Fusion in Signal Processing - Information Fusion in Signal and Image Processing - page 20

We can either plot the four distributions on the same graph, or produce the convolu-
tion of the distribution involving the measurement with each a priori distribution and
plot only the first order moment concerning the measurement on the resulting graph
(below). The intersection provides three likelihood values conditionally to each type
of moving object. The concept of likelihood, in this context, refers to what we believe
is true.

P (OV/V)

P (velocity/plane)

P (velocity/helicopter)

P (velocity/exhaust)

V OV

V (OV/plane)

V (OV/helicopter)

V (OV/exhaust)

OV observed velocity

Figure 2.3. Discriminating among several hypotheses

Let us assume that, at this stage, we have to pick one hypothesis out of three. Either
we have information at our disposal regarding the probability of a moving object in
this area and we are capable of producing a posteriori probabilities based on these a
priori probabilities and on the likelihoods after a phase of multiplicative combination,
followed by a renormalization (see Chapter 6). It is then natural to pick the type that
has the highest a posteriori probability. If this information is not available, we pick the
type with the highest likelihood. In both mechanisms, we have just added elements of
uncertainty that can be expressed using a confusion matrix in a probabilistic approach,
in other words there is a probability of picking type 1 when the type was actually H1,
H2, H3 or was just a false alarm. It is then possible to use the concept of Bayesian risk
to make a decision that minimizes a cost function we have to define.

Today, in the context of a multi-sensor system, we have to minimize the probability
of making a false decision while maintaining the highest possible level of operational