Exploring Phase Space with Nested Sampling

Exploring phase space?

Cross section = integral$_\text{(over kinematic variables)}$ ( Matrix Element )
 

Central challenge for many physics tasks:

• Total cross section $(\sigma)$ - Probability of process occuring
• Differential cross section $(d\sigma)$ - chunk integral into $d\Phi$ pieces
• Events - (unweight) and use as pseudo data

Workhorse in HEP on this set of problems is Importance Sampling

• Replace problem of sampling from unknown $P(\Phi)$ with a known $Q(\Phi)$
• Adjust importance of sample drawn from $Q$ by weighting, $w=\frac{P(\Phi)}{Q(\Phi)}$

Problem seemingly reduces to coming up with good mappings for target

However, Even in $D=\mathcal{O}(10)$ Dimensions this starts to break.

• Massless glue scattering, $D=3n_g-4$:
  • $gg\rightarrow 3g$, $D=5$
  • $gg\rightarrow 4g$, $D=8$

Even modern ML (normalising flows) won't save you [2001.05478]

A sampling problem? Anyone for MCMC?

Central problem:

• Convergent integral means you have good posterior samples
• Reverse not true, Samples from a convergent MCMC chain not guaranteed a good integral
• Multimodal targets well established failure mode.
  • Multichannel decompositions in MCMC HEP, (MC)$^3$ [1404.4328]

MCMC kicks in as we go to high dimensions, grey area between IS and MCMC, can ML help?

Where's the Evidence?

In neglecting the Evidence ($\mathcal{Z}$) we have neglected precisely the quantity we want,

• Mapping $\rightarrow$ Prior
• Matrix element $\rightarrow$ Likelihood
• Cross section $\rightarrow$ Evidence

Nested Sampling

Nested Sampling [Skilling 2006], implemented for in PolyChord [1506.00171]. Is a good way to generically approach this problem for $\mathcal{O}(10)\rightarrow \mathcal{O}(100)$ dimensions

• Primarily an integral algorithm (largely unique vs other MCMC approaches)

• Designed for multimodal problems from inception

• Requires construction that can sample under hard likelihood constraint

• Largely self tuning

  • Little interal hyperparameterization
  • More importantly, tunes any reasonable prior to posterior

Where do we go from here?

End to end stylised version of the problem demonstrated.

This is deeper than coming up with a new way of mapping phase space

Where do we go from here?

(dedicated section in paper)

• Physics challenges

• Variants of NS algorithm

• Prior information

• Fitting this together with modern ML

Physics challenges

The fundamental motivation for this work came from recognising not just an ML challenge but a physics challenge [2004.13687]

LO dijet isn't hard, NNNLO is. If your method isn't robust in these limits it doesn't solve the right problem. Unique features of NS open up interesting physics:

• No mapping required: NLO proposals generically harder, NNLO more so
• No channel decomposition: can we be really clever when it comes to counter events, negative events etc. with this?
• Computation scaling guaranteed to $\sim$ polynomial with $D$, other methods exponential: We can do genuinely high dimensional problems, $gg\rightarrow 10g$ anyone?

Conclusion

In my opinion (your milage may vary)

• The fundamental problem for LHC event generation trying to do Importance Sampling in high dimension.

• Machine learning can and will be useful, but this is not just a machine learning mapping problem.

• This is a Bayesian inference problem, precisely calculating Evidences or Posterior sampling.

• Nested Sampling is a high dimensional integration method, primarily from Bayesian Inference, that is an excellent choice for particle physics integrals