Comparing Likelihood and Bayesian Approaches to Statistics

In this section we graphically illustrate the similarities and differences between the likelihood and Bayesian approaches to inference specifically, how a parameter is estimated using each of the approaches. We will see that  

The likelihood and the posterior density are found in a similar manner, by cutting a surface with the same vertical hyperplane. However, the surfaces used in the two approaches have different interpretations and in most cases they will have different shapes. 

Even when the surfaces are the same (when flat priors are used) the estimators are chosen to satisfy different criteria. 

The two approaches have different ways of dealing with nuisance parameters. 

The observation(s) come from the observation density f(y 0) where 0 is the fixed parameter value. It gives the probability density over all possible observation values for the given value of the parameter. The parameter space, 0, is the set of all possible parameter values. The parameter space ordinarily has the same dimension as the total number of parameters, p. The sample space, S, is the set of all possible values of the observation(s). The dimension of the sample space is the number of observations n. Many of the commonly used observation distributions come from the one-dimensional exponential family of distributions. When we are in the one-dimensional exponential family, the sample space may be reduced to a single dimension due to the single sufficient statistic. 

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Confidence intervals nsing freqnentist and Bayesian approaches have been compared for the normal distribntion with mean p and standard deviation o (Aldenberg and Jaworska 2000). In particnlar, data on species sensitivity to a toxicant was fitted to a normal distribntion to form the species sensitivity distribution (SSD). Fraction affected (FA) and the hazardons concentration (HC), i.e., percentiles and their confidence intervals, were analyzed. Lower and npper confidence limits were developed from t statistics to form 90% 2-sided classical confidence intervals. Bayesian treatment of the uncertainty of p and a of a presupposed normal distribution followed the approach of Box and Tiao (1973, chapter 2, section 2.4). Noninformative prior distributions for the parameters p and o specify the initial state of knowledge. These were constant c and l/o, respectively. Bayes theorem transforms the prior into the posterior distribution by the multiplication of the classic likelihood fnnction of the data and the joint prior distribution of the parameters, in this case p and o (Fignre 5.4). 

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