Introduction to sbi: simulation & inference

About this document

This text corresponds to the slides of the Module 1 of the Simulation-based inference workshop held in 2021. Head on to the workshop page for the rest of the content.

🄯 Álvaro Tejero-Cantero for all the text, licensed under a CC-BY-SA license.

ℹ️ Practical parts are indicated in red background. Speaker's notes are under ▸ §.

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Module's learning goals

This is the initial list of learning goals we set ourselves for this introduction.

After this module, the learner should

• understand what is the problem that SBI tries to solve,
• have seen practical examples of a number of simulators,
• understand ABC and its limitations, and
• know how (neural) density estimation comes up in SBI.

1. Simulators

• understand the role of simulators in science and their limitations as a modelling paradigm
• Be able to identify a variety of simulation approaches, and categorize precisely (ODE, ABM...) their simulator of interest
• Recognize the need to represent stochasticity and identify ways in which random numbers can appear in models

2. Parameter inference

• Understand what is an inverse problem and that functions are not the right paradigm to represent the solution.
• Know examples and reflect about the interpretation of multiple parameter sets being compatible with a given observation.

3. Bayesian inference w/o likelihoods

• list non-Bayesian approaches to the inverse problem - grid/brute forcing, genetic, MLE + MAP (optimization
• Understand how Bayesian posterior inference provides a framework for the inverse problem
• Be able to articulate how this approach provides a basis for a decision-theoretic approach to simulation as part of a process.
• Identify sources of intractability for likelihoods that call for simulation-based inference.
• Be able to recognise scenarios of black-box inference; conversely, identify the availability of additional information that may make other approaches ($\neq$SBI) possible/desirable.
• Describe what is ABC and what is problematic about it.
• Define what are summary statistics.

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Table of contents

• Simulators for science
• Models in science
• Simple pendulum
• Simulators, everywhere
• Simulators galore
• Agent-based models
• Differential-equation-based models
• Simulator example: core-collapse supernova models
• Epistemic ambition of simulators
• Interlude: introduce your simulator
• Simulators vs. statistical models
• Models, and models
• Example: modelling counts vs. modelling processes
• Induction vs. deduction, an age-old dichotomy
• Inference on simulators
• An inverse problem
• Multiple and uncertain parameter sets
• Bayesian parameter inference
• Introducing the posterior
• Interlude: Bayesian inference
• The Bayesian workflow
• Bayesian inference
• Simulators as statistical models
• Stochasticity and simulators
• Simulator as generative models
• Anatomy of a simulator: notation
• Some notes about the parameters ⁍ → latents ⁍ → outputs ⁍
• Latent-path intractability
• Partial observability in the field
• Ways to make likelihoods tractable
• Levels of simulator access
• Simulation-based inference
• LFI, ABC, SBI, FBI, CIA, WHO?
• Interlude: from ABC to SBI
• Conditional density estimation
• Amortized SBI is density estimation
• Advantages derived from using networks
• Further materials
• Resources

Simulators for science

Models in science

• making models is part of the scientific method
• models capture only some aspects of reality
• when formalized, they enable quantitative, testable hypotheses
• model functionalities
• prediction — to support decisions
• understanding — to select interventions
• the structure that doesn't change is the model
• the malleable part are parameters
• parameters are 'tuned' based on observations
• multiple input parameter sets can lead to the same output prediction
• equifinality, degeneracy are key to resilience, homeostasis of complex systems

Simple pendulum

• Can predict angles $\theta(t)$ given $g/\ell$ and $\theta(0)$.
• Can infer $g/\ell$ from measured $\theta(t)$.
• for small amplitudes $\sin\theta\simeq\theta$ and timing one oscillation $T=2\pi \sqrt{\ell/g}$ approximately suffices to infer $g/\ell$ → $T$ summarises $\theta(t)$ for inference.
• And extract understading wrt. interventions and counterfactuals.

Simulators, everywhere

• 20th-century: explosion of digital, expressive simulators → "complex science"
1. "simulator as numerical solver" for an explicit model (e.g. PDEs) — based on discretization
2. "simulator as defined by code", an implicit model built from individual interaction rules
3. and anything in-between, e.g. pulse-coupled NNs: discrete + continuous dynamics

Simulators galore

• simulate - /ˈsɪm·jəˌleɪt/ (verb). (Cambridge English dictionary)

1. create conditions or processes similar to something that exists.

1. to produce a situation or event that seems real but is not real, especially in order to help people learn how to deal with such situations or events
• simulation might be a key ingredient of cognitive processing? cf. predictive coding. (imagination, speculation, platonic shadow)
• typology: continuous vs. discrete (regular vs. event-based), dynamic vs. steady-state, deterministic vs. stochastic...
• Let's look at a couple of examples

Agent-based models

Other examples: culture modelling, tumor growth, epidemics, ecology, traffic, percolation (oil through soil), voting...

Differential-equation-based models

• continuous models for dynamically changing phenomena
• relate system response to infinitesimal changes of state variable(s)
• classes of Diferential equation (DE):

ODE: ordinary DE: one variable $y$, $F\left (x,y,y',\ldots, y^{(n-1)} \right )=y^{(n)}$ PDE: partial DE: multiple variables, e.g. time and space SDE: stochastic: DE with stochastic fluctuations (→ stochastic process)

• Solution via integration is rarely possible in closed form

→ numerical approximations: discretization.

Simulator example: core-collapse supernova models

Observables. lightcurves, neutrinos, EM spectrum (🤯 SN remnants visible for centuries, but relevant dynamics on $\lesssim 1\,{\rm s}$)

Solution.

• PDEs → Finite Elements Model (FEM)
• "parameter-free"
• months on high-performance computers

Supernova simulation 400ms post-bounce, courtesy Thomas Janka →

Epistemic ambition of simulators

Interlude: introduce your simulator

• Key features of your use case (use up to 2 minutes):
• why: phenomenon and scientific meaning of parameters and observations. Do you expect degeneracies / equifinality? what would they mean?
• what: type, dimensionality and structure of parameters and observations, sources of noise
• how: type of simulator (ODE, ABM...), programming language
• when: what is the approximate runtime?
• questions from the audience (about 1m).

Simulators vs. statistical models

Models, and models

Models can either be mechanistic or convenient (Ripley 1987, Stochastic Simulation p3).

Example: modelling counts vs. modelling processes

RNN model of infection counts

Designed to fit data and predict

• RNN flexible approximator,
• specific architecture to fit time correlations
• many, $\mathcal{O}(10^4)$, opaque parameters
• model hard to debug and interpret
• data-hungry,
• poor extrapolation

Epidemic compartmental model

Designed for insight

• ODE, limited modelling repertoire
• use rate constants for population kinetics
• few, $\mathcal{O}(1)$ interpretable parameters
• expresses interaction rules, conservation laws, smoothness constraints.
• misspecification risk, but know how to formulate hypotheses as compartments

Induction vs. deduction, an age-old dichotomy

• Induction $\sim$ statistical models: from experimental data, infer the simplest (usually statistical) laws that account best for the data.
• Deduction $\sim$ mechanistic models: from first principles, obtain laws that produce predictions in specific settings. Experimental design seeks measurements that falsify the laws.

Inference on simulators

An inverse problem

• from parameters to measurements: modelization, simulation: forward problem
• from measurements to parameters: parameter estimation: inverse problem
• point estimates via optimization
• grid search (not scalable)
• derivative-based optimization (gradients generally not available)
• population methods: evolutionary strategies
• share: no principled ranking of point estimates

Multiple and uncertain parameter sets

• functions destroy information, when they are noninjective (underdetermination, equifinality)
• $\hat{\theta}$ replicates observation $x_{\rm o}$, i.e. $x_{\rm o} = \operatorname{sim} \hat{\theta}$. But what about $\operatorname{sim}(\hat{\theta} + \epsilon)$
• this could be a serious problem
• sensitivity analysis is the discipline dealing with the $\operatorname{sim} (\hat{\theta}+\epsilon)$ problem
• solution? estimate $\hat{\theta}(x)$, calculate confidence sets
• uncertainty estimate ✔️
• find the multiple $\hat{\theta}$ which could have generated $x_{\rm o}$ (and rank them) ❌

Bayesian parameter inference

• But how to get there?

simulator data + observation + inference 🪄 ($\sim$day 2 of the workshop)→ posterior

• And what to do with them?

posterior + validation / interpretation ($\sim$ day 3 of the workshop) 🪄 → hypotheses, results

Introducing the posterior

For $\theta \in \mathbb{R}^{D>2}$ we need strategies for representation. Ground truth $\orange{\bullet}\equiv\{\orange{\theta_1, \theta_2}\}$

1. marginal posteriors in 2D 🔨 integrate out $D-2$ parameter dimensions

$p(\theta_1, \theta_2|x_{\rm o})=\green{\int} p(\theta_1,\theta_2|x_{\rm o}, \green{\theta_{3:31}})\ \green{{\rm d}\theta_{3:31}}$

typically more like a blob!

Posterior plots courtesy Michael Deistler →

2. conditioned posteriors in two dimensions

🏖️ fix $D - 2$ parameter dimensions

$p(\theta_1,\theta_2|x_{\rm o}, \color{green}{\theta_{3:31}\leftarrow\theta^\ast})$

typically sharper!

Interlude: Bayesian inference

The Bayesian workflow

1. Modeling (day 1): formulate joint pdf $p(x,\theta)$
• product rule: $p(x,\theta) = p(x|\theta) p(\theta)=p(\theta|x)p(x)$.
• prior $p(\theta)$ summarizes our knowledge of the parameters (e.g. 'must be positive')
• likelihood $p(x|\theta)$ embodies our knowledge of how $x$ is generated from $\theta$ (→ model)
• rinse and repeat (update the prior: the posterior is the new prior)
2. Inference (day 2): Use the data $x_{1:N}$ to learn about the target variable $\theta$ by conditioning
• Typically, use $x_{1:N}$ to fix $\phi$ in $p_\phi(\theta|x)$
• Multiple computational approaches, depending on the problem
3. Validation and interpretation (day 3)
• check internal consistency of the posterior (SBC, posterior-predictive checks)
• check consistency with domain knowledge

Bayesian inference

• For very few models this equation yields a posterior in closed form.
• a closed-form posterior can be sampled from and evaluated. Ideal case!
• Usually the evidence integral is hard to compute
• Markov-Chain Monte-Carlo (stochastic, asymptotically exact), works with unnormalized joint. Implicit $\color{0B6E99}{p(\theta|x)}$: low bias / high-variance (slow). Only samples!
• Variational inference (approximate, deterministic). Assume explicit parametric posterior, fast.

Simulators as statistical models

Stochasticity and simulators

• Typically stochasticity models all the processes where we lack specific mechanistic hypotheses
• unless the process is itself stochastic at the physical level, e.g. $\beta$ decay in a nucleus.
• What are sources of stochasticity?
• unobserved latent variables
• stochastic program paths
• instrument noise (aleatoric)
• numerical approximations (→ probabilistic numerics)

Simulator as generative models

• $\color{E03E3E}p(x|\theta)$ usually not evaluatable for simulators, as not built for inference.
• generative models build a probability density over samples — often an implicit one! (e.g. GANs)
• simulators are generative models with implicit likelihood. Most general scenario for inference - inference just from samples - likelihood-free inference (LFI).
• let's look at simulators from a probabilistic perspective

Anatomy of a simulator: notation

• take parameter vector as input, $\theta$ — we treat $\theta$ as a random variable, and assign it prior $p(\theta)$
• compute internal states — latent variables $z_\ell \sim p_\ell(z_\ell|\theta,z_{<\ell})$
• produce result vectors $x$ comparable with experimental observations $x_{\rm o}\sim p(x|\theta_{\rm true})$

Some notes about the parameters $\theta$ → latents $z_{1:L}$ → outputs $x$

• $\theta$ fixed dimensionality, typically no structure
• $x$ can be structured (e.g. images, graphs...), and high-dimensional
• $z$ correspond to meaningful states, but are typically unobservable.
• continuous or discrete, changing dimensionality (even during simulation)
• updated deterministically, or stochastically
• we will NOT infer $z$ here
• but the existence of this unobserved state is problematic...

Latent-path intractability

Partial observability in the field

Ways to make likelihoods tractable

• model reduction / coarsening. different model / limits expressivity, model-specific ❌
• likelihood data augmentation. model-specific, iterative ❌

(...) introduce additional parameters $\psi$, which represent missing data, in such a way that the likelihood $p(x,\psi|\theta)$ is tractable. Inference then proceeds by estimating both $\theta$ and $\psi$, typically via EM or MCMC. (after O'Neill 2010)

• martingales. elegant ✔️ few specific models, hard ❌
• don't? likelihood-free inference: generally applicable, can target posterior✔️ performance❓, cutting edge❓

Levels of simulator access

Simulation-based inference

LFI, ABC, SBI, FBI, CIA, WHO?

Interlude: from ABC to SBI

Conditional density estimation

Amortized SBI is density estimation

• the likelihood (emulation). Prior-independent, but need still MCMC, VI for the evidence.
• the posterior directly.

Advantages derived from using networks

• Feature learning, can incorporate inductive biases
• Scales well with more data
• Interpolation properties
• Differentiable

Further materials

• Slides for the next sessions are available on GitHub
• https://github.com/mlcolab/sbi-workshop

Resources

• Review. Cranmer et al. (2020). The frontier of simulation-based inference. Proceedings of the National Academy of Sciences, 117(48), 30055–30062.
• Software. Tejero-Cantero et al., (2020). sbi: A toolkit for simulation-based inference. Journal of Open Source Software, 5(52), 2505 — 🧑🏽‍💻 https://github.com/mackelab/sbi
• Application (neuroscience). Gonçalves et al. (2020). Training deep neural density estimators to identify mechanistic models of neural dynamics. ELife, 9, e56261.