An Accurate Asymptotic Approximation for Experience Rated Premiums

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.

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An Accurate Asymptotic Approximation for Experience Rated Premiums

Astin Bulletin ◽

10.1017/s051503610001391x ◽

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

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

International Journal of Technology Assessment in Health Care ◽

10.1017/s0266462301104034 ◽

2001 ◽

Vol 17 (1) ◽

pp. 17-26 ◽

Cited By ~ 44

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.

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Metropolis-Hastings Algorithms for DSGE Models

Bayesian Estimation of DSGE Models ◽

10.23943/princeton/9780691161082.003.0004 ◽

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.

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

Jurnal Gaussian ◽

10.14710/j.gauss.v10i3.32801 ◽

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.

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Making Better Decisions: Can Minimizing Frequentist Risk Help?

International Journal of Statistics and Probability ◽

10.5539/ijsp.v5n3p80 ◽

2016 ◽

Vol 5 (3) ◽

pp. 80 ◽

Cited By ~ 1

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

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.30.22007 ◽

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.

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Elite Athletes Refine Their Internal Clocks: A Bayesian Analysis

Motor Control ◽

10.1123/mc.2014-0070 ◽

2016 ◽

Vol 20 (3) ◽

pp. 255-265

Author(s):

Yin-Hua Chen ◽

Isabella Verdinelli ◽

Paola Cesari

Keyword(s):

Bayesian Analysis ◽

Prior Distribution ◽

Likelihood Function ◽

Elite Athletes ◽

Time Intervals ◽

Design Data ◽

Data Set ◽

Short Intervals ◽

Additional Support ◽

The Bayesian Approach

This paper carries out a full Bayesian analysis for a data set examined in Chen & Cesari (2015). These data were collected for assessing people’s ability in evaluating short intervals of time. Chen & Cesari (2015) showed evidence of the existence of two independent internal clocks for evaluating time intervals below and above the second. We reexamine here, the same question by performing a complete statistical Bayesian analysis of the data. The Bayesian approach can be used to analyze these data thanks to the specific trial design. Data were obtained from evaluation of time ranges from two groups of individuals. More specifically, information gathered from a nontrained group (considered as baseline) allowed us to build a prior distribution for the parameter(s) of interest, and data from the trained group determined the likelihood function. This paper’s main goals are (i) showing how the Bayesian inferential method can be used in statistical analyses and (ii) showing that the Bayesian methodology gives additional support to the findings presented in Chen & Cesari (2015) regarding the existence of two internal clocks in assessing duration of time intervals.

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On Bayesian procedure for maximum earthquake magnitude estimation

Research in Geophysics ◽

10.4081/rg.2012.e7 ◽

2012 ◽

Vol 2 (1) ◽

pp. 7 ◽

Cited By ~ 5

Author(s):

Andrzej Kijko

Keyword(s):

Maximum Likelihood ◽

Posterior Distribution ◽

Ad Hoc ◽

Likelihood Function ◽

Fundamental Problem ◽

Likelihood Estimation ◽

Earthquake Magnitude ◽

Bayesian Estimator ◽

Bayesian Procedure ◽

Posterior Estimate

This work is focused on the Bayesian procedure for the estimation of the regional maximum possible earthquake magnitude mmax. The paper briefly discusses the currently used Bayesian procedure for mmax, as developed by Cornell, and a statistically justifiable alternative approach is suggested. The fundamental problem in the application of the current Bayesian formalism for mmax estimation is that one of the components of the posterior distribution is the sample likelihood function, for which the range of observations (earthquake magnitudes) depends on the unknown parameter mmax. This dependence violates the property of regularity of the maximum likelihood function. The resulting likelihood function, therefore, reaches its maximum at the maximum observed earthquake magnitude mobsmax and not at the required maximum possible magnitude mmax. Since the sample likelihood function is a key component of the posterior distribution, the posterior estimate of m^max is biased. The degree of the bias and its sign depend on the applied Bayesian estimator, the quantity of information provided by the prior distribution, and the sample likelihood function. It has been shown that if the maximum posterior estimate is used, the bias is negative and the resulting underestimation of mmax can be as big as 0.5 units of magnitude. This study explores only the maximum posterior estimate of mmax, which is conceptionally close to the classic maximum likelihood estimation. However, conclusions regarding the shortfall of the current Bayesian procedure are applicable to all Bayesian estimators, e.g. posterior mean and posterior median. A simple, ad hoc solution of this non-regular maximum likelihood problem is also presented.

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ANALISIS SISTEM PELAYANAN KERETA API DI STASIUN SEMARANG TAWANG MENGGUNAKAN PROSES BAYESIAN

Jurnal Gaussian ◽

10.14710/j.gauss.v9i4.29407 ◽

2020 ◽

Vol 9 (4) ◽

pp. 495-504

Author(s):

Lifana Nugraeni ◽

Sugito Sugito ◽

Dwi Ispriyanti

Keyword(s):

Poisson Distribution ◽

Posterior Distribution ◽

Public Transportation ◽

Bayesian Method ◽

Service System ◽

Likelihood Function ◽

Performance Standard ◽

Sample Distribution ◽

Service Patterns ◽

Transportation Facilities

Along with the times, transportation has progressed. Regarding the means of transportation, one of the phenomenon that is easily encountered in everyday life is the queue at public transportation facilities. One of the queues that occurred at public transportation facilities is the train queue at Semarang Tawang Station. The number of trains that passes the station can cause the train service at the station busy. This study aims to see whether the train service system of Semarang Tawang Station is good or not. This can be consider by the queues method, determining the distribution of arrival patterns and service patterns to obtain a queues system model and a system performance standard. In this study, the distribution of arrival patterns and service patterns are determined by calculating the posterior distribution using the Bayesian method. The bayesian method was chosen because it is able to combine the sample distribution in the current study with the previous information for the same cases. The prior distribution and the likelihood function are the elements needed to obtain the posterior distribution. The distribution of arrival patterns and service patterns obtained from previous information follows the Poisson distribution. Based on the calculation of the posterior distribution, the result shows that the distribution of the arrival pattern is a discrete uniform distribution and the distribution of the service pattern is a Poisson distribution. The result shows that the train service system at Semarang Tawang Station has a model (Uniform Discrete / Gamma / 7: GD / ~ / ~) and has good service based on the performance values obtained.

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An In-Class Demonstration of Bayesian Inference

10.31234/osf.io/d8bvn ◽

2019 ◽

Author(s):

Johnny van Doorn ◽

Dora Matzke ◽

Eric-Jan Wagenmakers

Keyword(s):

Bayesian Inference ◽

Posterior Distribution ◽

Prior Distribution ◽

Sequential Analysis ◽

Bayes Factor ◽

Classroom Setting ◽

Key Concepts

Sir Ronald Fisher's venerable experiment "The Lady Tasting Tea'' is revisited from a Bayesian perspective. We demonstrate how a similar tasting experiment, conducted in a classroom setting, can familiarize students with several key concepts of Bayesian inference, such as the prior distribution, the posterior distribution, the Bayes factor, and sequential analysis.

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