Information Technology Reference
In-Depth Information
for a number of years. We will explore how this func-
tional form maps onto the equilibrium membrane po-
tential equation (equation 2.9), but you can already see
that it has the same weighted average quality to it.
Another way of putting the objective of this analy-
sis is that we want to evaluate to what extent a rational
agent should believe in one of these hypotheses over
the other. This rephrasing can be important because, al-
though it is sometimes possible to actually go around
and measure the objective probabilities that some hy-
pothesis was true given a particular set of data, this is
typically impossible for a number of reasons. Thus, we
usually have to settle for a more subjective definition
of probability that refers to belief instead of objective
fact. In our analysis, we will start with the case where
we know the objective probabilities, so that everything
is perfectly clear. However, we will see that using such
probabilities quickly becomes intractable due to com-
binatorial explosion, so we have to rely on subjective
probabilities instead.
In either case, probabilities are simply numbers be-
tween 0 and 1, where 0 means something never happens
(is never true, should never be believed in), and 1 means
something always happens (is always true, should al-
ways be believed in). Intermediate values mean some-
thing in between, with .5 meaning that something hap-
pens half the time on average (like flipping a coin and
getting “heads”). These intermediate probabilities cor-
respond to intermediate truth or belief values, so that a
value of .5 means that something is half-true, or that it
should only be half-believed. 1 Another way of putting
this is that you should believe in something to a graded
extent p ,where p is the probability value.
Inputs
1/0
1/0
1/0
Vertical Line Detector
Figure 2.20: Simple vertical line detector that detects pres-
ence of vertical line (which amounts to all 3 inputs active).
Each input can either be on or off (1 or 0). It is assumed that
the inputs are driven in some way by visual signals, and that
when there is a vertical line in the visual inputs, all 3 inputs
will tend to light up. However, the system is noisy and inputs
can be spuriously active or inactive.
data
freq h h 1 2 3
_
3 0 1 0 0 0
2 0 1 1 0 0
2 0 1 0 1 0
2 0 1 0 0 1
1 0 1 1 1 0
1 0 1 0 1 1
1 0 1 1 0 1
0 0 1 1 1 1
0 1 0 0 0 0
1 1 0 1 0 0
1 1 0 0 1 0
1 1 0 0 0 1
2 1 0 1 1 0
2 1 0 0 1 1
2 1 0 1 0 1
3 1 0 1 1 1
24
2.7.1
Objective Probabilities and Example
Figure 2.21: All possible states of the world for the vertical
line detector. freq gives the frequency (number of times) the
state occurs in the world. h is the hypothesis that a line exists
in the world, n is the null hypothesis that it doesn't exist, and
the numbers are for the 3 inputs (data). The given frequencies
show that states where more inputs are active are more likely
to have the hypothesis true, and vice versa. The bottom line
contains the total number of states, which is used in comput-
ing probabilities.
For the purposes of concretely instantiating the proba-
bilistic hypothesis testing framework, we will use the
simple detector example shown in figure 2.20. This de-
tector receives inputs from three sources, which we as-
sume are driven by th e world, so that when a vertical
1 The distinction between half-believing something and fully be-
lieving it half the time is an important one, but we are ultimately con-
cerned here with real-valued, time-averaged numbers that are more
consistent with something more like half-belief.
Search WWH ::




Custom Search