# A tutorial on the Bayesian statistical approach to inverse problems

arXiv: https://arxiv.org/abs/2304.07610 · HTML: https://mlsanity.com/paper/2304.07610/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 122 references that are in the map.
- First posted on arXiv 15 Apr 2023.

## Builds on (most cited first, 6 of 6)

- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [Visualization in Bayesian workflow](https://mlsanity.com/paper/1709.01449/) (2017)
- [Prior knowledge elicitation: The past, present, and future](https://mlsanity.com/paper/2112.01380/) (2021)
- [What can be estimated? Identifiability, estimability, causal inference and ill-posed inverse problems](https://mlsanity.com/paper/1904.02826/) (2019)

## Abstract (arXiv)

Inverse problems are ubiquitous in the sciences and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input-output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a "solution" that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems, by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our tutorial aims to reach a wide range of scientists and engineers.