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So V(x) = exp(-pi(x)) and you use HMC to explore those potential energy level sets? But what happens when you change coordinates and get a different density, p

So V(x) = exp(-pi(x)) and you use HMC to explore those potential energy level sets? But what happens when you change coordinates and get a different density, pi(x), a different potential energy, V(x), and hence different level sets? And why does the gradient help as it points transverse to the level sets and not along them which you would need to move in the right direction? Maybe the momenta fixes that somehow? So you just add any IID gaussian variables to the target? You need special variables? Even so, why IID gaussians? The problem with the "hockey puck" intuition is that it is incredibly shallow and obfuscates the underlying principles that really make the method work. Unfortunately the false intuition also motivates bad tuning strategies and method extensions which have real practical consequences. Radford's review was really, really important but it was written when we really didn't know what was going on at a foundational level. The linked paper has the benefit of the last five years of research where we've figured those foundations out and consequently can identify the principled important to robust use of the method in practice.

soVeryTired 10y ago
> So V(x) = exp(-pi(x)) and you use HMC to explore those potential energy level sets? Not quite. You explore the level sets of the augmented system of kinetic-plus-potential energy. The use of a Gaussian distribution as auxiliary variable is arbitrary, so it's not surprising that it's suboptimal. That the gradient information is needed comes from physical intuition. I roll a marble with along an inclined plane with velocity V. What determines the trajectory of the marble? The gradient of the plane. Now since differentiable functions look locally like planes, that's more or less all you need, at least instantaneously. I'm sure you know more about this topic than I do, but you'll have to work quite a bit harder to convince me that the hockey puck intuition is "incredibly shallow".
- betanalpha 10y ago
  The point of those rhetorical questions is that HMC is _not_ just about adding augmenting the system with additional parameters and following the gradients of the larger system. Even in your marble system, for example, the gradient of the plane does not affect the position of the marble directly. The gradient affects the velocity which then moves the marble. If the interactions aren't just right then you no longer have a physical system with energy conservation and all the other nice properties one might expect. In both the physical system of the marble and the pseudo-physical system created by HMC there is a deeper mathematical structure that relates gradients to motion to probability in exactly the right way. If that structure is compromised, say in a naive implementation of HMC, then all of the magic is lost and you don't get the performance that you might expect. It may sound like I'm being pedantic here, but it's only be understanding and respecting the underlying math far beneath the "hockey puck" description that we've been able to take HMC to the next level, especially in tools like Stan, with faster and more robust performance as well as sensitive diagnostics of failures.
  - soVeryTired 10y ago
    Yep, I can absolutely agree with that. It's not just the intuition that matters, the deeper details are important too.
    - betanalpha 10y ago
      +1. As someone who has struggled to understand math for a long time, I have come to appreciate that it's not about finding any "intuitive" description of a system but rather the one that's compatible with the underlying mathematics.

soVeryTired 10y ago

> So V(x) = exp(-pi(x)) and you use HMC to explore those potential energy level sets? Not quite. You explore the level sets of the augmented system of kinetic-plus-potential energy. The use of a Gaussian distribution as auxiliary variable is arbitrary, so it's not surprising that it's suboptimal. That the gradient information is needed comes from physical intuition. I roll a marble with along an inclined plane with velocity V. What determines the trajectory of the marble? The gradient of the plane. Now since differentiable functions look locally like planes, that's more or less all you need, at least instantaneously. I'm sure you know more about this topic than I do, but you'll have to work quite a bit harder to convince me that the hockey puck intuition is "incredibly shallow".

betanalpha 10y ago
The point of those rhetorical questions is that HMC is _not_ just about adding augmenting the system with additional parameters and following the gradients of the larger system. Even in your marble system, for example, the gradient of the plane does not affect the position of the marble directly. The gradient affects the velocity which then moves the marble. If the interactions aren't just right then you no longer have a physical system with energy conservation and all the other nice properties one might expect. In both the physical system of the marble and the pseudo-physical system created by HMC there is a deeper mathematical structure that relates gradients to motion to probability in exactly the right way. If that structure is compromised, say in a naive implementation of HMC, then all of the magic is lost and you don't get the performance that you might expect. It may sound like I'm being pedantic here, but it's only be understanding and respecting the underlying math far beneath the "hockey puck" description that we've been able to take HMC to the next level, especially in tools like Stan, with faster and more robust performance as well as sensitive diagnostics of failures.
- soVeryTired 10y ago
  Yep, I can absolutely agree with that. It's not just the intuition that matters, the deeper details are important too.
  - betanalpha 10y ago
    +1. As someone who has struggled to understand math for a long time, I have come to appreciate that it's not about finding any "intuitive" description of a system but rather the one that's compatible with the underlying mathematics.