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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

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Table of contents

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

\frac{{\rm d}^2 \theta}{{\rm d} t^2} + \frac{g}{\ell}\, \sin\theta = 0

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.

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Simulators, everywhere

20th-century: explosion of digital, expressive simulators → "complex science"

"simulator as numerical solver" for an explicit model (e.g. PDEs) — based on discretization
"simulator as defined by code", an implicit model built from individual interaction rules
and anything in-between, e.g. pulse-coupled NNs: discrete + continuous dynamics

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Simulators galore

But what are simulators?

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

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

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

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Agent-based models

Microscale models "defined by code". Agents represented directly, not by density or concentration; possess internal state, interaction rules, and learning processes that determine state updates; they live in a topology embedded in an environment.

SIR ABM running (w. social distancing, isolated=0.8).

Parameters of SIR model in Agents.jl.

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

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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

Goal. Test understanding of physics. What are the mechanisms that make SN explode?

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Epistemic ambition of simulators

🖍️ Theory building — focus on process

🖥️ Hypothesis testing — focus on result

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Interlude: introduce your simulator

Share your simulator-based research with the group. You can re-use slides you have. Paste them here in this Google presentation.

Key features of your use case (use up to 2 minutes):