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https://github.com/htjb/beta-flows


https://github.com/htjb/beta-flows

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README 

        
# $\beta$-flows: Learning with the whole Nested Sampling run



|bflows | Temperature dependent Normalising Flows |

|---------|-----------|

|Author | Harry Bevins|

|Homepage | |



UNDER CONSTRUCTION



$\beta$-flows are a density estimation tool that were intially designed

to improve the accuracy of more traditional normalising flows.



They are built using conditional masked autoregressive flows and utilise the

thermodynamic description of nested sampling during training. The structure

of the $\beta$-flows class and the interface are largely based on

the code [margarine](https://github.com/htjb/margarine).



The nested sampling algorithm is used to perform Bayesian inference and estimate

the normalising factor on Bayes theorem known as the evidence



$$\mathcal{Z} = \int \mathcal{L}(\theta)\pi(\theta) d\theta$$



where $\mathcal{L}(\theta)$ is the postulated probability, known as the

likelihood, of the data given a set of parameters $\theta$ and $\pi(\theta)$

is the prior on the parameters. In the process of estimating the evidence the

nested sampling algorithm also returns samples on the posterior 

probability $P(\theta) = P(\theta|D, M)$.



An analogy can be made between the Bayesian evidence and the partition function

in statistical mechanics. If one writes the energy of a given state as

$E(\theta) = - \log \mathcal{L}(\theta)$ then the partition function can be 

written as



$$\mathcal{Z}(\beta) = \int e^{-\beta E} g(E) dE = \int \mathcal{L}(\theta)^\beta \pi(\theta)

d\theta$$



where $g(E)$ is the density of states with a given energy and $\beta$ is an inverse

temperature. It can be seen that when $\beta=1$ we recover the integral over the posterior

and when $\beta=0$ the integral is over the prior.



We can therefore track over the nested sampling algorihtm transforms the prior

into the posterior by varying $\beta$ between 0 and 1. Further we can

generate a set of samples from any distribution between the prior and posterior

by reweighting the returned posterior points according to



$$p_i^\beta = \frac{w_i \mathcal{L}_i^\beta} {\mathcal{Z}(\beta)}$$



where $w_i$ corresponds to the prior volume contraction at each iteration in the

run.



It has been shown that normalising flows can be used to emulate posteriors, priors

and likelihood functions and that researchers can take advantage of these emulators

in a number of different ways.



Typically normalising are trained on the $\beta=1$ posterior distribution or

the $\beta=0$ prior distribution. When learning the posterior researchers often

find that the flows perform poorly in the tails of the distribution.



The inovation behind $\beta$-flows is to train the flow conditioned on $\beta$ using 

appropriately weighted samples from a nested sampling run.



This has been seen to improve the accuracy of emulation of the $\beta=1$

distribution and has many other applications.



## Example



## Citation



Please cite this code base if you use Beta-flows in your work and the corresponding

paper [Accelerated nested sampling with  β -flows for gravitational waves](https://ui.adsabs.harvard.edu/abs/2024arXiv241117663P/abstract).



```bibtex

@ARTICLE{2024arXiv241117663P,

       author = {{Prathaban}, Metha and {Bevins}, Harry and {Handley}, Will},

        title = "{Accelerated nested sampling with $\beta$-flows for gravitational waves}",

      journal = {arXiv e-prints},

     keywords = {Astrophysics - Instrumentation and Methods for Astrophysics, Astrophysics - High Energy Astrophysical Phenomena, General Relativity and Quantum Cosmology},

         year = 2024,

        month = nov,

          eid = {arXiv:2411.17663},

        pages = {arXiv:2411.17663},

archivePrefix = {arXiv},

       eprint = {2411.17663},

 primaryClass = {astro-ph.IM},

       adsurl = {https://ui.adsabs.harvard.edu/abs/2024arXiv241117663P},

      adsnote = {Provided by the SAO/NASA Astrophysics Data System}

}

```



```bibtex

@article{beta-flows,

    year  = {2024},

    author = {Harry Bevins et al},

    title = {beta-flows: Temperature dependent Normalising Flows},

    journal = {In preparation}

}

```



## Contribution



Contributions are welcome. Please open up an issue to discuss.