Boomers Quantified Uncertainty. We Simulate It
[Video Credit: N-body simulation Francisco Villaescusa-Navarro]
IAIFI Fellow
Carolina Cuesta-Lazaro
IAIFI Summer School
Decision making
Decision making in science
Is the current Standard Model ruled out by data?
Mass density
Vacuum Energy Density
CMB
Supernovae
Observation
Ground truth
Prediction
Uncertainty
Is it safe to drive there?
Interpretable Simulators
Noise in features
+ correlations
Noise in finite data realization
Uncertain parameters
Limited model architecture
Imperfect optimization
Ensembling / Bayesian NNs
Forward Model
Observable
Dark matter
Dark energy
Inflation
Predict
Infer
Parameters
Inverse mapping
Fault line stress
Plate velocity
Variational
Posterior
Likelihood
Posterior
Prior
Evidence
Markov Chain Monte Carlo MCMC
Hamiltonian Monte Carlo HMC
Variational Inference VI
If can evaluate posterior (up to normalization), but not sample
Intractable
Unknown likelihoods
Amortized inference
Scaling high-dimensional
Marginalization nuisance
Image Credit: Chi Feng mcmc demo
["Polychord: nested sampling for cosmology" Handley et al]
["Fluctuation without dissipation: Microcanonical Langevin Monte Carlo" Robnik and Seljak]
Higher Effective Sample Size (ESS) = less correlated samples
Number of Simulator Calls
Known likelihood
Differentiable simulators
z: All possible trajectories
Maximize the likelihood of the training samples
Model
Training Samples
No implicit prior
Not amortized
Goodness-of-fit
Scaling with dimensionality of x
Implicit marginalization
Loss Approximate variational posterior, q, to true posterior, p
Image Credit: "Bayesian inference; How we are able to chase the Posterior" Ritchie Vink
KL Divergence
Need samples from true posterior
Run simulator
Minimize KL
Amortized Inference!
Run simulator
High-Dimensional
Low-Dimensional
s is sufficient iif
Maximise
Mutual Information
Need true posterior!
No implicit prior
Not amortized
Goodness-of-fit
Scaling with dimensionality of x
Amortized
Scales well to high dimensional x
Goodness-of-fit
Fixed prior
Implicit marginalization
Implicit marginalization
Just use binary classifiers!
Binary cross-entropy
Sample from simulator
Mix-up
Likelihood-to-evidence ratio
Likelihood-to-evidence ratio
Classifier logits
Classifier logits
log(Likelihood-to-evidence ratio)
No implicit prior
Not amortized
Goodness-of-fit
Scaling with dimensionality of x
Amortized
Scales well to high dimensional x
Goodness-of-fit
Fixed prior
Implicit marginalization
No need variational distribution
No implicit prior
Implicit marginalization
Approximately normalised
Not amortized
Implicit marginalization
Maximize the likelihood of the training samples
Model
Training Samples
Trained Model
Evaluate probabilities
Low Probability
High Probability
Generate Novel Samples
Simulator
Simulator
[Image Credit: "Understanding Deep Learning" Simon J.D. Prince]
Bijective
Sample
Evaluate probabilities
Probability mass conserved locally
Image Credit: "Understanding Deep Learning" Simon J.D. Prince
Neural Network
Sample
Evaluate probabilities
Continuity equation
[Image Credit: "Understanding Deep Learning" Simon J.D. Prince]
Need to solve this expensive integral at each step during training to maximise likelihood!
Very slow -> Difficult to scale to high dims
Can we avoid it?
Restricted trajectories: flow matching / diffusion
Regress the velocity field directly!
[Image Credit: "An Introduction to flow matchig" Tor Fjelde et al]
["Flow Matching for Generative Modeling" Lipman et al]
["Stochastic Interpolants: A Unifying framework for Flows and Diffusions" Albergo et al]
Assume a conditional vector field (known at training time)
The loss that we can compute
The gradients of the losses are the same!
["Flow Matching for Generative Modeling" Lipman et al]
["Stochastic Interpolants: A Unifying framework for Flows and Diffusions" Albergo et al]
Intractable
Continuity equation
[Image Credit: "Understanding Deep Learning" Simon J.D. Prince]
Sample
Evaluate probabilities
Reverse diffusion: Denoise previous step
Forward diffusion: Add Gaussian noise (fixed)
["A point cloud approach to generative modeling for galaxy surveys at the field level"
Cuesta-Lazaro and Mishra-Sharma
arXiv:2311.17141]
Siddharth Mishra-Sharma
No implicit prior
Not amortized
Goodness-of-fit
Scaling with dimensionality of x
Amortized
Scales well to high dimensional x
Goodness-of-fit
Fixed prior
Implicit marginalization
No need variational distribution
No implicit prior
Implicit marginalization
Approximately normalised
Not amortized
Implicit marginalization
Test log likelihood
["Benchmarking simulation-based inference"
Lueckmann et al
arXiv:2101.04653]
Posterior predictive checks
Observed
Re-simulated posterior samples
Real or Fake?
["Benchmarking simulation-based inference"
Lueckmann et al
arXiv:2101.04653]
["A Trust Crisis In Simulation-Based Inference? Your Posterior Approximations Can Be Unfaithful" Hermans et al
arXiv:2110.06581]
Much better than overconfident!
["A Trust Crisis In Simulation-Based Inference? Your Posterior Approximations Can Be Unfaithful" Hermans et al
arXiv:2110.06581]
Credible region (CR)
Not unique
High Posterior Density region (HPD)
Smallest "volume"
True value in CR with
probability
Empirical Coverage Probability (ECP)
["Investigating the Impact of Model Misspecification in Neural Simulation-based Inference" Cannon et al arXiv:2209.01845 ]
Underconfident
Overconfident
Expected Coverage Probability (ECP)
Hard to find in high dimensions!
U
Underconfident
Overconfident
["Sampling-Based Accuracy Testing of Posterior Estimators for General Inference" Lemos et al arXiv:2302.03026]
["Investigating the Impact of Model Misspecification in Neural Simulation-based Inference" Cannon et al arXiv:2209.01845 ]
Always look at information gain too
["A Trust Crisis In Simulation-Based Inference? Your Posterior Approximations Can Be Unfaithful" Hermans et al
arXiv:2110.06581]
["Calibrating Neural Simulation-Based Inference with Differentiable Coverage Probability" Falkiewicz et al
arXiv:2310.13402]
["A Trust Crisis In Simulation-Based Inference? Your Posterior Approximations Can Be Unfaithful" Hermans et al
arXiv:2110.06581]
["Benchmarking simulation-based inference"
Lueckmann et al
arXiv:2101.04653]
[Image credit: https://www.mackelab.org/delfi/]
["Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows" Papamakarios et al
arXiv:1805.07226]
Proposal (different from prior)
["Fast -free Inference of Simulation Models with Bayesian Conditional Density Estimation" Papamakarios et al
arXiv:1605.06376]
["Flexible statistical inference for mechanistic models of neural dynamics." Lueckmann et al
arXiv:1711.01861]
Sequential can't be amortized!
Proposal (different from prior)
["A Strong Gravitational Lens Is Worth a Thousand Dark Matter Halos: Inference on Small-Scale Structure Using Sequential Methods" Wagner-Carena et al arXiv:2404.14487]
["All-in-one simulation-based inference" Gloeckler et al arXiv:2404.09636]
Model all conditionals at once!
["All-in-one simulation-based inference" Gloeckler et al arXiv:2404.09636]
Score based diffusion model with sampled conditional masks
["Investigating the Impact of Model Misspecification in Neural Simulation-based Inference" Cannon et al arXiv:2209.01845]
More misspecified
"The frontier of simulation-based inference" Kyle Cranmer, Johann Brehmer, and Gilles Louppe
Github repos
Review
cuestalz@mit.edu
Book
"Probabilistic Machine Learning: Advanced Topics" Kevin P. Murphey