Simulation-based inference

• Comment: See WP:PREPRINTS. S0091 (talk) 21:59, 7 October 2026 (UTC)

• Comment: In preparing this draft, I disclose that AI assistance was used as follows: I wrote a draft that was pretty similar to the one submitted here and asked the chatbot for help with formulating and shortening the original text. All references have been added and checked by me, the chatbot did however suggest adding more links to wikipedia pages which I did as a result. Maxweso (talk) 20:39, 7 October 2026 (UTC)

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Simulation-based inference

Simulation-based inference (SBI) is a class of statistical inference methods that uses samples generated by a simulator to infer unknown parameters or other quantities of interest from observed data. It is particularly useful when the likelihood function of a statistical model is unavailable, computationally intractable, or too expensive to evaluate directly, while generating data from the model is feasible.

SBI is also referred to as likelihood-free inference, although the latter term can be misleading for methods that learn an approximation to the likelihood or likelihood ratio from simulated data. The methods are used in a range of fields including particle physics, gravitational wave astronomy, neuroscience and evolutionary biology

Statistical formulation

Consider a probabilistic model with parameters $\theta$ and observations $x$. In Bayesian inference, the objective is to determine the posterior distribution

$p(\theta \mid x_o)=\frac{p(x_o\mid \theta )p(\theta )}{p(x_o)},$

where $x_o$ denotes the observed data, $p(\theta )$ is the prior probability distribution, $p(x_o\mid \theta )$ is the likelihood function, and $p(x_o)$ is the marginal likelihood.

The likelihood is the probability density of the observed data conditional on a particular value of the model parameters. Many conventional Bayesian inference methods, such as Markov chain Monte Carlo methods, require repeated evaluation of the likelihood. For complex mechanistic models, however, the likelihood may be unavailable or computationally prohibitive to evaluate.

In SBI, the model is instead represented by a simulator or generative process. Given a parameter value $\theta$, the simulator generates synthetic observations according to

$x\sim p(x\mid \theta ).$

The simulator therefore provides samples from the model without necessarily providing an explicit expression for the likelihood. The objective is to use these simulated data to approximate the desired distribution, such as the posterior distribution.

The distinction between the forward and inverse problems is useful in this context. In the forward problem, parameters and a model are used to generate simulated observations. In the inverse problem, observed data are used to infer the parameters of the model. SBI addresses the inverse problem using repeated evaluations of the forward simulator.

Methods

SBI methods can be broadly divided according to which statistical quantity is learned from simulations. Three major approaches based on neural density estimation are neural posterior estimation, neural likelihood estimation, and neural ratio estimation. Methods that learn all statistical quantities have also been developed.

Amortized and sequential inference

SBI methods can also be distinguished by how simulations are allocated between training and inference.

In amortized inference, a model is trained using simulations over a broad range of parameter values before the observed data are analyzed. Once trained, the resulting estimator can perform inference for many observations with little additional computational cost. This can be advantageous when the same simulator is to be used for repeated inference tasks.

In sequential inference, simulations are generated in several rounds. Information obtained from earlier rounds is used to modify the distribution from which subsequent parameter values are simulated. This can concentrate computational effort around regions that are relevant to the observed data and thereby improve simulation efficiency.

Limitations

The main computational cost of SBI is often the simulation itself. When simulations are expensive, generating a sufficiently large training or reference data set can be a limiting factor. The performance of neural SBI methods can also depend on the choice of density estimator, simulation distribution and evaluation metric; benchmarking studies have found that there is no single method that performs best across all inference problems.

SBI is also dependent on the simulator being an adequate representation of the data-generating process. A posterior obtained from a highly accurate approximation of the simulated model does not necessarily provide an accurate description of the real-world system when the simulator is misspecified. Consequently, calibration, posterior predictive checks and other validation procedures are important components of SBI workflows.

See also

• Approximate Bayesian computation
• Bayesian inference
• Markov chain Monte Carlo
• Neural network

References

This article "Simulation-based inference" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Simulation-based inference. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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