__doc__
Project description
Bayesian average
Version:
0.1.7
Authors
Martino Trassinelli
CNRS, Institute of NanoSciences of Paris
email: trassinelli AT insp.jussieu.fr
email: m.trassinelli AT gmail.com
Marleen Maxton
Max Planck Institute Heidelberg
email:
Homepage
https://github.com/martinit18/bayesian_average
License
Type: X11, see LICENCE.txt
Short description
Calculation of robust weigted average from a set of data points and their uncertainties based on Bayesian statistical methods. The proposed weighted average is particularly adapted to inconsistent data and for the presence of outliers, which can both false the results of standard methods.
Basic principles
From a ntuple data
and sigma
corresponding to a set of data points $x_i$ and the associated uncertainties $\sigma_i$, this package calculate the corresponding weighted average particularly adapted for inconsistent data sets (with a spread larger than the asociated error bars) and/or the presence of outliers.
This robust weighted average is based on Bayesian statistics assuming a normal disrtribution for each $x_i$ and considering the values in sigma
as just a lower bound of the real possibly larger uncertainty $\sigma'$.
It is obtained by the marginalization on $\sigma'$, this result in a modified probability distribution for each $x_i$ that still depens on $\sigma_i$.
Two different priors are proposed for $\sigma'$: the non-informative Jeffreys' prior $p(\sigma') \propto 1/ \sigma'$ (more precisely its limit, see Ref.[1]), and a modified version of it $p(\sigma') \propto 1/ (\sigma')^2$ proposed in Ref.[2].
For both priors, the weighted average and its associated uncertainty are obtained numerically using basinhopping
minimisation algorithm.
In addition, the standard (inverse-variance) weighted average is also available for possible comparisons.
How to use it
For the calculation of the Jeffreys' weighted average (see below for other averages)
import bayesian_average as ba
ba.jwa(data,sigma)
where data
is a ntuple of data and sigma
is the associated uncertainty of the same dimension.
The typical output is
(6.6742395674538315, 9.74833292573106e-5)
where the first number is the weighted average and the second one is the estimated final uncertainty
To plot the resulting probability distribution, the final weighted average and the input data (optionally)
ba.plot_average(data,sigma,jwa_val=True,plot_data=True)
The option jwa_val=True
is on on as default. plot_data=True
show the input data in addition.
Details of the vailable weighted averages
jwa
: Jeffreys weighted average (main average, RECOMENDED, see Ref.[1]).
The priors of the real uncertainty value are non-informative Jeffeys' prior proportional to $1/\sigma'$. Because of the non-normalisability of the final probability distribution, this weighted average results correspond to the limit case with prior bounds $[\sigma, \sigma_\mathrm{max}]$ with $\sigma_\mathrm{max} \to \infty$ and where $\sigma$ is the value provided by the user. The final probability distribution is, however not a proper probability distribution.cwa
: Conservative weighted average (adapted for proper final probability distributions, see Ref.[2]).
The priors of the real uncertainty value are proportional to $\sigma/(\sigma')^2$, where $\sigma$ is the value provided by the user. The bounds of the prior are $[\sigma, \sigma_\mathrm{max}]$. This is a modified and normalisable version of the non-informative Jeffeys' prior.wa
: Standard weighted average
The standard inverse-variance weighted average useful for comparisons.
Refere articles:
[1] M. Trassinelli and M. Maxton, A minimalistic and general weighted average for inconsistent data, in preparation for Metrologia
[2] D. S. Sivia and J. Skilling, Data analysis: a Bayesian tutorial, 2nd ed 2006, Oxford Univ. Press
Project details
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