Simulation-Based Inference Intro

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Questions and Answers

Describe how insights and constraints from model evaluation can be used in mechanistic modeling.

Insights and constraints from model evaluation inform adjustments to the mechanistic model, refining its structure or parameters to better align with empirical data.

Explain how mechanistic models and machine learning differ in their approach to data and interpretation.

Mechanistic models are built on assumptions about mechanisms and provide interpretable parameters, while machine learning models focus on performance, built with computation and generalization in mind, often lacking direct interpretation.

What is the role of Bayesian inference in determining the compatibility of model parameters with observed data?

Bayesian inference quantifies the compatibility of mechanistic model parameters with data by calculating the posterior probability distribution, reflecting the plausibility of different parameter values given the data and prior knowledge.

Describe the components necessary to perform Bayesian inference.

<p>To perform Bayesian inference, you need parameters, a defined model, data, prior knowledge, and the ability to perform simulations.</p> Signup and view all the answers

Why is the likelihood function often considered 'not tractable' when working with mechanistic models, and how does this challenge simulation-based inference?

<p>The likelihood function, representing the probability of observed data given model parameters, is often intractable in mechanistic models because it involves complex computations or unknown forms, hindering direct evaluation in simulation-based inference.</p> Signup and view all the answers

Define 'likelihood-free' inference in the context of ABC methods.

<p>Likelihood-free inference refers to methods that do not require explicit knowledge or evaluation of the likelihood function, relying instead on simulations from the model to perform inference.</p> Signup and view all the answers

What does the parameter $\epsilon$ (epsilon) represent in the context of Approximate Bayesian Computation (ABC), and how does it influence the balance between computability and accuracy?

<p>In ABC, $\epsilon$ defines a threshold for accepting simulated data as 'close' to observed data, balancing computability and accuracy: a larger $\epsilon$ increases acceptance rate but reduces accuracy, while a smaller $\epsilon$ enhances accuracy but demands more computation.</p> Signup and view all the answers

Describe key challenges associated with ABC.

<p>Key challenges involve balancing the tolerance parameter (epsilon) to manage the ABC error and effectively choosing summary statistics to minimize information loss, further complicated by computational constraints and simulator expense.</p> Signup and view all the answers

What is meant by the 'curse of dimensionality' in the context of ABC, and how are summary statistics used to mitigate this issue?

<p>The 'curse of dimensionality' refers to the challenge of findingAcceptable simulations when data is high-dimensional. Summary statistics reduce dimensionality by condensing data into a few relevant features, thus increasing the chances of observing 'close' simulations.</p> Signup and view all the answers

Explain how summary statistics can lead to information loss in ABC.

<p>Summary statistics reduce the dimensionality of the data, which leads to loss of information because they may not capture all the nuances and dependencies present in the original data. Inference is based on the summary statistics rather than the full dataset.</p> Signup and view all the answers

Explain why ABC methods are more popular in genetics than other biological disciplines.

<p>ABC methods are popular in biological disciplines, particularly genetics, because they are simple to implement, intuitive, and embarrassingly parallelizable.</p> Signup and view all the answers

What are some ways to achieve more accurate summary statistics?

<p>Ways to achieve more informative summaries: expert knowledge and machine-learning tools.</p> Signup and view all the answers

What is the goal in combining Machine Learning and Mechanistic Models?

<p>The goal is to use the strengths of both approaches to build tools for data-driven science.</p> Signup and view all the answers

What are some advanced topics in SBI?

<p>Some advanced topics in SBI include normalizing flows, neural density estimation, and NPE.</p> Signup and view all the answers

What are some reasons that simulation-based inference is rapidly expanding.

<p>Simulation-based inference is rapidly expanding because to recent developments in probabilistic deep learning.</p> Signup and view all the answers

What is the purpose of neuro-simulation?

<p>Neuro-simulations can be used to see how neurons are influenced by neuromodulators and how that affects behavior.</p> Signup and view all the answers

How does simulation-based inference contribute to combining mechanistic models and data for mechanistic insights?

<p>Simulation-based inference combines mechanistic models and observed data to obtain mechanistic insights.</p> Signup and view all the answers

What role does probability theory play in reasoning with data and quantifying uncertainty in simulation-based inference?

<p>Probability allows to the user to reason with data with a mathematical framework that quanitifies our uncertainity.</p> Signup and view all the answers

Describe the goal of mechanistic models.

<p>The goal of mechanistic models is for understanding.</p> Signup and view all the answers

Describe the goal of Machine Learning models.

<p>The goal of machine learning model is for high performance.</p> Signup and view all the answers

What are some ABC algorithmns?

<p>Examples include: SMC-ABC, GP-ABC, Regression-ABC, and Hamiltonian-ABC.</p> Signup and view all the answers

Why are simulator important in science?

<p>Simulators are important in science because they allow researchers to study very complicated issues that would otherwise be impossible.</p> Signup and view all the answers

In the context of simulation-based inference, why is it important to validate a model against experimental data?

<p>Validating a model against experimental data is important to ensure it can accurately represent the system and behaviors being modeled.</p> Signup and view all the answers

Describe how prior knowledge or beliefs are incorporated into Bayesian inference, and explain why this is important.

<p>Prior knowledge is incorporated into Bayesian inference through a prior distribution, allowing existing information or beliefs to influence the estimation of model parameters, and reflecting existing insights or constraints.</p> Signup and view all the answers

In the context of simulation-based inference, how do the goals of understanding and performance influence the choice between mechanistic models and machine learning models?

<p>If the researcher values understanding, they should use a mechanistic model. If the researcher values performance, they should use a machine learning model.</p> Signup and view all the answers

Define the role of the posterior distribution in Bayesian inference, and explain how it is informed by both the prior distribution and the likelihood function.

<p>The posterior distribution represents the updated belief about model parameters after considering observed data. It's formed by combining the prior distribution (initial belief) with the likelihood function (evidence from data).</p> Signup and view all the answers

How do mechanistic models incorporate knowledge of dynamics, and how does this facilitate a deeper understanding of the underlying system?

<p>Mechanistic models include mathematical equations or rules that describe how the system changes over time. Including knowledge of dynamics helps understand the behavior of the system.</p> Signup and view all the answers

How does the trade-off between bias and variance manifest in the context of choosing summary statistics for ABC?

<p>Selecting too few summary statistics introduces bias by discarding relevant information, while using too many increases variance due to overfitting noise, requiring a trade-off to balance informativeness and robustness.</p> Signup and view all the answers

What is the difference between a Rejection Algorithm and a 'Mechanical' Rejection Algorithm?

<p>The Rejection Algorithm accepts hypothesis with a given probability; this contrasts with a 'Mechanical' Rejection Algorithm which requires simulating each hypothesis.</p> Signup and view all the answers

With the Rejection Algorithm, what do you get as a result?

<p>You get accepted $\theta$ values from the posterior distribution $p(\theta | x_o)$.</p> Signup and view all the answers

What is the approximate version of the Uniform Rejection Algorithm?

<p>Accept $\theta_n$ if $p(x_\eta, x_o) \leq \epsilon$.</p> Signup and view all the answers

How does the tolerance variable, epsilon, influence the results?

<p>As epsilon approaches infinity, we get observations from the prior $p(\theta)$.</p> Signup and view all the answers

How can neural networks enhance data analysis?

<p>Neural networks can analyze data to train data-compatible models.</p> Signup and view all the answers

How does ABC handle the fact that some data is too high dimensional?

<p>The data are reduced by using summary statistics, $S(x_o)$.</p> Signup and view all the answers

What is an advantage that mechanistic models have over machine learning?

<p>Mechanistic models have interpretability while machine learning may not.</p> Signup and view all the answers

Is ABC often a useful tool with data?

<p>ABC is often hard to use because it scales poorly given high dimensionality summary statistics and many parameters to infer.</p> Signup and view all the answers

In simulation-based inference, what does the mechanistic model do?

<p>Mechanistic models will generate predictions.</p> Signup and view all the answers

In simulation-based inference, what needs to be done with the models to ensure a relevant result?

<p>The data needs to be related back to insights or constraints.</p> Signup and view all the answers

Generally speaking, why do mechanistic insight and data need to be combined.

<p>Generally speaking, mechanistic insight and data need to be combined to gain insight.</p> Signup and view all the answers

How does summary statistic $S(x_o)$ control 'information loss'?

<p>Inference is based on $p(\theta | S(x_o))$ rather than $p(\theta | x_o)$.</p> Signup and view all the answers

What is the key question that can be answered via Bayesian inference?

<p>Which parameters of a mechanistic model are compatable with the data?</p> Signup and view all the answers

Flashcards

Mechanistic Model Cycle

A cycle where a mechanistic model is used to generate predictions, which are then evaluated against collected empirical data. Insights from this comparison can refine the model.

Mechanistic Models

Models built on assumptions about mechanisms to pursue understanding, using interpretable parameters and knowledge of dynamics.

Machine Learning Models

Models focused on performance. Built with computation and generalization in mind, using data and inductive bias, often lacking direct interpretation.

Model Predictions

The use of mechanistic models to generate expected outcomes, allowing comparison with real-world observations.

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Bayesian Inference

The process of using observed data to infer the parameters of a model.

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Simulate Data

Using models to create data similar to what's observed in real-world scenarios.

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Approximate Bayesian Computation (ABC)

A class of computational methods for performing inference on simulators without explicit knowledge of the likelihood function.

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ABC key aspects

Algorithms don't need explicit knowledge of the likelihood function and inference is done using simulations from the generative model.

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Accepted Parameters

Accepted parameters are considered draws from the posterior distribution, representing the plausible parameter values given the observed data.

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Tolerance (epsilon)

As tolerance (epsilon) trends to infinity, observations trend to be from the prior. As tolerance trends to zero, observations trend to be from p(theta |x0).

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Use Summary Statistics

Reduces the data's dimensions using summary statistics, helping with the curse of dimensionality.

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Simulation-based Inference

A toolkit for making sense of the real world with simulations.

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Mechanistic Insights

Combining mechanistic models and data to gain insights into the underlying processes.

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Probability Theory

A framework for reasoning with data and expressing the certainty about the process.

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Study Notes

  • Lecture 6 is about simulation-based inference and was presented in March 2025
  • The lecture is an introduction to the topic
  • Pedro Goncalves and team is the presenter, and can be reached at goncalveslab.sites.vib.be/en
  • Philipp Berens , Jonas Beck also contributed with more info available at https://hertie.ai/data-science/team
  • Parts of these lectures are adapted from AIMS SBI January 2024, JH Macke, C Schroder, PJ Goncalves
  • Some slides are by Álvaro Tejero-Cantero
  • Labs aim to combine Scientific Modelling and Machine Learning to build tools for data-driven scientific discovery
  • Applications focus on neuroscience
  • Mechanistic models are refined and simulated to generate inferred parameters interpreted through neural networks to identify parameters.
  • Refined mechanistic models can use the equation: Cm * dV/dt = SUM g_i(t) * (E_i − V) + I
  • C. Elegans worms have Neurons and Neuromodulators to influence behavior

Science and Simulators

  • Mechanistic models generate predictions which simulate data
  • Empirical data is collected and used evaluate the model
  • Insights are added to create constraints on the mechanistic model
  • Mechanistic models are built from assumptions about mechanisms with knowledge of dynamics and interpretable parameters
  • It is often hard to fit mechanistic model to data
  • Machine learning gives goal performance
  • Machine learning is built with computation and generalization in mind
  • Machine learning uses data + inductive bias and is often without direct interpretation, to fit data
  • By combining strengths of both approaches we can build data-drive science
  • Parameters of a mechanistic model define how compatible they are with given data
  • Bayesian inference is used to define the parameters

Bayesian Inference and Simulation

  • Bayesian inference finds model parameters consistent with data and prior knowledge.
  • p(θ|x) ∝ p(x|θ)π(θ) is the formula behind this
  • θ represents the parameters
  • x represents the data.
  • Calculating the posterior is hard
  • Likelihood is often not tractable
  • For many mechanistic models, it is possible to simulate x but not evaluate the likelihood p(x|θ)
  • Models are then defined through black-box simulators
  • Simulation-based inference is used as a solution to this problem

Approximate Bayesian Computation (ABC)

  • ABC provides an intuitive first approach.
  • It is based on a tutorial by Wilkinson 2016
  • ABC algorithms are a diverse collection of methods for performing inference on simulators which do not require knowledge of the likelihood function.
  • Inference is done using simulations from the model and is 'likelihood-free'
  • ABC methods are popular in biological disciplines like genetics for its simplicity, intuitive implementation and embarrassingly parallelizable nature
  • Rejection Algorithm: Sample θn ~ p(θ) then Accept θn with probability p(x0|θn)
  • 'Mechanical' Rejection Algorithm samples θn ~ p(θ) and then simulate xn from model (equivalently, sample xn ~ p(x|θn)
  • In approximate version in uniform Rejection Algorithm, for θn we sample xn ~ p(x|θn) and accept θn if p(xn , x0) ≤ ε
  • ε reflects the tension between computability and accuracy
  • As ε ∞, we get observations from the prior p(θ)
  • If ε = 0, we generate observations from p(θ|x)
  • If the data are too high dimensional, simulations will not be close to mesured data, resulting in the curse of dimensionality.
  • Dimensionality is reduced using summary statistics S(x0)
  • Summary statistics also have the equation: p(S(xn), S(x0)) ≤ ε

Key Challenges and Comments about ABC

  • Accuracy in ABC is determined by Tolerance ε, which control the 'ABC error'
  • Computation constrains also limit accuracy, and often rules out expensive simulators
  • Summary statistic S(x0)controls 'information loss'
  • Inference is based on p(θ|S(x0)) rather than p(θ|x0)
  • Expert-knowledge and machine-learning tools can be used to find informative summaries
  • It can be useful to get more accurate and efficient algorithms (scaling to more parameters and summary statistics)
  • It can be useful to figure out how to choose summary statistics or how to deal with expensive simulators
  • There are many ABC algorithms such as (SMC-ABC, Regression-ABC, GP-ABC, Hamiltonian-ABC...)

Simulation-based Inference with Neural Networks

  • Simulation-based inference is rapidly expanding due to recent advancements in probabilistic deep learning
  • Neural networks can train models to identify data-compatible parameters from a mechanistic model with a prior and simulated data
  • Simulation-based inference aims to combine simulators and data for new insights
  • Probability theory is a mathematical language for performing inference
  • Use advances in machine learning, particularly neural networks, for Bayesian inference

Course outline

  • Week 2 goes over: ABC, neural density estimation, NPE, normalising flows, advanced topics in SBI
  • Week 3: SBI hands-on tutorial, project work

Organizational matters

Closing remarks

  • For mechanistic insights, combine mechanistic models and data
  • Use probability theory as a framework for reasoning with data, to quantify our uncertainty about it.
  • Use Simulation-based inference for making sense of the real world with simulations
  • ABC-based methods are a popular method for doing simulation-based inference that is simple to implement, intuitive, and embarrassingly parallelizable.
  • ABC does not scale well with the number of parameters to infer or with the dimensionality of the summary statistics.

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