P(Anaplasmosis=TRUE, Anthrax=FALSE,... T.B.=FALSE, Abortion=FALSE,
Anaemia=TRUE, Change in Urine=FALSE,.... Wasting=FALSE)=0.007
P(Anaplasmosis=FALSE, Anthrax=TRUE,... T.B.=FALSE, Abortion=TRUE, Anaemia=TRUE,
Change in Urine=FALSE,.... Wasting=TRUE)=0.00002
and so on....
In the case of CaDDiS, where we consider 20 diseases
and 27 clinical signs,
we would require 2 to the power of 47 different probabilities to define
the joint probability distribution. The generation of these 140 trillion
probabilities would not be a trivial exercise! It is easier to define our
knowledge of the system in terms of conditional probabilities, but the
derivation of the required joint p.d.f. from conditionals is still computationally
prohibitive. The Bayesian Network uses the concept of Conditional
Independence to simplify our calculations, allowing us to derive the
information which we need from the joint distribution using a much smaller
number of conditional probabilities. The system is defined in terms of
probabilities such as
P(Anaemia | Anaplasmosis)="the probability of a cow exhibiting anaemia
given that it is infected with anaplasmosis" or
P(Anthrax | Constipation, Dyspnoea)= " the probability of a cow being
infected with anaemia given that it is exhibiting constipation and dyspnoea".
These equations can be simplified using conditional
independence where, for example
P( FMD | Lameness and foot lesions)=P(FMD | foot lesions) (the knowledge
that the animal is lame adds nothing to our understanding if we already
know that the cow has foot lesions),
or
P(Anthrax | Tongue lesions) = P(Anthrax) (tongue lesions have nothing
to do with anthrax.. their presence should not directly affect our estimate
of whether a cow has that disease.
The effect of these conditional probabilities can be summarised graphically.
Each disease or sign is represented as a point, and two points A and B
are only connected by lines if there are no other events C(i) such that
P(A | B, C(1), C(2),... C(n)) = P(A | C(1), C(2),... C(n)),
i.e. that A and B are never conditionally independent. The resulting
graph is known as a network.
Where a clinical sign is observed in an animal, that variable in the
network is specified: e.g., Coughing=TRUE is given a belief of 1. This
information will affect the conditional probabilities of those other events
which the Coughing node has lines into:
e.g. where P(ECF | coughing) is a valid conditional probability, the
observation of coughing will affect our belief in the presence of ECF in
the cow.
The effects of new information are propagated through the network using
Bayes's Theorem, changing
the belief values of other diseases and signs. Hence, the graph is a Bayesian
Belief Network.
It is possible to break the overall graph down into smaller sub-sets
within which information flows are largely self-contained. This approach
allows the propagation of information to proceed much more efficiently.
More details of this topic can be found in Lauritzen
and Spiegelhalter (1988).