On this, the 20th day of April, I for some reason get the itch to vamp on some Jefferson Airplane

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Modular synthesizers are truly the nerd's instrument. I say this as a complete and total nerd. I'm loving this!

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More synth stuff. This time exploiting the fact that two sequences of different prime numbers won't repeat too often

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If you want to follow my music stuff, I post it all here: www.facebook.com/blandmusicis...

In the low light I managed to forget which buttons where which on my sequencer, so I ended up with a much more random melody than I was planning for, but it all worked out OK. 😂

I had lots of fun playing my synth for the first time in public last night at an open mic night. I learned heaps, and will take it with me for next time.

Now, the sound is a nice one, and 88-note polyphony comes standard, but a grand piano's worth of money buys a lot of synth, and you don't have to buy it all at once.

Also more on the cost, I think it needs to be put into context. I *could* be a pianist and want a new piano. That's thousands of dollars for an instrument that makes exactly one kind of sound.

And sometimes synth-ing can be *really* cheap if you lower your standards enough and build it yourself: www.facebook.com/share/r/1CjD...

Only exceptions in my rack are the Beringer synth voice, which was my first module, and the unity gain mixer, which I couldn't find a good kit for.

Ah yes. It isn't a cheap hobby. My cost-containment device is that I only buy stuff I have to assemble myself. They're considerably cheaper that way and I like the soldering process.

Hello to you too!

Do you have any of your music posted?

I've had so much fun with this since it become my COVID hobby. There's a lot to like for a nerdy amateur musician who loved learning about sound and electronics in physics in high school.

Even more musical stuff today 😁😁

More music stuff

In the case of (2) if DE can't find that mode, then PT might be worth a try. It might also be worth flattening out the off-equilibrium penalty but not the likelihood function like this

2. If I am trying to estimate a model, then my belief about the posterior is that it will most likely have one global mode, and that's what I want to find. Maybe two in very knife-edge cases.

I think it also depends on what you're trying to do.

1. If I am trying to find equilibria, then ideally each chain will get stuck in different equilibria. HMC does just fine here, and Stan will give you lots of errors that the chains haven't converged (which is a good thing).

I will check it out then, thanks!

Do you think Claude is making sense here?

It makes sense that the PT sampler would explore the whole simplex at high temperatures, which might be inefficient. Whereas DE-MH should jump between modes once it finds them.

Not yet, but I'm always on the lookout for a better sampler for this problem. Thanks!

I'm just learning about these other samplers now. The learning curve is steep.

If parallel tempering works and it can work in Stan then that would be great for this project!

But I think the thing I'm most excited about is that the estimation part. It can handle multiple branches. So if the game you're studying has multiple branches, this might be a good estimation strategy.

It could also be helpful for finding points on the non-principal branch, you could then use these points as starting values for the predictor-corrector algorithm.

So when should you use this technique? Firstly, *don't* use it if you know for sure you have just one branch. The predictor-corrector algorithm will do just fine here.

If you want a diagnostic for multiple equilibria, then this could be helpful.

So instead, I implemented a Differential Evolution Markov Chain, which is very good at moving between modes. This sampler had no trouble finding the global modes, even if they started close to the wrong branch.

And here's the new bit for this updated version: I found a sampler that can handle multiple modes! The problem was with the Hamiltonian MC sampler I was using, not the posterior distribution itself. Basically, HMC (Stan in my case) has trouble jumping between the branches.

But here I ran into a problem: Some chains settled into *local* modes of the posterior. These are typically the most likely QRE to have generated the data on the wrong branch. That is, chains have a hard time moving between the branches.

There's more! QRE isn't just something we want to compute. It's also something we want to *estimate*. With data from an economic experiment, it is not hard to modify the target distribution with a likelihood.

So this can be used to diagnose whether a game has multiple equilibria. Just make sure you spread out your initial conditions for the MCMC chains, and run lots of chains

Furthermore, if you find multiple equilibria and you're using something like Stan to do your simulation, then your convergence diagnostics will scream at you that the chains haven't converged, because they will settle in to different modes.

Start your MCMC chains at different points and the sampler will have a good chance of finding multiple equilibria.

Here you can adjust a tuning parameter (w in the Figure) to decide how much to penalize off-equilibrium mixed strategies.