835 
The apriori values for each crop type p(X_^) can be obtained from recent 
agricultural statistics pertaining to the local area of interest. (It is 
almost unique to agriculture to have sources of published statistics from 
which to obtain the respective values of p(X.) in order to construct the 
key for local situations). 
Summarizing to this Point : The Bayesian Formula (1) revises the 
apriori probability p(X^) to the posterior conditional probability P(X^jT) 
in accordance to the observed photo-density values t^ and t£* Thus 
P(XjJT) expresses the probability that the two measured photo-density 
values, one value from each of two image dates, came from a specified crop 
type. By choosing to call each unknown field in accordance to the highest 
posterior probability, the probability of error is mathematically minimized 
(Duda and Hart, 1973). If one desired to minimize the risk of mis-identifi- 
cation (assuming a zero-one loss function) he should choose to identify the 
crop in accordance to greatest value of P(X i |T). 
The P(tjJXj_) P(X-j^) values for all t and x possibilities in this example 
are shown in Figure 6. Values from this matrix are subsequently used to 
calculate P(X^jT) values for each crop type t^, t 2 combination. A working 
version of a probabilistic key would pre-calculate all possible values of 
P(X^|T) and arrange the answers in a matrix whereby they could be "located" 
using the observed values of t-^ and t£ as pointers. The internal structure 
of a probabilistic key is illustrated by completing the calculation for only 
one set of hypothetically "observed" photo-densities. 
RESULTS 
Example of Calculations 
Suppose that a value of .38 was obtained from 
densitométrie measurement of a field on a 1000 degree day image and that 
later in the growing season (at 2000 degree days) a densitométrie value of 
.20 was obtained for the same field. Applying the hypothetical probabilistic 
key valid for the region, a set of probabilities is obtained (Figure 7 ) 
The P(XjJT) values indicate that the identification involving the least 
error would be to call the unknown field spring wheat. The alternative 
identifications are presented also in Figure 7 along with their respective 
probabilities in this example. 
Probabilistic Key Style : A section of a simple probabilistic key is 
shown in Figure 8. Only two photo-derived pieces of information are required 
in order to use this particular key. When more than two inputs are used, the 
key mechanism would be more complex than the simple matrix. Using a computer 
to pre-calculate all possible posterior probabilities, the arranging of these 
values in a practical manner is a technical problem where several alternatives 
can usually be found, such as using a series of look-up matrices.