We discuss hypothesis testing and compare different theories in light of observed or experimental data as fundamental endeavors in the sciences. Issues associated with the p-value approach and null hypothesis significance testing are reviewed, and the Bayesian alternative based on the Bayes factor is introduced, along with a review of computational methods and sensitivity related to prior distributions. We demonstrate how Bayesian testing can be practically implemented in several examples, such as the t-test, two-sample comparisons, linear mixed models, and Poisson mixed models by using existing software. Caveats and potential problems associated with Bayesian testing are also discussed. We aim to inform researchers in the many fields where Bayesian testing is not in common use of a well-developed alternative to null hypothesis significance testing and to demonstrate its standard implementation.

Gaussian process models underlie many spatial transcriptomics tools but typically assume stationary covariance. To test this assumption, we use approximate Bayes factors from R-INLA to compare stationary and non-stationary Matérn covariance functions across 12 Visium datasets; covariance non-stationarity is substantially evident as it appears in 3% to 50% of genes across datasets. We find that gene sets associated with effector functions are enriched among genes favoring non-stationary spatial covariance.