What is the significance of multiplying 2 Gaussian PDFs? I've been reading on Kalman Filter and came across the following statement:
The best estimate we can make of the location is achieved by multiplying the 2 corresponding PDFs together.
But I don't really understand why this is true. Are there any statistic theory that you can point me to that validate the above statement of multiplying PDFs?
Multiplying PDFs is it the same as saying "What is the probability that you will get a head from a coin throw AND get a 6 on a dice throw?"
$~~~~~\Rightarrow 1/2 \times 1/6 = 1/12$
Is it the same analogy ?
 A: Yes, it is the same analogy.
The meaning of the pdf is as follows. Let a random variable $X$ have a pdf $f_X$ and let an arbitrary interval $[a,b]$ be given. The meaning of the pdf is completely described by the following formula:
$$P(X\in [a,b])=\int_a^b f_X(x) \ dx$$
Consider another random variable $Y$ having a pdf $f_Y$. The meaning of the pdf is the same; for any interval $[c,d]$ 
$$P(Y\in [c,d])=\int_c^d f_Y(x) \ dx.$$
If $X$ and $Y$ are independent then their joint pdf $f_{X,Y}$ is the product of their pdf's, in the sense that
$$f_{X,Y}(x,y)=f_X(x)f_Y(y).$$
The meaning of the common pdf is analogous to the individual pdf's. If a (measurable) set $A$ in $R^2$ is given then (in the independent case) we have
$$P((X,Y)\in A)=\iint_Af_X(x)f_Y(y)\ dxdy.$$
For instance, if $A$ is a rectangle, that is, $A=[a,b]\times[c,d]$ then (in the independent case)
$$P((X,Y)\in [a,b]\times[c,d])=\int_a^bf_X(x)\left[\int_c^d f_Y(y) \ dy \right] \ dx=\int_a^bf_X(x)\ dx\cdot \int_c^d f_Y(y) \ dy.$$
A: The analogy is true but still there might be some confusion. The reason why multiplication is used in the update step is because of Bayes' theorem which basically tells us to multiply the probabilities to update a prior distribution with new information to get a (then more accurate) posterior distribution.
What might be confusing is that in the Kalman filter example we are interested in the best estimate of the new location, in the head/dice example you gave we are interested in the probability itself. So in the first case we are talking about parameters of the new distribution (mean and variance) and perhaps even about the form of the new distribution but conveniently enough multiplying two Gaussians gives another Gaussian.
Think about it as with the Kalman filter we are talking about the resulting x-values and in your example we are talking about the resulting y-values of the pdfs.
You can also find quite clear explanations here: https://www.udacity.com/wiki/cs373/unit-1 andhttps://www.udacity.com/wiki/cs373/unit-2
