Update of Prior Probabilities by Minimal Divergence

The present paper investigates the update of an empirical probability distribution with the results of a new set of observations. The update reproduces the new observations and interpolates using prior information. The optimal update is obtained by minimizing either the Hellinger distance or the quadratic Bregman divergence. The results obtained by the two methods differ. Updates with information about conditional probabilities are considered as well.

Encoding a Categorical Independent Variable for Input to TerrSet’s Multi-Layer Perceptron

The profession debates how to encode a categorical variable for input to machine learning algorithms, such as neural networks. A conventional approach is to convert a categorical variable into a collection of binary variables, which causes a burdensome number of correlated variables. TerrSet’s Land Change Modeler proposes encoding a categorical variable onto the continuous closed interval from 0 to 1 based on each category’s Population Evidence Likelihood (PEL) for input to the Multi-Layer Perceptron, which is a type of neural network. We designed examples to test the wisdom of these encodings. The results show that encoding a categorical variable based on each category’s Sample Empirical Probability (SEP) produces results similar to binary encoding and superior to PEL encoding. The Multi-Layer Perceptron’s sigmoidal smoothing function can cause PEL encoding to produce nonsensical results, while SEP encoding produces straightforward results. We reveal the encoding methods by illustrating how a dependent variable gains across an independent variable that has four categories. The results show that PEL can differ substantially from SEP in ways that have important implications for practical extrapolations. If users must encode a categorical variable for input to a neural network, then we recommend SEP encoding, because SEP efficiently produces outputs that make sense.

Asymptotic Theory Of Stochastic Choice Functionals For Prospects With Embedded Comotonic Probability Measures

We introduce a monotone class theory of Prospect Theory's value functions, which shows that they can be replaced almost surely by a topological lifting comprised of a class of compact isomorphic maps that embed weakly co-monotonic probability measures, attached to state space, in outcome space. Thus, agents solve a signal extraction problem to obtain estimates of empirical probability weights for prospects under risk and uncertainty. By virtue of the topological lifting, we prove an almost sure isomorphism theorem between compact stochastic choice operators, and well defined outcomes which, under Brouwer-Schauder theory, guarantees fixed point convergence in convex choice sets. Along the way we introduce a risk operator in the Hoffman-Jorgensen class of lifting operators, and value function [averaging] operators with respect to Radon measure. In that set up, suitable binary operations on gain-loss space show that our risk operator is isometric for gains and skewed for losses. The point spectrum from this operator constitutes the range of admissible observations for loss aversion index in a well designed experiment.

Asymptotic Theory Of Stochastic Choice Functionals For Prospects With Embedded Comotonic Probability Measures

We introduce a monotone class theory of Prospect Theory's value functions, which shows that they can be replaced almost surely by a topological lifting comprised of a class of compact isomorphic maps that embed weakly co-monotonic probability measures, attached to state space, in outcome space. Thus, agents solve a signal extraction problem to obtain estimates of empirical probability weights for prospects under risk and uncertainty. By virtue of the topological lifting, we prove an almost sure isomorphism theorem between compact stochastic choice operators, and well defined outcomes which, under Brouwer-Schauder theory, guarantees fixed point convergence in convex choice sets. Along the way we introduce a risk operator in the Hoffman-Jorgensen class of lifting operators, and value function [averaging] operators with respect to Radon measure. In that set up, suitable binary operations on gain-loss space show that our risk operator is isometric for gains and skewed for losses. The point spectrum from this operator constitutes the range of admissible observations for loss aversion index in a well designed experiment.

Hierarchical Computational Anatomy: Unifying the Molecular to Tissue Continuum via Measure Representations of the Brain

This paper presents a unified representation of the brain based on mathematical functional measures integrating the molecular and cellular scale descriptions with continuum tissue scale descriptions. We present a fine-to-coarse recipe for traversing the brain as a hierarchy of measures projecting functional description into stable empirical probability laws that unifies scale-space aggregation. The representation uses measure norms for mapping the brain across scales from different measurement technologies. Brainspace is constructed as a metric space with metric comparison between brains provided by a hierarchy of Hamiltonian geodesic flows of diffeomorphisms connecting the molecular and continuum tissue scales. The diffeomorphisms act on the brain measures via the 3D varifold action representing "copy and paste" so that basic particle quantities that are conserved biologically are combined with greater multiplicity and not geometrically distorted. Two applications are examined, the first histological and tissue scale data in the human brain for studying Alzheimer's disease, and the second the RNA and cell signatures of dense spatial transcriptomics mapped to the meso-scales of brain atlases. The representation unifies the classical formalism of computational anatomy for representing continuum tissue scale with non-classical generalized functions appropriate for molecular particle scales.

Potential impact of a nonavalent anti HPV vaccine in Italian men with and without clinical manifestations

Human papilloma virus infection (HPV) is the most common sexually transmitted disease. Little is known about male infection. Nonavalent vaccine against types 6/11/16/18/31/33/45/52/58 was approved and neutral gender immunization programs have been proposed. This study evaluates the potential impact of nonavalent vaccine compared to quadrivalent in male living in Sicily (Italy). 58.7% of samples were HPV positive and forty-four types of HPV were identified. A significant higher estimated coverage of nonavalent vaccine than quadrivalent was observed (64.3% vs. 45.8%), with absolute and relative additional impact of 20.1% and 47.2%, respectively. Low impact of the vaccine were calculated as the empirical probability of HPV genotypes 6/11/16/18/31/33/45/52/58 alone or in combination; the high impact as empirical probability of HPV6/11/16/18/31/33/45/52/58 genotypes alone or in association with other genotypes. The potential impact of the nonavalent vaccine vs quadrivalent was significant for low and high impact (29.7% > 18:8%; 34:6% > 26.6%, respectively). Particularly, in men with lesions and risky sexual contact was significant only for low impact (35.5% > 29.7%; 31.4% > 19.7%, respectively). In partners with positive females was significant for low impact (26.3% > 15.1%) and high impact (33.7% > 23.2%). Nonavalent vaccine offers broader protection in men with HPV positive partners, who would have a potential role in the transmission of the infection.