Hello, a lot of questions in this list are about the spam : ham ratio to be trained and how much mails should be trained. One continuously read myth is the 1 : 1 ratio.
I read an article about the best ratio as 1 : 1 and it was expirienced by a
test and later on derived from the bayesian theorem. Unfortunately I didn't
copy this article and can't remember enough to find the article by googling.
The problem is the conclusion of the article was wrong.
What I will try to show in the next steps - which unfortunately require a
little bit algebra - is: Train bayes filter in accordance with your real spam
ham ratio and train as much as possible. But never train to less ham or train
only spam!
Here my argument follows:
In short the bayes theorem says
P(Spam|Token) = P(Token|Spam)*P(Spam)/P(Token)
that means: the probability of a message being Spam under the contition that
a token is in the message
is equal to
the propability of the Token contained in a Spam message
multiplied by
the propability of a message being spam
devided by the propability of any message containig the token.
So if you have received s spam messages and h ham messages where the token is
in S spams and in H ham messages then you get:
s = number of spam messages
h = number of ham messages
S = number of spam messages containing the token
H = number of ham messages containing the token
s+h= number of messages
S+H = number of messages containing the token
Therefor
P(Spam) = s/(s+h)
is an aproximation of the probability of a random message being spam.
And for:
P(Token) = (S+H)/(s+h)
P(Token|Spam) = S/s
that leads to
P(Spam|Token) = S/s *s/(s+h) / ((S+H)/(s+h))
= S / (S+H)
That means, that the probability of a given message being spam when it
contians a token is independend of the number of messages trained.
Lets say in your real spam ham ratio is 10 to 1 and your messge body contains
1100 messages. 100 spam and also 50 ham messages should contain a certain
token. Lets say "[EMAIL PROTECTED]@".
Total Messages: 1100
Spam (trained): 1000
Ham: 100
[EMAIL PROTECTED]@: in 100 spam and 50 ham
If you train all messages you will get a propability of 100 / (100+50) = 66.6%
for the next message containing the token of being spam. Which isn't a high
probability but works fine for this example.
If you train only 10% of your spam to get the spam ham ration of 1:1 you will
supposably count only 10 spam messages with the token.
Spam (trained): 100
Ham: 100
[EMAIL PROTECTED]@: in 10 (=10% of 100) spam and 50 ham
Which leads to a spam probability of only 10 /(10+50) = 16.6%
Which is a little bit low.
What happens when you train less ham?
Lets assume you train only 50% of your ham but all your spam. You will
supposably count only 25 ham messages with the token.
Spam (trained): 1000
Ham (50% trained): 50
[EMAIL PROTECTED]@: in 100 spam and 25 (= 50% of 50) ham
Which leads to a spam probability of 100 /(100+25) = 80%.
What happens when your ham spam ratio is 10 to 1?
Ham = 1000
Spam = 100
[EMAIL PROTECTED]@: in 100 ham and 50 spam
=> 50 / (50+100) = 33.3%
Ham (10% trained) = 100
Spam = 100
[EMAIL PROTECTED]@: in 10 (=10% of 100) ham and 50 spam
=> 50 / (50+10) = 83.3%
OOps!!!
So if you train to less spam you will get a higher False Negative rate, if you
train to less ham you will get a higher False Positive rate.
Because a False Positive is more harmfull than a False Negative my conclusion
is:
train iaw your real spam ham ratio, train as much as possible (= train
all
messages), but never train to less ham or train only spam!
(BTW: The risk of a False Positives is the reason why Paul Graham multiplied
his token counts for ham with 2)
Another lesson should be: Never train whitelisted mails as ham!!!
Best regards
Thomas Arend
PS: I hope I made no mistakes.
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