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The posterior pdf of Vc alone can be obtained by integrating the joint posterior pdf with respect to a over all values of a. The result, graphed in Figure 3, is a continuous function defined for all values of Vc from minus to plus infinity. A result ex pressed as a function is difficult to deal with when one is accustomed to handling a single number. Like any other mathematical function, a posterior pdf can be characterized by a model and a number of param eters. At least two parameters will be needed, one for position and one for scale, and more if the function is complicated. The posterior pdf of V0 is proportional to the reciprocal of the square root of a quadratic function of Vc times the natural ex ponent of the ratio of two quad ratics. Although six parameters consist ing of sums and products of observa tions will provide an exact descrip tion of the posterior pdf of Vc, a Gaussian approximation with a mean and standard deviation may be accurate enough for many pur poses. The mean and standard deviation of the posterior pdf of Vc are 3.01 and 0.12 ml for the observa tions in Table I. These values are generally difficult to calculate, how ever. It is easier to find the maxi mum of the posterior pdf and calcu late an approximate standard devi ation from an equivalent of Equa tion 8 (14)- The most probable value of V0 is 3.00 ml, and the approximate standard deviation is 0.11 ml. Requiring a method to present evidence about true values entails the use of Bayes' theorem and prior pdf's. In the problem considered here, the prior pdf of Vc cannot be defined by frequencies of different
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Equiv vol, v„ ml
Method Graphical
2.9
Minmax
2.93 ± 0 . 1 6
Least squares
3.00, σ ^ ~ 0 . 1 1
Least squares, weighted
3.00,0·^. r ^ O . l l
Maximum likelihood
3.00, σφ, =i0.11 Posterior pdf, Figure 3
Mean value
3.01, standard deviation = 0.12
Most probable value
3.00, * j > c ~ 0.11
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(Continued on page 38 A)
Table II. Summary of Results
Bayes
36A
true values because we have postu lated that there is only one true value. This value has a frequency probability of one, and all other values have a probability of zero. The prior pdf, in describing the state of the analyst's personal knowl edge about Vc, describes subjective degrees of belief. That a prior pdf is subjective is the reason why the Bayesian approach has not been developed greatly. Note, however, that after prescribing a prior pdf, the posterior pdf can be interpreted objectively. Although methods of inference restricted to frequencies give objective results, the results must be interpreted subjectively to answer questions about reality. Results obtained by the different methods discussed above are sum marized in Table II. That the numerical results are all practically the same is a coincidence, owing to both the particular observations and the models, not a justification for any of the methods. The inter pretation of the results obtained from the methods differs greatly. As presented here, the methods are capable of using an increasingly de tailed picture of reality. Unfortu nately, the numerical procedures are, in general, increasingly labori ous also. The method of choice in any particular situation depends on a compromise between numerical difficulty and the detail in which one will desire to examine the re sults. Apart from practical con siderations of the use of the methods described here, a knowledge of the different methods, particularly the Bayesian, helps in constructing and understanding the implications of models.
ANALYTICAL CHEMISTRY, VOL. 44, NO. 11, SEPTEMBER 1972
