Questions tagged [hamiltonian-monte-carlo]

Tag for questions related to Hamiltonian Monte Carlo.

Filter by
Sorted by
Tagged with
1 vote
0 answers
45 views

Hamiltonian Monte Carlo vs. "Metropolis-Hastings with a Hamiltonian step"

In Hamiltonian Monte Carlo the proposal is accepted with probability: $$ \alpha\left(\mathbf{x}_n(0),\mathbf{x}_n(L\Delta t)\right) = \min\left(1, \frac{\exp\left[-H\left(\mathbf{x}_n(L\Delta t),\...
user avatar
  • 1,580
0 votes
0 answers
16 views

Monte Carlo for Dirichlet Multinomial Model

Problem I am trying to implement Markov Chain Monte Carlo for the Dirichlet Multinomial mixture, described in this reference (where one used the expectation maximization algorithm). The model is as ...
user avatar
  • 1,580
1 vote
0 answers
24 views

Hamiltonian trajectory stays in the typical set?

I'm currently studying Hamiltonian MCMC by reading Betancourt's 2014 and Neal's 2011 pedagogical papers, but I still don't understand why following a Hamiltonian trajectory for our proposed update ...
user avatar
0 votes
0 answers
25 views

Hamiltonian Monte Carlo - Is the auxiliary momentum variable the time derivative of the position variable?

In Hamiltonian Monte Carlo the position $q$ and velocity $p$ variables follow Hamilton's dynamic \begin{align} \dot{q} &= \partial_p H(q, p) \\ \dot{p} &= -\partial_q H(q, p) \end{align} where ...
user avatar
  • 1,458
1 vote
0 answers
24 views

volume preservation in MCMC

In the paper of MCMC using Hamiltonian dynamics, there is the following statement on volume preservation. What does it mean exactly? I am not very clear about the ...
user avatar
  • 2,627
0 votes
0 answers
20 views

what is the advantage of using Hamilton dynamics in sampling methods? [duplicate]

I am wondering apart form being gradient based sampling methods, what is the advantages of using Hamiltonian MCMC?
user avatar
  • 133
0 votes
0 answers
21 views

Use Monte Carlo to produce new 'p' correlated data from existing data [duplicate]

As mentioned above, I have a problem where I need to generate new data Y from an existing data X such that Y is p correlated to X. I know their are several ways to do it but I want to know if monte ...
user avatar
1 vote
0 answers
37 views

Step-size adaptation of NUTS within Gibbs

I am trying to solve a hierarchical problem with a Gibbs sampler. I do not have closed-form expressions for the conditionals, thus I have to use another MCMC method within the Gibbs scheme to sample ...
user avatar
  • 33
0 votes
1 answer
112 views

Regarding Gibbs sampling and HMC in fitting Bayesian model, their differences and advantages

I have a question regarding the two MCMC algorithms, Gibbs sampling and Hamiltonian Monte Carlo (HMC) for performing the Bayesian analysis. If using Gibbs sampling, my understanding is that we need to ...
user avatar
  • 4,672
8 votes
1 answer
404 views

MCMC sampling for a model with a multinomial choice--so the parameters need to sum to 1

this is a head-scratcher for me, but a very interesting problem. So I have a stochastic simulation model for a hiring process. Basically different groups get hired into a company with different ...
user avatar
  • 1,118
2 votes
0 answers
114 views

Hamiltonian Monte Carlo (or Langevin Monte Carlo) on a Sphere

I want to perform Hamiltonian Monte Carlo (HMC) or Langevin Monte Carlo (LMC) on a spherical domain $\mathbb{S}^{D-1}$ embedded in a Euclidean space $\mathbb{R}^D$. My energy function is a deep neural ...
user avatar
1 vote
0 answers
40 views

NUTS Sampler: runaway stepsize

I've been trying to implement a NUTS sampler according to the Gelman 2014 paper, and I've been finding that my log-stepsize $\log \epsilon$ runs away towards $-\infty$ for reasonable values of $\delta=...
user avatar
  • 21
2 votes
1 answer
152 views

Can someone explain how dual averaging helps the No U-Turn Sampler (NUTS) choose step-size adaptively?

I have read both the original NUTS paper and also the dual averaging paper by Nesterov but due to my lack of background knowledge in optimisation, I don't really understand how dual averaging works. ...
user avatar
10 votes
1 answer
328 views

How does Hamiltonian Monte Carlo work?

I made the below graphic to explain how I currently understand the HMC algorithm. I'd like verification from a subject matter expert if this understanding is or isn't correct. The text in the below ...
user avatar
  • 107
1 vote
0 answers
44 views

What is the inference behind the momentum variable and the Kinetic energy for a weakly non-linear inverse problems in the HMC method?

We generate an auxiliary momentum variable in the HMC method to provide gradient for the propagation of trajectory (m, p) (model or position, momentum) in the phase space. If we look into Newton's ...
user avatar
  • 11
0 votes
1 answer
51 views

Can I do HMC with the wrong Hamiltonian?

I am a novice HMC user. I am reading Neal's chapter in the Handbook of MCMC. I think I can present the HMC algorithm as : Sample a new momentum Propose a new momentum and a new position using a ...
user avatar
3 votes
0 answers
55 views

Deriving a momentum proposal distribution for Hamiltonian Monte Carlo -- non-Gaussian kinetic energies

I am trying to understand how to derive the optimal way to generate momenta in HMC. In the gaussian case, I think the approach is that if one samples proportional to the Gaussian, the log likelihoods ...
user avatar
  • 203
2 votes
1 answer
35 views

For Hamiltonian Monte Carlo, what should be done when one of the steps in the leapfrog path yields no solution?

When estimating a very complex (potentially discontinuous) model with Hamiltonian Monte Carlo, what should be done when one of the steps in the leapfrog path yields no solution? The issue is that ...
user avatar
0 votes
1 answer
45 views

Strange substitution in HMC

I try to read paper, MCMC using Hamiltonian dynamics). The author, Neal states(P28): To begin, Cruetz nodes that the following relationship holds when any Metropolis-style algorithm is used to ...
user avatar
2 votes
0 answers
34 views

Optimal Scaling HMC proof

I'm reading the paper https://arxiv.org/pdf/1001.4460.pdf I get very confused when reading the author proof of the theorem (4.2) Here are few points. (1) The expected squared jump distance is ...
user avatar
  • 669
4 votes
1 answer
309 views

Why does Stan initialize an MCMC chain with a random value generated uniformly from [-2, 2] instead of a random value generated from the prior?

From Stan reference, The default is to randomly generate initial values between -2 and 2 on the unconstrained support It seems to me that it makes more sense to randomly generate initial values ...
user avatar
  • 7,484
7 votes
2 answers
416 views

Reconciling Langevin MC methods as one-step HMC versus as diffusion or brownian motion

I have a basic understanding of Hamiltonian monte carlo and why it works. I've read that Langevin MC is basically a special case of HMC when you only step the dynamics forward a single timestep before ...
user avatar
  • 23k
2 votes
1 answer
67 views

NUTS algorithm efficient transition kernel

I'm reading this paper, but I'm struggling to understand the following transition kernel. $T(w^{'}|w,\mathcal{C})=\left\{\begin{matrix} \frac{\mathbb{I}[w^{'}\in\mathcal{C}^{new}]}{|\mathcal{C}^{new}|...
user avatar
  • 75
1 vote
0 answers
85 views

Does specifying normalizing constant significantly improves Hamiltonian Monte Carlo?

From my understanding the energy function needs only be specified such that it is proportional to the log density, and not specifying the normalizing constant should not greatly impact the sampling ...
user avatar
5 votes
1 answer
68 views

Is there an HMC algorithm that estimates a model with noncontinuous parameters?

Is there an HMC algorithm that estimates a model with noncontinuous parameters? All of the intuition I have for how HMC surfs around in the phase space is based on examples for posterior distributions ...
user avatar
  • 18.6k
1 vote
0 answers
146 views

Why volume preservation is important for Metropolis update? [duplicate]

I think my question is naive but I would like to ask why why volume preservation is important for MCMC and specifically Metropolis update.I'm reading the following paper https://arxiv.org/pdf/1206....
user avatar
  • 75
0 votes
0 answers
22 views

Hamiltonian MCMC information gathering [duplicate]

I started gathering information about Hamiltonian MCMC and I would like to ask if someone knows some good papers or books.If it possible notes that give a detailed explanation of Hamiltonian MCMC. ...
user avatar
  • 75
3 votes
0 answers
138 views

Is the MC produced by HMC reversible?

I know that the deterministic dynamics in Hamiltonian Monte Carlo/Hybrid Monte Carlo are reversible and the numerical integrators one uses to approximate them are reversible too. But HMC consists of 2 ...
user avatar
  • 41
8 votes
1 answer
4k views

What is the purpose of "transformed variables" in Stan?

I find references to transformed values in the Stan Reference and User Guides, and example code but no clear tutorial explanation. I'd be grateful for a link. Michael Betancourt, in his Stan ...
user avatar
4 votes
0 answers
106 views

Adaptive selection of Mass values in Hamiltonian Monte-Carlo?

I know there are good solutions for adaptive selection of path lengths and step-size for Hamiltonian Monte-Carlo (e.g. the NUTS sampler), but for the sampler to work efficiently we also require that ...
user avatar
  • 583
1 vote
1 answer
58 views

Can I use Hamiltionian Monte Carlo when my likelihood is not a direct function of my parameters?

By "not a direct function of my parameters" I mean the following. I have some observed K-dimensional data and a model that can generate synthetic data based on 6 free parameters. I use this model to ...
user avatar
  • 3,134
10 votes
0 answers
1k views

No-U-Turn Sampler (NUTS) for Hamiltonian Monte Carlo (HMC): how do I understand the doubling process?

I'm reading the original NUTS paper by Hoffman and Gelman, but couldn't fully understand the recursively doubling process. The following figure is taken from the paper. The NUTS process starts ...
user avatar
  • 729
13 votes
1 answer
2k views

Hamiltonian Monte Carlo for dummies

Could you provide a step-by-step for dummies explanation of how Hamiltonian Monte Carlo work? PS: I've already read the answers here, Hamiltonian monte carlo, and here, Hamiltonian Monte Carlo vs. ...
user avatar
13 votes
1 answer
2k views

Hamiltonian Monte Carlo: how to make sense of the Metropolis-Hasting proposal?

I am trying to understand the inner working of Hamiltonian Monte Carlo (HMC), but can't fully understand the part when we replace the deterministic time-integration with a Metropolis-Hasting proposal. ...
user avatar
  • 729
10 votes
1 answer
843 views

Understanding the Typical Set for Markov chain Monte Carlo sampling

I started reading "A Conceptual Introduction to Hamiltonian Monte Carlo" today, and I've gotten stuck on understanding Betancourt's explanation of what a "typical set" is. If $q_1, q_2, \ldots, q_n$ ...
user avatar
  • 18.6k
11 votes
1 answer
2k views

Hamiltonian Monte Carlo (HMC): what's the intuition and justification behind a Gaussian-distributed momentum variable?

I am reading an awesome introductory HMC paper by Prof. Michael Betancourt, but getting stuck in understanding how do we go about the choice of the distribution of the momentum. Summary The basic ...
user avatar
  • 729
13 votes
2 answers
1k views

For Hamiltonian Monte Carlo, why does negating the momentum variables result in a symmetric proposal?

I have been going through Radford Neal's excellent HMC book chapter in detail. However, there is one detail that I'm really obsessing with now, and I'm not sure if I'm thinking about it right. When ...
user avatar
5 votes
2 answers
608 views

Proposal distribution in Hamiltonian Monte Carlo

I have been reading A Conceptual Introduction to Hamiltonian Monte Carlo by Betancourt (https://arxiv.org/abs/1701.02434), which is a great introduction to HMC, but there is one part that I can't get ...
user avatar
  • 550
5 votes
1 answer
753 views

Plotting the typical set of a Gaussian distribution

There is this article where the author Michael Betancourt uses this image to convey the concept of the typical set in a distribution. I would like to plot the typical set of a univariate or a ...
user avatar
  • 2,666
4 votes
1 answer
251 views

How to know if the derivatives exist in Hamiltonian Monte Carlo?

In section 3.2 of Radford Neal's take on HMC he says: We must also be able to compute the partial derivatives of the log of the density function. These derivatives must therefore exist, except ...
user avatar
  • 825
25 votes
1 answer
4k views

Hamiltonian Monte Carlo vs. Sequential Monte Carlo

I am trying to get a feel for the relative merits and drawbacks, as well as different application domains of these two MCMC schemes. When would you use which and why? When might one fail but the ...
user avatar
  • 825
16 votes
2 answers
1k views

Hamiltonian monte carlo

Can someone explain the main idea behind Hamiltonian Monte Carlo methods and in which cases they will yield better results than Markov Chain Monte Carlo methods ?
user avatar
1 vote
0 answers
87 views

Hamiltonian Monte Carlo with large parameter values fail to converge

I'm trying to learn about Hamiltonian Monte Carlo. Therefore I tried to infer the Parameters of a Multivariate Normal given some samples. My procedure is the following: Define $\mu$ and $\Sigma$ ...
user avatar
  • 296
3 votes
1 answer
940 views

Interesting / strange behavior of one chane on different [unrelated] variables in STAN

I have a quite complex hierarchical model for which I'm estimating parameters and producing posterior predictive using STAN (rstan) for some psychophyiscal data. I'm (sometimes) observing some ...
user avatar
7 votes
1 answer
352 views

Hamiltonian Monte-Carlo with piecewise differentiable log likelihood

This is a bit of a curious situation. I have an energy function $E=S+N$ which is the sum of a smooth differentiable function $S$ and a piecewise constant "noise" function $N$. This means that on ...
user avatar
  • 2,510