A Comparison of Bayesian and Sampling Theory Inferences in a Probit Model

HE PROBIT MODEL IS A POPULAR DEVICE for explaining binary choice decisions in econometrics. It has been used to describe choices such as labor force participation, travel mode, home ownership, and type of education. These and many more examples can be found in papers by Amemiya (1981) and Maddala (1983). Given the contribution of economics towards explaining such choices, and given the nature of data that are collected, prior information on the relationship between a choice probability and several explanatory variables frequently exists. Bayesian inference is a convenient vehicle for including such prior information. Given the increasing popularity of Bayesian inference it is useful to ask whether inferences from a probit model are sensitive to a choice between Bayesian and sampling theory techniques. Of interest is the sensitivity of inference on coefficients, probabilities, and elasticities. We consider these issues in a model designed to explain choice between fixed and variable interest rate mortgages. Two Bayesian priors are employed: a uniform prior on the coefficients, designed to be noninformative for the coefficients, and an inequality restricted prior on the signs of the coefficients. We often know, a priori, whether increasing the value of a particular explanatory variable will have a positive or negative effect on a choice probability. This knowledge can be captured by using a prior probability density function (pdf) that is truncated to be positive or negative. Thus, three sets of results are compared:those from maximum likelihood (ML) estimation, those from Bayesian estimation with an unrestricted uniform prior on the coefficients, and those from Bayesian estimation with a uniform prior truncated to accommodate inequality restrictions on the coefficients.

Partitioning of reproduction in mother-daughter versus sibling associations : a test of optimal skew theory

A critical feature of cooperative animal societies is the reproductive skew, a shorthand term for the degree to which a dominant individual monopolizes overall reproduction in the group. Our theoretical analysis of the evolutionarily stable skew in matrifilial (i.e., mother-daughter) societies, in which relatednesses to offspring are asymmetrical, predicts that reproductive skews in such societies should tend to be greater than those of semisocial societies (i.e., societies composed of individuals of the same generation, such as siblings), in which relatednesses to offspring are symmetrical. Quantitative data on reproductive skews in semisocial and matrifilial associations within the same species for 17 eusocial Hymenoptera support this prediction. Likewise, a survey of reproductive partitioning within 20 vertebrate societies demonstrates that complete reproductive monopoly is more likely to occur in matrifilial than in semisocial societies, also as predicted by the optimal skew model.

Which is the optimal sampling strategy for habitat suitability modelling

Designing an efficient sampling strategy is of crucial importance for habitat suitability modelling. This paper compares four such strategies, namely, 'random', 'regular', 'proportional-stratified' and 'equal -stratified'- to investigate (1) how they affect prediction accuracy and (2) how sensitive they are to sample size. In order to compare them, a virtual species approach (Ecol. Model. 145 (2001) 111) in a real landscape, based on reliable data, was chosen. The distribution of the virtual species was sampled 300 times using each of the four strategies in four sample sizes. The sampled data were then fed into a GLM to make two types of prediction: (1) habitat suitability and (2) presence/ absence. Comparing the predictions to the known distribution of the virtual species allows model accuracy to be assessed. Habitat suitability predictions were assessed by Pearson's correlation coefficient and presence/absence predictions by Cohen's K agreement coefficient. The results show the 'regular' and 'equal-stratified' sampling strategies to be the most accurate and most robust. We propose the following characteristics to improve sample design: (1) increase sample size, (2) prefer systematic to random sampling and (3) include environmental information in the design'

Optimal control theory for a unitary operation under dissipative evolution

We show that optimizing a quantum gate for an open quantum system requires the time evolution of only three states irrespective of the dimension of Hilbert space. This represents a significant reduction in computational resources compared to the complete basis of Liouville space that is commonly believed necessary for this task. The reduction is based on two observations: the target is not a general dynamical map but a unitary operation; and the time evolution of two properly chosen states is sufficient to distinguish any two unitaries. We illustrate gate optimization employing a reduced set of states for a controlled phasegate with trapped atoms as qubit carriers and a iSWAP gate with superconducting qubits.

Conceptuals Mentefactos as a Didactic-Pedagogical Strategy of the Basic Concepts in the Sampling Theory Applyed to the Health Research

In populational sampling it is vitally important to clarify and discern: first, the design or sampling method used to solve the research problem; second, the sampling size, taking into account different components (precision, reliability, variance); third, random selection and fourth, the precision estimate (sampling errors), so as to determine if it is possible to infer the obtained estimates from the target population. The existing difficulty to use concepts from the sampling theory is to understand them with absolute clarity and, to achieve it, the help from didactic-pedagogical strategies arranged as conceptual “mentefactos” (simple hierarchic diagrams organized from propositions) may prove useful. This paper presents the conceptual definition, through conceptual “mentefactos”, of the most important populational probabilistic sampling concepts, in order to obtain representative samples from populations in health research.

Optimal tax theory and the taxation of housing in the US and the UK

First, we survey recent research in the application of optimal tax theory to housing. This work suggests that the under-taxation of housing for owner occupation distorts investment so that owner occupiers are encouraged to over-invest in housing. Simulations of the US economy suggest that this is true there. But, the theoretical work excludes consideration of land and the simulations exclude consideration of taxes other than income taxes. These exclusions are important for the US and UK economies. In the US, the property tax is relatively high. We argue that excluding the property tax is wrong, so that, when the property tax is taken into account, owner occupied housing is not undertaxed in the US. In the UK, property taxes are relatively low but the cost of land has been increasing in real terms for forty years as a result of a policy of constraining land for development. The price of land for housing is now higher than elsewhere. Effectively, an implicit tax is paid by first time buyers which has reduced housing investment. When land is taken into account over-investment in housing is not encouraged in the UK either.

On some extension of optimal control theory

Some problems of Calculus of Variations do not have solutions in the class of classic continuous and smooth arcs. This suggests the need of a relaxation or extension of the problem ensuring the existence of a solution in some enlarged class of arcs. This work aims at the development of an extension for a more general optimal control problem with nonlinear control dynamics in which the control function takes values in some closed, but not necessarily bounded, set. To achieve this goal, we exploit the approach of R.V. Gamkrelidze based on the generalized controls, but related to discontinuous arcs. This leads to the notion of generalized impulsive control. The proposed extension links various approaches on the issue of extension found in the literature.

On the allowance for support costs in Prager-Rozvany’ optimal layout theory

A recent study by Rozvany and Sokól discussed an important topic in structural design: the allowance for support costs in the optimization process. This paper examines a frequently used kind of support —that of simple foundation with horizontal reaction by friction— that appears no covered for the Authors’ approach. A simple example is examined to illustrate the case and to apply the Authors’ method and the standard design method.

3P sampling theory, examples, and rationale /

Originally published in Spanish in 1977 in "Proceedings of the (1975) 4th National Conference on CFI" by the Mexican Government.

Prospective evaluation of a D-optimal designed population pharmacokinetic study

Recently, methods for computing D-optimal designs for population pharmacokinetic studies have become available. However there are few publications that have prospectively evaluated the benefits of D-optimality in population or single-subject settings. This study compared a population optimal design with an empirical design for estimating the base pharmacokinetic model for enoxaparin in a stratified randomized setting. The population pharmacokinetic D-optimal design for enoxaparin was estimated using the PFIM function (MATLAB version 6.0.0.88). The optimal design was based on a one-compartment model with lognormal between subject variability and proportional residual variability and consisted of a single design with three sampling windows (0-30 min, 1.5-5 hr and 11 - 12 hr post-dose) for all patients. The empirical design consisted of three sample time windows per patient from a total of nine windows that collectively represented the entire dose interval. Each patient was assigned to have one blood sample taken from three different windows. Windows for blood sampling times were also provided for the optimal design. Ninety six patients were recruited into the study who were currently receiving enoxaparin therapy. Patients were randomly assigned to either the optimal or empirical sampling design, stratified for body mass index. The exact times of blood samples and doses were recorded. Analysis was undertaken using NONMEM (version 5). The empirical design supported a one compartment linear model with additive residual error, while the optimal design supported a two compartment linear model with additive residual error as did the model derived from the full data set. A posterior predictive check was performed where the models arising from the empirical and optimal designs were used to predict into the full data set. This revealed the optimal'' design derived model was superior to the empirical design model in terms of precision and was similar to the model developed from the full dataset. This study suggests optimal design techniques may be useful, even when the optimized design was based on a model that was misspecified in terms of the structural and statistical models and when the implementation of the optimal designed study deviated from the nominal design.

Optimal sampling strategy for estimation of spatial genetic structure in tree populations

Fine-scale spatial genetic structure (SGS) in natural tree populations is largely a result of restricted pollen and seed dispersal. Understanding the link between limitations to dispersal in gene vectors and SGS is of key interest to biologists and the availability of highly variable molecular markers has facilitated fine-scale analysis of populations. However, estimation of SGS may depend strongly on the type of genetic marker and sampling strategy (of both loci and individuals). To explore sampling limits, we created a model population with simulated distributions of dominant and codominant alleles, resulting from natural regeneration with restricted gene flow. SGS estimates from subsamples (simulating collection and analysis with amplified fragment length polymorphism (AFLP) and microsatellite markers) were correlated with the 'real' estimate (from the full model population). For both marker types, sampling ranges were evident, with lower limits below which estimation was poorly correlated and upper limits above which sampling became inefficient. Lower limits (correlation of 0.9) were 100 individuals, 10 loci for microsatellites and 150 individuals, 100 loci for AFLPs. Upper limits were 200 individuals, five loci for microsatellites and 200 individuals, 100 loci for AFLPs. The limits indicated by simulation were compared with data sets from real species. Instances where sampling effort had been either insufficient or inefficient were identified. The model results should form practical boundaries for studies aiming to detect SGS. However, greater sample sizes will be required in cases where SGS is weaker than for our simulated population, for example, in species with effective pollen/seed dispersal mechanisms.

Passive air sampling theory for semivolatile organic compounds

The mathematical modelling underlying passive air sampling theory can be based on mass transfer coefficients or rate constants. Generally, these models have not been inter-related. Starting with basic models, the exchange of chemicals between the gaseous phase and the sampler is developed using mass transfer coefficients and rate constants. Importantly, the inter-relationships between the approaches are demonstrated by relating uptake rate constants and loss rate constants to mass transfer coefficients when either sampler-side or air-side resistance is dominating chemical exchange. The influence of sampler area and sampler volume on chemical exchange is discussed in general terms and as they relate to frequently used parameters such as sampling rates and time to equilibrium. Where air-side or sampler-side resistance dominates, an increase in the surface area of the sampler will increase sampling rates. Sampling rates are not related to the sampler/air partition coefficient (K-SV) when air-side resistance dominates and increase with K-SV when sampler-side resistance dominates.

Accuracy and optimal sampling in Monte Carlo solution of population balance equations

Implementation of a Monte Carlo simulation for the solution of population balance equations (PBEs) requires choice of initial sample number (N0), number of replicates (M), and number of bins for probability distribution reconstruction (n). It is found that Squared Hellinger Distance, H2, is a useful measurement of the accuracy of Monte Carlo (MC) simulation, and can be related directly to N0, M, and n. Asymptotic approximations of H2 are deduced and tested for both one-dimensional (1-D) and 2-D PBEs with coalescence. The central processing unit (CPU) cost, C, is found in a power-law relationship, C= aMNb0, with the CPU cost index, b, indicating the weighting of N0 in the total CPU cost. n must be chosen to balance accuracy and resolution. For fixed n, M × N0 determines the accuracy of MC prediction; if b > 1, then the optimal solution strategy uses multiple replications and small sample size. Conversely, if 0 < b < 1, one replicate and a large initial sample size is preferred. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2394–2402, 2015

Sobolev Type Decomposition of Paley-Wiener-Schwartz Space with Application to Sampling Theory

2000 Mathematics Subject Classification: 94A12, 94A20, 30D20, 41A05.