COVID-19 testing has become a standard approach for estimating prevalence which then assist in public health decision making to contain and mitigate the spread of the disease. The sampling designs used are often biased in that they do not reflect the true underlying populations. For instance, individuals with strong symptoms are more likely to be tested than those with no symptoms. This results in biased estimates of prevalence (too high). Typical post-sampling corrections are not always possible. Here we present a simple bias correction methodology derived and adapted from a correction for publication bias in meta analysis studies. The methodology is general enough to allow a wide variety of customization making it more useful in practice. Implementation is easily done using already collected information. Via an example and a real dataset, we show that the bias corrections can provide dramatic reductions in estimation error.

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The fine-tuning of the universe for life, the idea that the constants of nature (or ratios between them) must belong to very small intervals in order for life to exist, has been debated by scientists for several decades. Several criticisms have emerged concerning probabilistic measurement of life-permitting intervals. Herein, a Bayesian statistical approach is used to assign an upper bound for the probability of tuning, which is invariant with respect to change of physical units, and under certain assumptions it is small whenever the life-permitting interval is small on a relative scale. The computation of the upper bound of the tuning probability is achieved by first assuming that the prior is chosen by the principle of maximum entropy (MaxEnt). The unknown parameters of this MaxEnt distribution are then handled in such a way that the weak anthropic principle is not violated. The MaxEnt assumption is “maximally noncommittal with regard to missing information.” This approach is sufficiently general to be applied to constants of current cosmological models, or to other constants possibly under different models. Application of the MaxEnt model reveals, for example, that the ratio of the universal gravitational constant to the square of the Hubble constant is finely tuned in some cases, whereas the amplitude of primordial fluctuations is not.

Principal Components Analysis is a widely used technique for dimension reduction and characterization of variability in multivariate populations. Our interest lies in studying when and why the rotation to principal components can be used effectively within a response-predictor set relationship in the context of mode hunting. Specifically focusing on the Patient Rule Induction Method (PRIM), we first develop a fast version of this algorithm (fastPRIM) under normality which facilitates the theoretical studies to follow. Using basic geometrical arguments, we then demonstrate how the PC rotation of the predictor space alone can in fact generate improved mode estimators. Simulation results are used to illustrate our findings.

Philosophers frequently define knowledge as justified, true belief. In this paper we build a mathematical framework that makes possible to define learning (increased degree of true belief) and knowledge of an agent in precise ways. This is achieved by phrasing belief in terms of epistemic probabilities, defined from Bayes' Rule. The degree of true belief is then quantified by means of active information $I^+$, that is, a comparison between the degree of belief of the agent and a completely ignorant person. Learning has occurred when either the agent's strength of belief in a true proposition has increased in comparison with the ignorant person ($I^+>0$), or if the strength of belief in a false proposition has decreased ($I^+<0$). Knowledge additionally requires that learning occurs for the right reason, and in this context we introduce a framework of parallel worlds, of which one is true and the others are counterfactuals. We also generalize the framework of learning and knowledge acquisition to a sequential setting, where information and data is updated over time. The theory is illustrated using examples of coin tossing, historical events, future events, replication of studies, and causal inference.