USING FULL PROBABILITY MODELS TO COMPUTE PROBABILITIES OF ACTUAL INTEREST TO DECISION MAKERS

2001 ◽  
Vol 17 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Frank E. Harrell ◽  
Ya-Chen Tina Shih
Keyword(s):  
Bayesian Approach ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Likelihood Function ◽  
Scientific Evidence ◽  
A Priori ◽  
Relevant Information ◽  
Decision Makers ◽  
Probability Models ◽  
The Bayesian Approach

The objective of this paper is to illustrate the advantages of the Bayesian approach in quantifying, presenting, and reporting scientific evidence and in assisting decision making. Three basic components in the Bayesian framework are the prior distribution, likelihood function, and posterior distribution. The prior distribution describes analysts' belief a priori; the likelihood function captures how data modify the prior knowledge; and the posterior distribution synthesizes both prior and likelihood information. The Bayesian approach treats the parameters of interest as random variables, uses the entire posterior distribution to quantify the evidence, and reports evidence in a “probabilistic” manner. Two clinical examples are used to demonstrate the value of the Bayesian approach to decision makers. Using either an uninformative or a skeptical prior distribution, these examples show that the Bayesian methods allow calculations of probabilities that are usually of more interest to decision makers, e.g., the probability that treatment A is similar to treatment B, the probability that treatment A is at least 5% better than treatment B, and the probability that treatment A is not within the “similarity region” of treatment B, etc. In addition, the Bayesian approach can deal with multiple endpoints more easily than the classic approach. For example, if decision makers wish to examine mortality and cost jointly, the Bayesian method can report the probability that a treatment achieves at least 2% mortality reduction and less than $20,000 increase in costs. In conclusion, probabilities computed from the Bayesian approach provide more relevant information to decision makers and are easier to interpret.

scholarly journals An Accurate Asymptotic Approximation for Experience Rated Premiums

2004 ◽  
Vol 34 (01) ◽  
pp. 113-124
Author(s):  
Riccardo Gatto
Keyword(s):  
Numerical Methods ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Asymptotic Approximation ◽  
Good Accuracy ◽  
Likelihood Function ◽  
Analytical Calculation ◽  
Expected Loss ◽  
Exact Calculations ◽  
The Bayesian Approach

In the Bayesian approach, the experience rated premium is the value which minimizes an expected loss with respect to a posterior distribution. The posterior distribution is conditioned on the claim experience of the risk insured, represented by a n-tuple of observations. An exact analytical calculation for the experience rated premium is possible under restrictive circumstances only, regarding the prior distribution, the likelihood function, and the loss function. In this article we provide an analytical asymptotic approximation as n → ∞ for the experience rated premium. This approximation can be obtained under more general circumstances, it is simple to compute, and it inherits the good accuracy of the Laplace approximation on which it is based. In contrast with numerical methods, this approximation allows for analytical interpretations. When exact calculations are possible, some analytical comparisons confirm the good accuracy of this approximation, which can even lead to the exact experience rated premium.

scholarly journals An Accurate Asymptotic Approximation for Experience Rated Premiums

2004 ◽  
Vol 34 (1) ◽  
pp. 113-124
Author(s):  
Riccardo Gatto
Keyword(s):  
Numerical Methods ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Asymptotic Approximation ◽  
Good Accuracy ◽  
Likelihood Function ◽  
Analytical Calculation ◽  
Expected Loss ◽  
Exact Calculations ◽  
The Bayesian Approach

In the Bayesian approach, the experience rated premium is the value which minimizes an expected loss with respect to a posterior distribution. The posterior distribution is conditioned on the claim experience of the risk insured, represented by a n-tuple of observations. An exact analytical calculation for the experience rated premium is possible under restrictive circumstances only, regarding the prior distribution, the likelihood function, and the loss function. In this article we provide an analytical asymptotic approximation as n → ∞ for the experience rated premium. This approximation can be obtained under more general circumstances, it is simple to compute, and it inherits the good accuracy of the Laplace approximation on which it is based. In contrast with numerical methods, this approximation allows for analytical interpretations. When exact calculations are possible, some analytical comparisons confirm the good accuracy of this approximation, which can even lead to the exact experience rated premium.

scholarly journals Hierarchical Bayesian Estimation for Stationary Autoregressive Models Using Reversible Jump MCMC Algorithm

2018 ◽  
Vol 7 (4.30) ◽  
pp. 64
Author(s):  
Supar Man ◽  
Mohd Saifullah Rusiman
Keyword(s):  
Bayesian Approach ◽  
Posterior Distribution ◽  
Autoregressive Model ◽  
Prior Distribution ◽  
Likelihood Function ◽  
Simulated Data ◽  
Bayesian Estimator ◽  
Reversible Jump ◽  
Hierarchical Bayesian ◽  
Mcmc Algorithm

The autoregressive model is a mathematical model that is often used to model data in different areas of life. If the autoregressive model is matched against the data then the order and coefficients of the autoregressive model are unknown. This paper aims to estimate the order and coefficients of an autoregressive model based on data. The hierarchical Bayesian approach is used to estimate the order and coefficients of the autoregressive model. In the hierarchical Bayesian approach, the order and coefficients of the autoregressive model are assumed to have a prior distribution. The prior distribution is combined with the likelihood function to obtain a posterior distribution. The posterior distribution has a complex shape so that the Bayesian estimator is not analytically determined. The reversible jump Markov Chain Monte Carlo (MCMC) algorithm is proposed to obtain the Bayesian estimator. The performance of the algorithm is tested by using simulated data. The test results show that the algorithm can estimate the order and coefficients of the autoregressive model very well. Research can be further developed by comparing with other existing methods.

Metropolis-Hastings Algorithms for DSGE Models

2015 ◽  
Author(s):  
Edward P. Herbst ◽  
Frank Schorfheide
Keyword(s):  
Random Walk ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Large Scale ◽  
Likelihood Function ◽  
Dsge Models ◽  
Posterior Distributions ◽  
Dsge Model ◽  
Alternative Proposal ◽  
Posterior Moments

This chapter talks about the most widely used method to generate draws from posterior distributions of a DSGE model: the random walk MH (RWMH) algorithm. The DSGE model likelihood function in combination with the prior distribution leads to a posterior distribution that has a fairly regular elliptical shape. In turn, the draws from a simple RWMH algorithm can be used to obtain an accurate numerical approximation of posterior moments. However, in many other applications, particularly those involving medium- and large-scale DSGE models, the posterior distributions could be very non-elliptical. Irregularly shaped posterior distributions are often caused by identification problems or misspecification. In lieu of the difficulties caused by irregularly shaped posterior surfaces, the chapter reviews various alternative MH samplers, which use alternative proposal distributions.

A Predictive Bayesian Approach to Multistate Reliability Analysis

2003 ◽  
Vol 10 (03) ◽  
pp. 221-234 ◽  
Author(s):  
T. Aven ◽  
A. Hjorteland
Keyword(s):  
Reliability Analysis ◽  
Bayesian Approach ◽  
Probability Models ◽  
Bayesian Paradigm ◽  
Bayesian Procedures ◽  
Starting Point ◽  
The World ◽  
The Bayesian Approach

In this paper we discuss how to implement a Bayesian thinking for multistate reliability analysis. The Bayesian paradigm comprises a unified and consistent framework for analysing and expressing reliability, but in our view the standard Bayesian procedures gives too much emphasis on probability models and inference on fictional parameters. We believe that there is a need for a rethinking on how to implement the Bayesian approach, and in this paper we present and discuss such a rethinking for multistate reliability analysis. The starting point of the analysis should be observable quantities, expressing states of the world, not fictional parameters.

scholarly journals ANALISIS METODE BAYESIAN PADA SISTEM ANTREAN RAWAT JALAN DI RSUP Dr. KARIADI DENGAN DISTRIBUSI SAMPEL POISSON DAN GEOMETRIK

2021 ◽  
Vol 10 (3) ◽  
pp. 413-422
Author(s):  
Nur Azizah ◽  
Sugito Sugito ◽  
Hasbi Yasin
Keyword(s):  
Internal Medicine ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Bayesian Method ◽  
Likelihood Function ◽  
Queuing System ◽  
Hospital Service ◽  
Queuing Model ◽  
Sample Distribution ◽  
New Information

Hospital service facilities cannot be separated from queuing events. Queues are an unavoidable part of life, but they can be minimized with a good system. The purpose of this study was to find out how the queuing system at Dr. Kariadi. Bayesian method is used to combine previous research and this research in order to obtain new information. The sample distribution and prior distribution obtained from previous studies are combined with the sample likelihood function to obtain a posterior distribution. After calculating the posterior distribution, it was found that the queuing model in the outpatient installation at Dr. Kariadi Semarang is (G/G/c): (GD/∞/∞) where each polyclinic has met steady state conditions and the level of busyness is greater than the unemployment rate so that the queuing system at Dr. Kariadi is categorized as good, except in internal medicine poly. 

scholarly journals A Bayesian Approach to Estimating Reciprocal Effects with the Bivariate STARTS Model using Markov Chain Monte Carlo

2021 ◽  
Author(s):  
Oliver Lüdtke ◽  
Alexander Robitzsch ◽  
Esther Ulitzsch
Keyword(s):  
Bayesian Approach ◽  
Prior Distribution ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Reciprocal Effects ◽  
Ml Estimation ◽  
Model Parameterization ◽  
Lagged Effects ◽  
The Bayesian Approach ◽  
Over Time

The bivariate Stable Trait, AutoRegressive Trait, and State (STARTS) model provides a general approach for estimating reciprocal effects between constructs over time. However, previous research has shown that this model is difficult to estimate using the maximum likelihood (ML) method (e.g., nonconvergence). In this article, we introduce a Bayesian approach for estimating the bivariate STARTS model and implement it in the software Stan. We discuss issues of model parameterization and show how appropriate prior distributions for model parameters can be selected. Specifically, we propose the four-parameter beta distribution as a flexible prior distribution for the autoregressive and cross-lagged effects. Using a simulation study, we show that the proposed Bayesian approach provides more accurate estimates than ML estimation in challenging data constellations. An example is presented to illustrate how the Bayesian approach can be used to stabilize the parameter estimates of the bivariate STARTS model.

scholarly journals BUILDING MODELS OF FREIGHT CARS REFUSALS INVOLVING BAYESIAN APPROACH

2016 ◽  
Vol 1 ◽  
pp. 54-60
Author(s):  
Leontii Muradian
Keyword(s):  
Mechanical Properties ◽  
Bayesian Approach ◽  
Posterior Probability ◽  
Likelihood Function ◽  
A Priori ◽  
Physical And Mechanical Properties ◽  
Total Probability ◽  
Knowledge Process ◽  
Building Models ◽  
Additional Value

Based on the theoretical analysis and Bayesian statistics ordinary shown that Bayesian analysis begins with the known data from the following consideration changes in knowledge process of obtaining new information and mathematical statistics methods of sample observation comes only with the knowledge of some group of objects. Using Bayesian formula, we can determine the probability of any event, provided that there was another statistically correlated with it an event that counted with greater accuracy the likelihood. This used previously known information and data obtained as a result of new observations. The study of failures of freight cars, the Bayesian approach allows you to evaluate the occurrence of each failure of parts or assemblies separately, as well as through changes in the formula for the total probability. The paper, based on Bayesian method was done combining two models: the failures of freight cars and the changing physical and mechanical properties of composite materials. This posterior probability determined a priori probability of failures given using the model change of physical and mechanical properties and the likelihood function that takes into account the additional value failures. Using the expression for the posterior probability held specification mentioned developments (run) freight wagon to failure.

scholarly journals Updating Knowledge in The Estimation of The Genetics Parameters Multi-trait and Multi-Environment Bayesian Analysis in Rice (Oryza Sativa L.)

2021 ◽  
Author(s):  
Camila Ferreira Azevedo ◽  
Cynthia Barreto ◽  
Matheus Suela ◽  
Moysés Nascimento ◽  
Antônio Carlos Júnior ◽  
...  
Keyword(s):  
Bayesian Approach ◽  
Prior Distribution ◽  
Variance Components ◽  
Genetic Correlations ◽  
Bayesian Framework ◽  
Prior Distributions ◽  
Informative Prior ◽  
Informative Prior Distribution ◽  
Informative Prior Distributions ◽  
The Bayesian Approach

Abstract Among the multi-trait models used to jointly study several traits and environments, the Bayesian framework has been a preferable tool for using a more complex and biologically realistic model. In most cases, the non-informative prior distributions are adopted in studies using the Bayesian approach. Still, the Bayesian approach tends to present more accurate estimates when it uses informative prior distributions. The present study was developed to evaluate the efficiency and applicability of multi-trait multi-environment (MTME) models under a Bayesian framework utilizing a strategy for eliciting informative prior distribution using previous data from rice. The study involved data pertained to rice genotypes in three environments and five agricultural years (2010/2011 until 2014/2015) for the following traits: grain yield (GY), flowering in days (FLOR) and plant height (PH). Variance components and genetic and non-genetic parameters were estimated by the Bayesian method. In general, the informative prior distribution in Bayesian MTME models provided higher estimates of heritability and variance components, as well as minor lengths for the highest probability density interval (HPD), compared to their respective non-informative prior distribution analyses. The use of more informative prior distributions makes it possible to detect genetic correlations between traits, which cannot be achieved with the use of non-informative prior distributions. Therefore, this mechanism presented for updating knowledge to the elicitation of an informative prior distribution can be efficiently applied in rice genetic selection.

scholarly journals Making Better Decisions: Can Minimizing Frequentist Risk Help?

2016 ◽  
Vol 5 (3) ◽  
pp. 80 ◽  
Author(s):  
Rose D. Baker ◽  
Ian G. McHale
Keyword(s):  
Decision Making ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Likelihood Function ◽  
Bayesian Decision Theory ◽  
Bayesian Decision ◽  
Risk Minimization ◽  
Likelihood Principle ◽  
Complete Class ◽  
Bet Size

The concept of shrinking bet size in Kelly betting to minimize estimated frequentist risk has recently been mooted. This rescaling appears to conflict with Bayesian decision theory through the likelihood principle and the complete class theorem; the Bayesian solution should already be optimal. We show theoretically and through examples that when the modeldetermining the likelihood function is correct, the prior distribution (if not dominated by data) is `correct' in a frequentist sense, and the posterior distribution is proper, then no further rescaling is required. However, if the model or the prior distribution is incorrect, or the posterior distribution improper, frequentist risk minimization can be a useful technique. We discuss how it might best be exploited. Another example, from maintenance, is used to show the wider applicability of the methodology; these conclusionsapply generally to decision-making.

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