]

Probability Mass Function (for discrete variables) and Probability Density Function (for continuous variables)

[@pelemanov]

Nice video, but why not focus more on the name of the property? I find that understanding the name makes it easier to understand (and remember!) the property. It also reduces the notions of aperiodicity to a very simple concept. I don't really understand the concept of stationarity though...

[@hahahaha4444]

If a mk other than m1 cotains ra then it is not aperiodic?

[@hongyangli5811]

No, it is not aperiodic. In other words, it is periodic.

[@AV1461]

I think I understood the aperiodicity property ok, but I don't understand why that will be important. Maybe next videos will explain.

[@mikewbma]

Good Job. Too bad the more deeply we head in to stochastic process. The less views you will have :(.