YHK - yet another hacker news clone

Those interested in this should read about DSMC: https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo. It’s the same idea for low density gas interactions without having to actually simulate all the particles involved while still getting accurate results. Cool stuff!

> After running 10 million simulated draws, I got an answer of 0.0323, which matches the real answer very closely.

If you enjoyed reading about how you can approximate complex statistical values using basic programming, I highly recommend Jake Vanderplas "Statistics for Hackers" [0][1].

It converts the sometimes hard to grok statistics terms and formulas into some very simple loops and basic operations.

Can't recommend it enough.

0 - https://speakerdeck.com/pycon2016/jake-vanderplas-statistics...

1 - https://www.youtube.com/watch?v=Iq9DzN6mvYA

There's been a lot of interesting work in the last ~decade on improving monte-carlo methods for solving PDEs, the field is really fascinating!

https://github.com/rohan-sawhney/mcgp-resources

> Can't differentiate the function? Fuck gradients. Here's the entire fucking posterior.

- Monte Carlo

I use a Monte-Carlo for my retirement planning. This did a good job of telling me not to trust it but didn't really give me a better option. Anyone have one?

I was working on a retirement planning simulation this week with Opus and it added this simulation into my tool! Had never heard of Monte-Carlo before but was fairly impressed.

If you are doing combinatorial sampling like estimating pi or doing probabilities with SET, stratification is a great idea

Here's a few related ideas:

1) rejection sampling - this is the name to the general technique used in the first example, to estimate pi by sampling from a uniform distribution over a unit square (easy) and then rejecting the samples that fall outside of the distribution you're actually interested in (unit disk).

Another simple example of rejection sampling is if you want to randomly pick between 5 ordered options and only have a d6. Roll the d6. If you roll a 6, reject the roll & re-roll. If you roll 1-5, that's your choice.

2) the Metropolis-Hastings algorithm -- this is a Markov chain Monte Carlo algorithm. It can be useful in some applications to sample from a distribution that is difficult to sample from.

Suppose you have some system described by a state vector x & you know that the probability of the system being in state x, p(x), is proportional to q(x), where q is some function you can compute. If you could integrate q over x, to give Z := integral_{x in X} q(x) dx then you'd have p(x) = Z^-1 q(x)

For a concrete example, x might be a state vector of a physical system -- say a bunch of molecules in a box, where you can compute the energy of any given state u(x) and know the probability of x, p(x), is proportional to exp(-u(x)).

Metropolis-Hastings is neat as it gives you a way to sample from p even if you cannot compute Z or sample from q --- you just need to have some way to sample a proposed a new candidate state x' from x (this proposal distribution doesn't need to have any relationship to the distributions p & q you're actually interested in). Can often think of the proposed candidate state x' as some kind of local mutation of x. Provided you can compute q(x') / q(x), you can use that ratio as a probability of accepting the move to replace x with candidate x'.

By the algorithm depending on the ratio of probabilities of two different states p(x') / p(x) = q(x') / q(x), it lets you sample from p even if you can't compute Z or have any direct way to sample from p or q. *

Metropolis-Hastings & Markov Chain Monte Carlo methods are definitely in the territory of clever 'what to do if you don't know what to do' methods --- if there's some other more direct & practical way of computing the thing you're trying to compute, use that instead! But if there's not, & you have some gnarly problem, maybe MCMC might be the best option available.

* caveat caveat caveat: provided you let the sampler run long enough for it to 'burn in', & perhaps restart it a few times from different initial states x_0 to reduce the chance of it getting trapped in some unrepresentative local optima

As a statistician, I’d be remiss if I didn’t mention bootstrapping - theoretically distinct, but another demonstration of the power of randomization techniques.

I (and Claude) have been working on an open source monte carlo based retirement tool for quite a while now. Its definitely geared towards people with more individual financial ownership and tech knowledge but I feel it gives good flexibility for me to model my situation. I had initially gravitated to this project because I think Monte Carlo and numerical methods are just so dang fun.

https://github.com/jgrazian/finplan

Try it live: https://finplan.rayknot.com

I wrote a Monte-Carlo simulation while working on a Pokemon-Go style game to simulate people walking all over the world. I used statistics for average walking speed and times played for similar games. To my amazement when the game launched, my predictions were within +/- 2%! A career highlight.

That brings back memories!

I wrote Monte-Carlo simulations to calculate the thermodynamics of oxygen ordering in a high temperature superconductor way back in 1993.

Calculating thermodynamic fluctuations on a 256x256 lattice took all night. And that, even though it ran on an HP 9000/750 with a 66 MHz PA-7000 (PA-RISC 1.1) 32-bit CPU.

While waiting, I used to look at weather satellite images downloaded from CERN using NCSA Mosaic. It was impressive. All the information available at your fingertips by using arrow keys and pressing return on underlined text to navigate to the next page! Emmm, link.

If you want to look further on this fascinating topic

[1]: https://austinrochford.com/posts/intro-prob-prog-pymc.html [2]: https://vincentk1991.github.io/Bayesian-regression-tutorial/

I can't recommend enough

[3]: https://academic.oup.com/book/54722

the best book on the subject with a Coursera series of top quality