Answers provided by Charles Dalrymple-Fraser.
If a hypothesis predicts an effect which can be described by E, then Prob(E|H)=1. If we sub that
into the theorem above, we get that Prob(H|E)=Prob(H)/Prob(E). If Prob(E) is less than 1, then
Prob(H) is less than Prob(H|E).
So, if H predicts something which can be described by E, and it occurs, then that successful
prediction increases the probability of H. If H predicts E, then Prob(E|H)=1 and Prob(H|E) is
stringent upon Prob(H) and Prob(E). If Prob(E) is less than one, then Prob(H|E)>Prob(H) and so,
when E is observed, the probability of H increases. Note that the less likely that E will occur [e.g.
Prob(E)], the greater the probability of H increases. If Prob(E)=1, then it was determined to occur
absolutely and the probability of H doesn’t change since E was 100% likely to occur anyways.
For example, if H predicts that the watermelon I bought at Sobeys will explode when I finish
typing this sentence, then, if the watermelon does explode, I will say that H is a lot more
probable because it is very unlikely that my watermelon will explode.1 Likewise, if H predicts that
something will get wet if it rains, and it rains and something gets wet, we wouldn’t really say that
H is that much more probable because, hey, stuff gets wet when it rains already.
This stuff seems odd at times with all the Probs floating about, but try to think of it in real life
terms and it should help clear things up.
The empirical support of a hypothesis given evidence is represented as S(E,H) and is calculated
via the equation S(E,H)=Prob(H|E)-Prob(H). The support is positive when Prob(H|E)>Prob(H),
negative when Prob(H)>Prob(H|E), and zero when they are equal. In words, the support is a
positive support when, evidence considered, the probability of H increases. The support is
negative when the probability of H decreases given the evidence (for example, if my H predicts
with absolute certainty that my watermelon will explode after that sentence up there, but my
watermelon did not explode, then the probability of my H being true is obviously knocked down a
fair bit: the support is negative to my H). The support for the hypothesis is zero when the
evidence doesn’t change our probability of H (for example, if my H predicts that when a coin is
flipped on the moon, there will be half heads and half tails and I flip a coin on the moon once and
get a heads, that one flip doesn’t really support my hypothesis at all, though two flips might).
Now onto some questions…
1 For those concerned, I note that my watermelon is fine and dandy having concluded that sentence. Watermelon > H.