Statistical inference models

In the following sections we propose typical methods of unsupervised learning and pattern recognition, the aim of which is to detect patterns in chemical, physicochemical and biological data, rather than to make predictions of biological activity. These inductive methods are useful in generating hypotheses and models which are to be verified (or falsified) by statistical inference. Cluster analysis has... 

Once we have estimated the unknown parameter values in a linear regression model and the underlying assumptions appear to be reasonable, we can proceed and make statistical inferences about the parameter estimates and the response variables. 

Other statistical inferences are possible, however, about characteristics of a population of cotton samples from a study of six randomly selected from the population. We conclude that the particulate-reflectance relationship is a very strong one and that the model of Equations 6 and 7 is a "highly likely candidate" even though there is no evidence in this paper that it is superior to the model obtained by replacing In (1/R) with R. 

There is a growing literature that addresses the transferability of a study s pooled results to subgroups. Approaches include evaluation of the homogeneity of different centers and countries results use of random effects models to borrow information from the pooled results when deriving center-specific or country-specific estimates direct statistical inference by use of net monetary benefit regression and use of decision analysis. 

We can use various methods of statistical inference to arrive at these conclusions. Because the original data are subject to experimental error and may not exactly fit our presumed model, we can draw a conclusion only within specified limits of certainty. We can never make completely unequivocal inferences about a population by using statistical procedures. 

Frequentist methods are fundamentally predicated upon statistical inference based on the Central Limit Theorem. For example, suppose that one wishes to estimate the mean emission factor for a specific pollutant emitted from a specific source category under specific conditions. Because of the cost of collecting measurements, it is not practical to measure each and every such emission source, which would result in a census of the actual population distribution of emissions. With limited resources, one instead would prefer to randomly select a representative sample of such sources. Suppose 10 sources were selected. The mean emission rate is calculated based upon these 10 sources, and a probability distribution model could be fit to the random sample of data. If this process is repeated many times, with a different set of 10 random samples each time, the results will vary. The variation in results for estimates of a given statistic, such as the mean, based upon random sampling is quantified using a sampling distribution. From sampling distributions, confidence intervals are obtained. Thus, the commonly used 95% confidence interval for the mean is a frequentist inference... 

As Morgan Henrion (1990) point out, for many quantities of interest in models used for decision-making, there may not be a relevant population of trials of similar events upon which to perform frequentist statistical inference. For example, some events may be unique or in the future, for which it is not possible to obtain empirical sample data. Thus, frequentist statistics are powerful with regard to their domain of applicability, but the domain of applicability is limited compared with the needs of analysts attempting to perform studies relevant to the needs of decision-makers. 

Aside from the continuity assumption and the discrete reality discussed above, deterministic models have been used to describe only those processes whose operation is fully understood. This implies a perfect understanding of all direct variables in the process and also, since every process is part of a larger universe, a complete comprehension of how all the other variables of the universe interact with the operation of the particular subprocess under study. Even if one were to find a real-world deterministic process, the number of interrelated variables and the number of unknown parameters are likely to be so large that the complete mathematical analysis would probably be so intractable that one might prefer to use a simpler stochastic representation. A small, simple stochastic model can often be substituted for a large, complex deterministic model since the need for the detailed causal mechanism of the latter is supplanted by the probabilistic variation of the former. In other words, one may deliberately introduce simplifications or errors in the equations to yield an analytically tractable stochastic model from which valid statistical inferences can be made, in principle, on the operation of the complex deterministic process. 

Whatever the geologic causes, there are several purely statistical inferences to be drawn from Figure 16 which bear directly on the issue of reservoir simulation. The size of grid four may be a natural choice for the grid block size in a deterministic simulation model. Such a selection would minimize the variation between blocks and may, in fact, make stochastic assignments of secondary importance (thus, reducing the differences between realizations). The variation of the fifth scale would be incorporated as pseudo functions or megascopic dispersivity into individual blocks. 

Bois FY (2009) GNU MCSim Bayesian statistical inference for SBML-coded systems biology models. Bioinformatics 25 1453-1454... 

Chatfield, C. Model uncertainty, data mining, and statistical inference. Journal of the Royal Statistical Society, Series A 1995 158 419 166. 

Optimal parameter estimation, statistical inference, and model predictions... 

Shapiro, A. (1996), Simulation-Based Optimization Convergence Analysis and Statistical Inference, Stochastic Models, Vol. 12, pp. 425- 54. 

The only approximation made in the Bayesian time-domain approach is that the system response at a particular time step estimated by its entire history is essentially the same as conditioning on a significantly smaller number of previous time steps. In practice, the time-domain approach provides virtually an exact solution in the sense that the Bayesian approach utilizes the complete information inherited in the measurement. Therefore, the Bayesian time-domain approach provides more accurate statistical inference of the model parameters with the information in the data. 

If the causal model did not include the identifiability assumptions, then it is unknown if /q equals the desired causal effect ipo- A variety of sensitivity analyses could be employed, which involve posing a new causal model that still allows identification of the desired causal effect but that represents a deviation from the original causal model. Such a causal model is indexed by a sensitivity parameter a that is assumed known, and for each a-specific causal model, one redevelops the identifiability result and estimator with corresponding statistical inference (Robins et al., 1999 Scharfstein et al., 1999 Rotnitzky et al., 2001). In a recent article by Dfaz and van der Laan (2013), we develop a much simpler sensitivity analysis that simply defines the sensitivity parameter as the bias il o - o or an upper bound thereof, and for each plausible value of this bias, it reports the estimator and possibly conservative confidence interval for the causal effect rpS. The latter method relies on fewer assumptions, and does not involve any extra work. 

Robins, J.M. Robust estimation in sequentially ignorable missing data and causal inference models. Proc. Am. Statist. Assoc. Sect. Bayesian Statist. Sci. 2000 6-10. 

Robins, J.M., A. Rotnitzky, and D.O. Scharfstein. Sensitivity analysis for selection bias and unmeasmied confoimding in missing data and causal inference models. In M.E. HaUoran and D. Berry (eds.). Statistical Models in Epidemiology, the Environment and Clinical Trials, IMA Volumes in Mathematics and Its Applications. Springer, New York, 1999. 

Stapleton, J. M. 2009. Models for Probability and Statistical Inference. John Wiley and Sons, New Jersey. 

A very powerful idea behind Bayesian inference is that statistical inference is simply updating a previous knowledge, assessed by a prior distribution. The obtained posterior distribution, which encodes the current state of knowledge, can be sequentially updated by adding more and more data. To exemphfy this idea, let us consider a very simple problem the Bayesian assessment of an exponential lifetime model, of failure rate X ... 

Benjamin/Cummings, Menlo Park Hogg RV, Tanis EA (1988) Probability and statistical inference, 3rd edn. Macmillan, New York Janossy L (1965) Theory and practice of the evaluation of measurements. Clarendon, Oxford Karlin S, Taylor HM (1975) A first course in stochastic processes. Academic, New York Karlin S, Taylor HM (1984) An introduction to stochastic modeling. Academic, Orlando Knoll GF (1979) Radiation detection and measurement. Wiley, New York, pp 104-147... 

The methodology to answering these parameter estimation and set-based questions relies on different mathematical approaches. In principle, the parameter identification of chemical kinetic models can be posed as classical statistical inference [17,19-21] given a mathematical model and a set of experimental observations for the model responses, determine the best-fit parameter values, usually those that produce the smallest deviations of the model predictions from the measurements. The validity of the model and the identification of outliers are then determined using analysis of variance. The general optimizations are computationally intensive even for well-behaved, well-parameterized algebraic functions. Further complications arise from the highly ill-structured character... 

Several methods have been suggested to deal directly with the differential equation reaction models and the experimental data [21-23], including the recently developed method of global optimization [15,16]. Yet another class of methods [24] pursues Bayesian statistical inference [5,25]. 

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