Inference Of Model Parameters Research Articles

Empty confidence sets for epidemics, branching processes and Brownian motion

SUMMARY This paper treats some examples where likelihood-based inference for certain model parameters may produce empty confidence sets. The first example concerns epidemics, and the parameter of interest is the basic reproduction number R0, which is to be estimated from the final size of an epidemic in a finite population. The second example treats estimation of the mean of the offspring distribution in a branching process, based on observing the total progeny, i.e. the total number of individuals ever born in the branching process. The final example considers estimation of the linear drift in a Brownian motion, based on observing the first hitting time of some horizontal barrier.

The Anatomy of Inverse Problems

A major task of geophysics is to make quantitative statements about the interior of the earth. For this reason, inverse problems are an important area of geophysical research and industrial application. Figure 1 shows how many texts present inverse problems. The earth model is an element of a mathematical space that contains all allowable parameterizations of the earth’s properties (or at least those properties relevant to a given experiment); this space is referred to as model space . The physics of the problem determines which data d correspond to a given model m . The problem of computing the model response (synthetic “data”) given a model is called the forward problem . The corresponding data reside in a mathematical space that is called data space . In many applications, one records the data, and the goal is to find the corresponding model. The task is called the inverse problem , as shown in Figure 1. FIG. 1. The conventional view of inverse problems: find the model that predicts the measurements. Unfortunately, Figure 1 is wrong. There is a simple reason for this. In general the model that one seeks is a continuous function of the space variables with infinitely many degrees of freedom. For example, the 3-D velocity structure in the earth has infinitely many degrees of freedom. On the other hand, the data space is always of finite dimension because any real experiment can only result in a finite number of measurements. A simple count of variables shows that the mapping from the data to a model cannot be unique; or equivalently, there must be elements of the model space that have no influence on the data. This lack of uniqueness is apparent even for problems involving idealized, noise-free measurements. The problem only becomes worse when the uncertainties of real …

Estimation and Inference Are Missing Data Problems: Unifying Social Science Statistics via Bayesian Simulation

Bayesian simulation is increasingly exploited in the social sciences for estimation and inference of model parameters. But an especially useful (if often overlooked) feature of Bayesian simulation is that it can be used to estimate any function of model parameters, including “auxiliary” quantities such as goodness-of-fit statistics, predicted values, and residuals. Bayesian simulation treats these quantities as if they were missing data, sampling from their implied posterior densities. Exploiting this principle also lets researchers estimate models via Bayesian simulation where maximum-likelihood estimation would be intractable. Bayesian simulation thus provides a unified solution for quantitative social science. I elaborate these ideas in a variety of contexts: these include generalized linear models for binary responses using data on bill cosponsorship recently reanalyzed in Political Analysis, item—response models for the measurement of respondent's levels of political information in public opinion surveys, the estimation and analysis of legislators' ideal points from roll-call data, and outlier-resistant regression estimates of incumbency advantage in U.S. Congressional elections

Do the Spectra of Soft X‐Ray Transients Reveal Bulk‐Motion Inflow Phenomenon?

We present our analysis of the high-energy radiation from black hole (BH) transients using archival data obtained primarily with RXTE and a comprehensive test of the bulk-motion Comptonization (BMC) model for the high-soft state continuum. The emergent spectra of over 30 separate measurements of the GRO J1655-40, GRS 1915+105, GRS 1739-278, 4U 1630-47 XTE J1755-32, and EXO 1846-031 X-ray sources are successfully fitted by the BMC model, which has been derived from basic physical principles in previous work. This in turn provides direct physical insight into the innermost observable regions, where matter impinging on the event horizon can effectively be directly viewed. The BMC model is characterized by three parameters: the disk color temperature, a geometric factor related to the illumination of the BH site by the disk, and a spectral index related to the efficiency of the bulk-motion upscattering. For the case of GRO J1655-40, where there are distance and mass determinations, a self-consistency check of the BMC model has been made, in particular of the assumption regarding the dominance of gravitational forces over the pressure forces within the inner few Schwarzschild radii. We have also examined the time behavior of these parameters, which can provide information on the source structure. Using our inferred model parameters: color temperature, spectral index, and an absolute normalization, we present new, independently derived constraints on the BH mass, mass accretion rate, and the distance for the aforementioned sources. Also notable is the relationship between the color temperature and flux, which for GRO J1655-40 is entirely distinct from a simple T4 dependence and strikingly consistent with the disk model we have invoked: a standard Shakura-Sunyaev disk with the modification to the electron scattering. This provides insight into the origin of the seed soft photons and allows us to impose an important estimation of the hardening parameter, Th, which is the ratio of the color temperature to the effective temperature: we find Th2.6, higher than previous estimates used in the literature.

A case study of model selection and parameter inference by maximum likelihood with application to uncertainty analysis

One of the uses of geostatistical conditional simulation is as a tool in assessing the spatial uncertainty of inputs to the Monte Carlo method of system uncertainty analysis. Because the number of experimental data in practical applications is limited, the geostatistical parameters used in the simulation are themselves uncertain. The inference of these parameters by maximum likelihood allows for an easy assessment of this estimation uncertainty which, in turn, may be included in the conditional simulation procedure. A case study based on transmissivity data is presented to show the methodology whereby both model selection and parameter inference are solved by maximum likelihood.

Pulmonary gas exchange kinetics during exercise: physiological inferences of model order and parameters

1. 1. The ventilatory and pulmonary gas exchange responses during moderate exercise can be appropriately modelled with first-order dynamics. 2. 2. A delay term, reflecting tissue-to-lung transit time, is needed for accurate characterization, however. 3. 3. The O 2 uptake time constant ( τ V ̇ O 2 reflects the enzymatically controlled tissue O 2 utilization. 4. 4. τ V ̇ CO 2 is appreciably longer than τ V ̇ O 2 , consequent to the tissue CO 2 capacitance. 5. 5. As τ V ̇ E typically longer than τ V ̇ CO 2 , transient errors in alveolar and arterial blood gas tensions are predicted: small for P CO 2 but much larger for P O 2. 6. 6. At work rates above the lactate threshold, a slow and delayed component of V̇ O 2 induces an additional V̇ component (“excess” V̇ O 2), leading to more rapid fatigue. 7. 7. The ventilatory compensation for the metabolic acidemia at these work rates is slow, with compensation being poor for rapid-incremental exercise. 8. 8. A justifiable control model of the coupling of ventilation to metabolism must cohere with these demonstrable physiological characteristics.

Regression analysis of data with repeated measurements using the method of successive differences

This paper discusses the first-order differencing approach to longitudinal data with measurements on individual subjects taken at irregular time points. Unlike the two-stage approach to this problem, which first fits a line or curve to each individual's growth pattern and then regresses the fitted line or curve on the covariates of interest, subjects with only two-observations can still contribute to the estimation of parameters. Another major advantage is that any time trend of the response or any covariate effects on the response over a time interval within the total study period can be detected. Such an analysis can be very useful in clinical trials studies, especially in trials involving testing a new drug or therapy. The therapeutic effect of a drug or therapy may decrease with time and it is important to know the time period in which the treatment is most effective so that the shortest treatment duration can be determined to reduce the risks of toxic side-effects. Since estimation and inference of model parameters are performed under the framework of the multiple regression model, the procedures are easily implemented using popular statistical software packages. The estimators are asymptotically normal even when the assumed correlation structure is not met by the underlying data. The methods are illustrated with data from a study of primary anaplastic brain tumors.

An application of Bayesian nonlinear regression to hydrologic models

A suite of FORTRAN 77 computer programs implementing Bayesian nonlinear regression is described. These programs, developed to cope with complex hydrologic models, are used to infer model parameters, test model structure and make predictions of future response. The programs are fully interactive with features including: interactive parameter optimization using the Gauss-Marquardt algorithm; run-time editing of user options; availability of a general error model; use of prior information; and joint fitting of multiple-response data.

Autocorrelation and regularization in digital images. I. Basic theory

Spatial structure occurs in remotely sensed images when the imaged scenes contain discrete objects that are identifiable in that their spectral properties are more homogeneous within than between them and other scene elements. The spatial structure introduced is manifest in statistical measures such as the autocovariance function and variogram associated with the scene, and it is possible to formulate these measures explicitly for scenes composed of simple objects of regular shapes. Digital images result from sensing scenes by an instrument with an associated point spread function (PSF). Since there is averaging over the PSF, the effect, termed regularization, induced in the image data by the instrument will influence the observable autocovariance and variogram functions of the image data. It is shown how the autocovariance or variogram of an image is a composition of the underlying scene covariance convolved with an overlap function, which is itself a convolution of the PSF. The functional form of this relationship provides an analytic basis for scene inference and eventual inversion of scene model parameters from image data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improved parameter inference in catchment models: 2. Combining different kinds of hydrologic data and testing their compatibility

Often some of the parameters of catchment models fitted to runoff data are poorly determined thereby making the task of developing useful regionalization relationships more difficult. The Bayesian methodology developed in part 1 is extended to utilize several kinds of hydrologic data in parameter inference, the goal being to improve the precision of poorly determined parameters. The concept of compatibility is developed using statistical hypothesis tests. Different kinds of data are said to be compatible if differences between their fitted parameters are not statistically significant. The pooling of incompatible data may undermine the model's ability to predict runoff and also induce bias in the parameters. A hierarchy of three levels of information is introduced to enable systematic checking for compatibility. Finally, a case study is presented. Using data on runoff, soil moisture, and interception, it is shown that substantial reductions in parameter uncertainty can be realized; also the importance of compatibility testing is demonstrated.