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The production of complex astronomical data is accelerating, especially with newer telescopes producing ever more large-scale surveys. The increased quantity, complexity, and variety of astronomical data demand a parallel increase in skill and sophistication in developing, deciding, and deploying statistical methods. Understanding limitations and appreciating nuances in statistical and machine learning methods and the reasoning behind them is essential for improving data-analytic proficiency and acumen. Aiming to facilitate such improvement in astronomy, we delineate cautionary tales in statistics via six maxims, with examples drawn from the astronomical literature. Inspired by the significant quality improvement in business and manufacturing processes by the routine adoption of Six Sigma, we hope the routine reflection on these Six Maxims will improve the quality of both data analysis and scientific findings in astronomy.

We present a new method to distinguish between different states (e.g., high and low, quiescent and flaring) in astronomical sources with count data. The method models the underlying physical process as latent variables following a continuous-space Markov chain that determines the expected Poisson counts in observed light curves in multiple passbands. For the underlying state process, we consider several autoregressive processes, yielding continuous-space hidden Markov models of varying complexity. Under these models, we can infer the state that the object is in at any given time. The continuous state predictions from these models are then dichotomized with the help of a finite mixture model to produce state classifications. We apply these techniques to X-ray data from the active dMe flare star EV Lac, splitting the data into quiescent and flaring states. We find that a first-order vector autoregressive process efficiently separates flaring from quiescence: flaring occurs over 30-40% of the observation durations, a well-defined persistent quiescent state can be identified, and the flaring state is characterized by higher plasma temperatures and emission measures., Comment: 27 pages, 13 figures, 12 tables

We estimate the coronal density of Capella using the O VII and Fe XVII line systems in the soft X-ray regime that have been observed over the course of the Chandra mission. Our analysis combines measures of error due to uncertainty in the underlying atomic data with statistical errors in the Chandra data to derive meaningful overall uncertainties on the plasma density of the coronae of Capella. We consider two Bayesian frameworks. First, the so-called pragmatic-Bayesian approach considers the atomic data and their uncertainties as fully specified and uncorrectable. The fully-Bayesian approach, on the other hand, allows the observed spectral data to update the atomic data and their uncertainties, thereby reducing the overall errors on the inferred parameters. To incorporate atomic data uncertainties, we obtain a set of atomic data replicates, the distribution of which captures their uncertainty. A principal component analysis of these replicates allows us to represent the atomic uncertainty with a lower-dimensional multivariate Gaussian distribution. A $t$-distribution approximation of the uncertainties of a subset of plasma parameters including a priori temperature information, obtained from the temperature-sensitive-only Fe XVII spectral line analysis, is carried forward into the density- and temperature-sensitive O VII spectral line analysis. Markov Chain Monte Carlo based model fitting is implemented including Multi-step Monte Carlo Gibbs Sampler and Hamiltonian Monte Carlo. Our analysis recovers an isothermally approximated coronal plasma temperature of $\approx$5 MK and a coronal plasma density of $\approx$10$^{10}$ cm$^{-3}$, with uncertainties of 0.1 and 0.2 dex respectively.

We present a new method for joint likelihood deconvolution (Jolideco) of a set of astronomical observations of the same sky region in the presence of Poisson noise. The observations may be obtained from different instruments with different resolution, and different point spread functions. Jolideco reconstructs a single flux image by optimizing the posterior distribution based on the joint Poisson likelihood of all observations under a patch-based image prior. The patch prior is parameterised via a Gaussian Mixture model which we train on high-signal-to-noise astronomical images, including data from the James Webb Telescope and the GLEAM radio survey. This prior favors correlation structures among the reconstructed pixel intensities that are characteristic of those observed in the training images. It is, however, not informative for the mean or scale of the reconstruction. By applying the method to simulated data we show that the combination of multiple observations and the patch-based prior leads to much improved reconstruction quality in many different source scenarios and signal to noise regimes. We demonstrate that with the patch prior Jolideco yields superior reconstruction quality relative to alternative standard methods such as the Richardson-Lucy method. We illustrate the results of Jolideco applied to example data from the Chandra X-ray Observatory and the Fermi-LAT Gamma-ray Space Telescope. By comparing the measured width of a counts based and the corresponding Jolideco flux profile of an X-ray filament in SNR 1E 0102.2-721} we find the deconvolved width of 0.58+- 0.02 arcsec to be consistent with the theoretical expectation derived from the known width of the PSF., Comment: 25 pages, 11 figures

Spectral fitting of X-ray data usually involves minimizing statistics like the chi-square and the Cash statistic. Here we discuss their limitations and introduce two measures based on the cumulative sum (CuSum) of model residuals to evaluate whether model complexity could be increased: the percentage of bins exceeding a nominal threshold in a CuSum array (pct$_{CuSum}$), and the excess area under the CuSum compared to the nominal (p$_\textit{area}$). We demonstrate their use with an application to a $\textit{Chandra}$ ACIS spectral fit., Comment: 3 pages, 1 figure, published in the Research Notes of the American Astronomical Society (RNAAS)