#
Galilean and Hamiltonian Monte Carlo^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

Question: | How does a mathematician find a needle in a haystack? |

Answer: | Keep halving the haystack and discarding the “wrong” half. |

## 2. Compression by Nested Sampling

Question: | How does a programmer find a small target in a big space? |

Answer: | …… |

## 3. Exploration by Galilean Monte Carlo

- Mistakenly, the author’s earlier introduction of GMC in 2011 [4] reduced the possibilities by eliminating West, but at the cost of allowing the particle to escape the constraint temporarily, which damaged the performance and cancelled its potential superiority.

## 4. Compression and Exploration

## 5. Exploration by Hamiltonian Monte Carlo

## 6. Compression versus Simulated Annealing

## 7. Conclusions

Nested sampling | Simulated Annealing |

Steady compression | Arbitrary cooling schedule for $\beta $ |

Invariant to relabelling Q | Q has fixed form ${L}^{\beta}$ |

Can deal with phase changes | Cannot deal with phase changes |

Evidence $Z=\int L\phantom{\rule{0.166667em}{0ex}}dX$ with uncertainty | Evidence $Z=exp{\int}_{0}^{1}{\langle logL\rangle}_{{\scriptscriptstyle \beta}}\phantom{\rule{0.166667em}{0ex}}d\beta $ |

Galilean Monte Carlo | Hamiltonian Monte Carlo |

No rejection | Trajectories can be rejected |

Any metric is OK | Riemannian metric required |

Invariant to relabelling Q | Trajectory explores nonuniformly |

Quality function $Q\left(\mathbf{x}\right)$ is arbitrary | Quality $Q\left(\mathbf{x}\right)$ must be differentiable |

Step functions OK (nested sampling) | Can not use step functions |

Can sample any probability distribution | Probability distribution must be smooth |

Needs 2 work vectors | Needs 3 work vectors |

## Funding

## Conflicts of Interest

## References

- Skilling, J. Nested Sampling for general Bayesian computation. J. Bayesian Anal.
**2006**, 1, 833–860. [Google Scholar] [CrossRef] - Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P. Optimization by Simulated Annealing. Science
**1983**, 220, 671–680. [Google Scholar] [CrossRef] [PubMed] - Duane, S.; Kennedy, A.D.; Pendleton, B.J.; Roweth, D. Hybrid Monte Carlo. Phys. Lett. B
**1987**, 195, 216–222. [Google Scholar] [CrossRef] - Skilling, J. Bayesian computation in big spaces—Nested sampling and Galilean Monte Carlo. AIP Conf. Proc.
**2012**, 1443, 145–156. [Google Scholar]

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**MDPI and ACS Style**

Skilling, J.
Galilean and Hamiltonian Monte Carlo. *Proceedings* **2019**, *33*, 19.
https://doi.org/10.3390/proceedings2019033019

**AMA Style**

Skilling J.
Galilean and Hamiltonian Monte Carlo. *Proceedings*. 2019; 33(1):19.
https://doi.org/10.3390/proceedings2019033019

**Chicago/Turabian Style**

Skilling, John.
2019. "Galilean and Hamiltonian Monte Carlo" *Proceedings* 33, no. 1: 19.
https://doi.org/10.3390/proceedings2019033019