Clinical Development in the Light of Bayesian Statistics
Emmanuel Lesaffre, Gianluca Baio, Bruno Boulanger in Bayesian Methods in Pharmaceutical Research, 2020
Spiegelhalter et al. (2004) define Bayesian thinking, in the context of health technology assessment, as “The explicit quantitative use of external evidence in the design, monitoring, analysis, interpretation and reporting of a health-care evaluation”. In his thought provoking paper on learning versus confirming in drug development, Sheiner (1997) described the Bayesian view as being particularly suited to the learning phases of drug development. He notes: “The Bayesian view is well suited to this task because it provides a theoretical basis for learning from experience; that is, for updating prior beliefs in the light of new evidence”. The work goes on to emphasize the term Bayesian is adopted to describe a point of view or a thought process, where prior knowledge (i.e. validated scientific theory) is to be incorporated into the analysis of current data. A clear distinction is made between this thought process and formal Bayesian inference (i.e. a statistical method involving the use of a prior probability distribution when analyzing data), with the former being considered the key concept behind Bayes and the latter more the technical details.
About What to Think in Step-by-Step Clinical Work and Care: Risk, Diagnosis, Treatment, Prognosis 1
Milos Jenicek in How to Think in Medicine, 2018
The Bayesian approach to biostatistics in the health sciences domain is one of the paradigms that uses observed data to draw inferences. Gustafson specifies correctly that the Bayesian approach distinguishes itself from other approaches and paradigms, “At the heart of Bayesian analysis is Bayes theorem, which describes how knowledge is updated on observing data.” 20 Bayes theorem differs then from probabilistic statements about mechanistically random processes using probabilistic statements about fixed, but unknown quantities of interest. Bayesian analysis has been used in a variety of contexts ranging from marine biology to medicine (diagnostic and screening testing), philosophy of science (relationships between evidence and theory), and even in the development of “Bayesian” spam blockers for email systems.20
Bayes Factor
Albert Vexler, Alan D. Hutson in Statistics in the Health Sciences, 2018
A formal Bayesian concept for analyzing the collected data involves the specification of a prior distribution for all parameters of interest, say , denoted by , and a distribution for , denoted by . Then the posterior distribution of given the data is which provides a basis for performing formal Bayesian analysis. Note that maximum likelihood estimation (Chapter 3) in light of the posterior distribution function, where is expressed using the likelihood function based on , has a simple interpretation as the mode of the posterior distribution, representing a value of the parameter that is “most likely” to have produced the data.
Exponentiated odd Chen-G family of distributions: statistical properties, Bayesian and non-Bayesian estimation with applications
Published in Journal of Applied Statistics, 2021
M. S. Eliwa, M. El-Morshedy, Sajid Ali
This section is concerned with the Bayesian analysis of some selected distributions of the proposed class. In particular, the MLE is conducted for EOChFr and EOChN and thus we focus on these distributions. The Bayesian analysis uses the Bayes theorem to combine the prior information with the observed information. It is to be noted that prior can be non-informative in the sense that it provides less accurate information than the informative prior. The likelihood of the EOChFr distribution can be written as b, i.e.
Bayesian inference for quantum state tomography
Published in Journal of Applied Statistics, 2018
D. S. Gonçalves, C. L. N. Azevedo, C. Lavor, M. A. Gomes-Ruggiero
In conclusion, both Bayesian and bootstrap (resampling) methods circumvent the problems related to the problematic data. Nevertheless, notice that the bootstrap estimates (both punctual and interval ones) are ‘contaminated’, since in some of the bootstrap samples, the MLE estimate corresponds to unacceptable values. In other words, the bootstrap distributions of the estimators present unacceptable values, whereas the respective Bayesian posterior distributions do not. Also, Bayesian inference allows one to incorporate prior information and easily considers other likelihoods, providing a general overview in terms of inference (estimation, model selection, and hypothesis test) through the posterior distribution. In addition, some credible intervals of interest, for example for the purity of a density matrix,
Simple and flexible Bayesian inferences for standardized regression coefficients
Published in Journal of Applied Statistics, 2019
However, one clear advantage of Bayesian inference is it affords a general workable framework for incorporating meaningful prior information into a statistical data analysis. From this perspective, the use of the improper priors in the above needs to be justified in order that applied researchers and practitioners can appreciate the benefits of our proposed approaches. For the conditional-x case, Murphy [33] indicates that the improper prior is essentially the reference prior for a proper Normal-Scaled-Inverse-χ2 prior distribution for θ, σ2, θ0,v0, g is a pre-specified positive constant. Hence, Algorithms c.1 and c.2 can be justified by considering they produce a suitable limit of posteriors obtained from the proper priors, according to Berger et al. [2]. It is also worth highlighting that 47]. In the case of unconditional-x, notice that the improper prior (22) is the reference prior for a Normal-Inverse-Wishart prior distribution μ0, v0 and g is a pre-specified positive constant.
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- Decision Theory
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- Radical Probabilism
- Sampling Distribution
- Posterior Predictive Distribution
- Frequentist Inference
- Bernstein–Von Mises Theorem
- Central Tendency