Feature Learning for Bayesian Inference

September 1, 2022
In Progress
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The goal of this project is to use machine learning to find low-dimensional features in high-dimensional noisy data generated by stochastic models or real systems. In the first case, we are interested in the features imprinted on simulated data by the parameters of the stochastic model. In the latter, the interesting features depend on the particular system. In hydrology, one of the domains considered in this project, they are fingerprints of catchment properties in observed time-series of river-runoff. In astrophysics, another considered domain, the features are the parameters of solar dynamo models that govern the magnetic field strength of the sun. In each case, the problem is to disentangle the effect of high-dimensional disturbances from the effects of relevant characteristics (i.e., parameters, fingerprints, etc.). This problem is reminiscent of the problem of finding collective quantities characterizing states of interacting particle systems in statistical mechanics. Variational autoencoders, have been proven to be capable of learning such quantities in the form of order parameters or collective variables, which could then be used to, e.g., enhance Markov chain Monte Carlo samplers or molecular dynamics simulations.



SDSC Team:
Simon Dirmeier
Fernando Perez-Cruz

PI | Partners:

Swiss Federal Institute of Aquatic Science and Technology, Mathematical Methods in Environmental Research:

  • Dr. Carlo Albert
  • Dr. Simone Ulzega
  • Alberto Bassi

More info

University of Lugano, Institute of Computational Science:

  • Prof. Dr. Antonietta Mira

More info



Computational Bayesian inference of parameters or summary statistics is complicated in high-dimensional parameter- or data-spaces. In that case, conventional methods, such as approximate Bayesian computation or sequential Monte Carlo versions thereof, fail to infer correct posterior distributions for a measurement which necessitates the development of efficient methods for these scenarios. A recent approach utilizes deep learning to approximate posterior distributions with neural networks, however, these approaches still do not scale well to high-dimensional data, such as time series data of the magnetic field of solar cycle.

Proposed Approach / Solution

For this project, we develop novel methods for high-dimensional approximate inference, both in parameter- and data-space, utilizing recent developments in neural density estimation and dimensionality reduction. We a) propose several theoretical approaches to solve high-dimensional approximate inference and b) develop high-quality Python libraries implementing these algorithms for applied researchers to use.
One of our recent methods, which we call surjective sequential neural likelihood estimation, uses dimensionality-reducing normalizing flows to more efficiently estimate probability densities which allows for increased accuracy in inferring posterior distributions. On several experimental benchmarks our method outperforms baseline methods while being more computationally efficient w.r.t. the number of trainable parameters and computational speed (Figures 1 and 2).


We expect of our methods to be of great value for Bayesian inference, since current methodology is not suitable for high-dimensional time series data, for instance time series observations from solar dynamos or other ODE/PDE-models.

Figure 1: Assessing the performance of SSNL on several experimental benchmarks. We compared our method, SSNL, against several baseline methods (SNL, SNRE-C, SNPE-C) in four benchmark models. The x-axis shows the sample size used for training the model, while the y-axis shows the divergence to the true posterior (the lower the better). SSNL either outperforms the baselines or is on par while requiring less trainable parameters due to the dimensionality reduction.
Figure 2: Assessing the performance of SSNL on a complicated solar dynamo model. Applying our dimensionality-reducing method, SSNL, to a solar dynamo shows its advantage over a dimensionality-preserving method (SNL). With only 1000 data points (called “round 1” above) SSNL achieves to locate the posterior distributions correctly around the true parameter value (black dot), while SNL infers a biased posterior distribution. When using 15,000 data points (called “round 15”), both methods achieve similar results.




  • Dirmeier, S., Albert, C., & Perez-Cruz, F. (2023). Simulation-based inference using surjective sequential neural likelihood estimation. arXiv preprint arXiv:2308.01054
  • Dirmeier, S. (2024). Surjectors: surjection layers for density estimation with normalizing flows. Journal of Open Source Software, 9(94), 6188.
  • Bassi, A., Höge, M., Mira, A., Fenicia, F., & Albert, C. (2024). Learning Landscape Features from Streamflow with Autoencoders. Hydrology and Earth System Sciences Discussions, 2024, 1-30.
  • Di Noia, A., Macocco, I., Glielmo, A., Laio, A., & Mira, A. (2024). Beyond the noise: intrinsic dimension estimation with optimal neighbourhood identification. arXiv preprint arXiv:2405.15132.

Additional resources


  1. Albert, Carlo, Hans R. Künsch, and Andreas Scheidegger. "A simulated annealing approach to approximate Bayes computations." Statistics and computing 25 (2015): 1217-1232.
  2. Greenberg, David, Marcel Nonnenmacher, and Jakob Macke. "Automatic posterior transformation for likelihood-free inference." International Conference on Machine Learning. PMLR, 2019.
  3. Papamakarios, George, David Sterratt, and Iain Murray. "Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows." The 22nd international conference on artificial intelligence and statistics. PMLR, 2019.


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