 do that. To come up with our answer, we're going to do some statistical work, which will ultimately lead us to an intuitively pleasing answer. First off, we want to make sure the decision device makes as few errors as possible. We'll start to build the decision device by requiring that it minimize the P(E), the probability of it making an error. Another way to say this is: we'll require that, given the decision device sees r, it must choose the Si that is most likely to have occurred. Statistically, this is stated simply as S,= argmax p(s, r r) = (5.34) Let me explain to you what that math/stats equation means. It states two things: (1) the S = argmax part: this is shorthand for the words "Have the decision device choose to output S the value of S that maximizes ": Chapter Five (2) the p () part: this is shorthand for the words "how likely Si is to occur, given I see r." Put together, this equation says "Have the decision device choose the output Si, the value of Si that is most likely to occur, given I see r." For the next little bit, we're going to manipulate Equation (5.34) until we get it into a nice and easy form. First, we apply Bayes' Rule, which lets us rewrite the equa- tion according to (5.35) Next, look at that term on the bottom (in the denominator). It's not a function of Si. That term will be the same regardless of Si. . It plays no role in choosing which Si makes biggest. It can be dropped. So we drop it, and end up with S = = argmax p(r's,)p(s).). (5.36) Now, the term p(s) is the probability of having a particular r given an S was sent. Now r=s+n, SO if r = (10,10) and Si = (6,7) , then n = r-si = (10,10)-(6,7) = (4,3). In other words, the probability of having r given Si is the probability that N=r-s, or mathematically, p(rs) =p(n=r-sj). Stuffing this equality in our equation leads to S = argmax p(n=r-s,)p(s) (5.37) Let's look at that noise term n It corresponds to n = (N1,...,NN): and, for conve- nience, we'll say here that it consists of two terms, that is, n=(n1,12). In Section 5.4.3, we talk