systematic errors

Ah ha~ Once I questioned, “what is systematic error?” (see [Q] systematic error.) Thanks to L. Lyons’ work discussed in [ArXiv] Particle Physics, I found this paper, titled Systematic Errors describing the concept and statistical inference related to systematic errors in the field of particle physics. It, gladly, shares lots of similarity with high energy astrophysics.

Systematic Errors by J. Heinrich and L.Lyons
in Annu. Rev. Nucl. Part. Sci. (2007) Vol. 57 pp.145-169 [http://adsabs.harvard.edu/abs/2007ARNPS..57..145H]

The characterization of two error types, systematic and statistical error is illustrated with an simple physics experiment, the pendulum. They described two distinct sources of systematic errors.

…the reliable assessment of systematics requires much more thought and work than for the corresponding statistical error.
Some errors are clearly statistical (e.g. those associated with the reading errors on T and l), and others are clearly systematic (e.g., the correction of the measured g to its sea level value). Others could be regarded as either statistical or systematic (e.g., the uncertainty in the recalibration of the ruler). Our attitude is that the type assigned to a particular error is not crucial. What is important is that possible correlations with other measurements are clearly understood.

Section 2 contains a very nice review in english, not in mathematical symbols, about the basics of Bayesian and frequentist statistics for inference in particle physics with practical accounts. Comparison of Bayes and Frequentist approaches is provided. (I was happy to see that χ2 is said to not belong to frequentist methods. It is just a popular method in references about data analysis in astronomy, not in modern statistics. If someone insists, statisticians could study the χ2 statistic under some assumptions and conditions that suit properties of astronomical data, investigate the efficiency and completeness of grouped Poission counts for Gaussian approximation within the χ2 minimization process, check degrees of information loss, and so forth)

To a Bayesian, probability is interpreted as the degree of belief in a statement. …
In contast, frequentists define probability via a repeated series of almost identical trials;…

Section 3 clarifies the notion of p-values as such:

It is vital to remember that a p-value is not the probability that the relevant hypothesis is true. Thus, statements such as “our data show that the probability that the standard model is true is below 1%” are incorrect interpretations of p-values.

This reminds me of the null hypothesis probability that I often encounter in astronomical literature or discussions to report the X-ray spectral fitting results. I believe astronomers using the null hypothesis probability are confused between Bayesian and frequentist concepts. The computation is based on the frequentist idea, p-value but the interpretation is given via Bayesian. A separate posting on the null hypothesis probability will come shortly.

Section 4 describes both Bayesian and frequentist ways to include systematics. Through its parameterization (for Gaussian, parameterization is achieved with additive error terms, or none zero elements in full covariance matrix), systematic uncertainty is treated as nuisance parameters in the likelihood for both Bayesian and frequentist alike although the term “nuisance” appears in frequentist’s likelihood principles. Obtaining the posterior distribution of a parameter(s) of interest requires marginalization over uninteresting parameters which are seen as nuisance parameters in frequentist methods.

The miscellaneous section (Sec. 6) is the most useful part for understanding the nature and strategies for handling systematic errors. Instead of copying the whole section, here are two interesting quotes:

When the model under which the p-value is calculated has nuisance parameters (i.e. systematic uncertainties) the proper computation of the p-value is more complicated.

The contribution form a possible systematic can be estimated by seeing the change in the answer a when the nuisance parameter is varied by its uncertainty.

As warned, it is not recommended to combine calibrated systematic error and estimated statistical error in quadrature, since we cannot assume those errors are uncorrelated all the time. Except the disputes about setting a prior distribution, Bayesian strategy works better since the posterior distribution is the distribution of the parameter of interest, directly from which one gets the uncertainty in the parameter. Remember, in Bayesian statistics, parameters are random whereas in frequentist statistics, observations are random. The χ2 method only approximates uncertainty as Gaussian (equivalent to the posterior with a gaussian likelihood centered at the best fit and with a flat prior) with respect to the best fit and combines different uncertainties in quadrature. Neither of strategies is superior almost always than the other in a general term of performing statistical inference; however, case-specifically, we can say that one functions better than the other. The issue is how to define a model (distribution, distribution family, or class of functionals) prior to deploying various methodologies and therefore, understanding systematic errors in terms of model, or parametrization, or estimating equation, or robustness became important. Unfortunately, systematic descriptions about systematic errors from the statistical inference perspective are not present in astronomical publications. Strategies of handling systematic errors with statistical care are really hard to come by.

Still I think that their inclusion of systematic errors is limited to parametric methods, in other words, without parametrization of systematic errors, one cannot assess/quantify systematic errors properly. So, what if such parametrization of systematics is not available? I thought that some general semi-parametric methodology possibly assists developing methods of incorporating systematic errors in spectral model fitting. Our group has developed a simple semi-parametric way to incorporate systematic errors in X-ray spectral fitting. If you like to know how it works, please check out my poster in pdf. It may be viewed too conservative as if projection since instead of parameterizing systemtatics, the posterior was empirically marginalized over the systematics, the hypothetical space formed by simulated sample of calibration products.

I believe publications about handling systematic errors will enjoy prosperity in astronomy and statistics as long as complex instruments collect data. Beyond combining in quadrature or Gaussian approximation, systematic errors can be incorporated in a more sophisticated fashion, parametrically or nonparametrically. Particularly for the latter, statisticians knowledge and contributions are in great demand.