Bayes rule for continuous hypotheses

In the case of continuous hypotheses, continuous probability distributions are used to represent different possible values for parameters. Bayes' rule is then expressed as:

where H represents a particular value for the parameter. The integral in the denominator substitutes for the summation in the discrete case, and the limits of the integration are over all the possible values of the parameter (x), which in this case is assumed to be positive. This integral makes Bayesian methods difficult to conduct analytically, because in most cases it cannot be determined.

Readers who are uncomfortable with mathematics may look at the above equation and decide that they can never solve those sorts of problems and decide that Bayesian methods are too hard. The complexity of the equation should not be discouraging because in most cases it is impossible to solve, regardless of a person's mathematical skills. Fortunately, software is available so users do not need to evaluate or even construct the integral.

As with the case when there were a finite number of hypotheses (Box 1.3), the denominator simply acts as a scaling constant, because it is the same for all possible values for the parameter H. As with discrete hypotheses, the posterior probability is simply proportional to the prior probability (Pr(H)) multiplied by the likelihood (Pr(D | H)). The main analytical task of Bayesian analyses is to determine the constant of proportionality.

remnants (Harper et al., 2005). After measuring so many trees in the study area, he has a good idea about likely values for the mean diameter of trees in the previously unmeasured remnant. Frequentist analyses do not permit this additional information to be used in determining the mean diameter of trees in the new remnant, but a Bayesian analysis does.

Based on data from the other 43 remnants, the mean diameter of trees in remnants is 53 cm and the mean varies among remnants with a standard deviation of approximately 5 cm. Assuming the mean diameter of trees follows a normal distribution, we would expect approximately 95% of remnants to have a mean tree diameter that is within 1.96 standard deviations of the overall average. Therefore, prior to collecting the data there is a 95% chance that the mean diameter of trees in the new remnant will be between approximately 43 and 63 cm. This prior reflects the researchers' expectation of the mean size of trees in a newly measured remnant based on his previous experience in the study area. A plot of the prior shows the range of likely values (Fig. 1.4).

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