15.2 Combining Continuous Information 359
15.2 Combining Continuous Information
One can also combine continuous random variables. The only difference
is that one must be careful to ensure that the combined distribution can be
rescaled to be a PD.
Continuous Information CombinationTheorem
LetXandYbe two continuous random variables that represent two independent
observations of the same phenomenon. Letf(x)andg(x)be the probability den-
sity functions ofXandY, respectively. Iffandgare bounded functions, and
if
∫
f(y)g(y)dyis positive, then there is a random variableZthat combines the
information of these two observations, whose probability density function is
h(x)=
∫f(x)g(x)
f(y)g(y)dy
.
ProofThe proof proceeds as in the discrete case except that one must check
that
∫
f(x)g(x)dxconverges. Nowf(x)was assumed to be bounded. Let
Bbe an upper bound of this function. Thenf(x)g(x)≤Bg(x)for everyx.
Since
∫
g(x)dxconverges, it follows that
∫
f(y)g(y)dyalso converges. The
result then follows as in the discrete case.
As with discrete random variables, information combination requires that
the observations be independent and they measure the same phenomenon.
Ensuring that the observations measure the same phenomenon can be dif-
ficult, as the observations can use different calibrations. Uncoordinated or
miscalibrated observations should not be combined unless they can be recal-
ibrated.
In both of the information combination theorems, the last step is to nor-
malize the distribution. Consequently, one can combineunnormalized distri-
butionsas long as the combined distribution is normalizable. In particular,
it makes sense to combine a uniformly distributed random variableUwith
a random variableX, even when the uniform distribution cannot be nor-
malized. It is easy to see that the combination ofUwithXis the same as
X. In other words, a uniform distribution adds no new information to any
distribution.
Normal distributions are an especially important special case which fol-
lows easily from the general case:
Combining Normal Distributions
IfXandYare independent normally distributed random variables with meansm,
