I've seen the following formula being used in various places:
$P(a|b) = \sum_z P(a|z) \times P(z|b)$
So essentially they sum over all the variables z to get the original conditional probability. Is this correct, and does it make any strong assumptions?
I am confused by the following example:
$P(good\ weather | month) = \\ \sum_{z \in \{T,F\}} P(good\ weather| universe\ exists = z) * P(universe\ exists = z| month) = \\ P(good\ weather|universe\ exists = T) $
It starts off with the probability of having good weather, given the current month. We then introduce an additional variable (does the universe exist), and sum over all values (T,F). Let's say that the universe exists with probability 1.0, then:
$P(universe\ exists = T| month) = 1.0$
$P(universe\ exists = F| month) = 0.0$
Therefore, we end up with:
$P(good\ weather | month) = P(good\ weather | universe\ exists = T)$
Now, somewhere I have clearly lost some important information, because I'm showing that the conditional probability of good weather, depending on the month, is equal to the probability of good weather, given that the universe exists. What am I missing?