200.

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(7.5)

Equation (7.5) is only valid under the circumstance that the data sources are mutually independent. The practical value of this assumption is considered further in Section 7.3.5.

When processing multisource information it is natural to consider the inclusion of a reliability (or uncertainty) measure for each different data source. In other words, one can assign different weights to different sources in order to achieve a more satisfactory multisource consensus. One simple way to adjust the contribution for each data source is to append exponents to the source-specific posterior probabilities (Lee et al., 1987). This modification to Equation (7.5) gives:

(7.6)

where α_i [0,1] is the source-specific weighting parameter which allows one to adjust the contribution for the ith source, and a measure of the sensitivity of P(ω_j|x_l, x₂,…, x_n) to changes of the ith posterior probability, which reduces to following expression:

(7.7)

Equation (7.7) shows that the difference in the posterior probability P(ω_j|x_i) for source i leads to a difference of α_i in P(ω_j|x₁, x₂,…, x_n). The weighting parameter α_i thus represents the level of sensitivity of the data source i.

Given the above definition of α_i it is clear that if α_i is equal to the value 1 then the data source i is fully reliable. As α_i tends to 0 then the source i is considered to be less reliable. Note that, in the case α_i=0, one should not include any contribution from source i. Equation (7.6) cannot provide a satisfactory mechanism to cope with this situation. For instance, if there is a total of five data sources, and only the weighting parameters α₁ and α₂ for the first and second data sources are non-zero, the posterior probability derived form Equation (7.6) will use P(w_j)⁻⁴ (where the exponent −4 is obtained from 1−n, and the number of data sources n=5 in this case) as prior probability. A refined Bayesian multisource classification mechanism was proposed by Benediktsson et al. (1990) to deal with this difficulty.

7.3.3 A refined multisource Bayesian model

Consider that one more source is being added to the classification analysis. Equation (7.5) then becomes (Benediktsson et al., 1990):

(7.8)

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