January 2003  
  
 _(This article was given as a talk at the 2003 Spam Conference. It describes
the work I've done to improve the performance of the algorithm described in[A
Plan for Spam](spam.html), and what I plan to do in the future.)_  
  
The first discovery I'd like to present here is an algorithm for lazy
evaluation of research papers. Just write whatever you want and don't cite any
previous work, and indignant readers will send you references to all the
papers you should have cited. I discovered this algorithm after ``A Plan for
Spam'' [1] was on Slashdot.  
  
Spam filtering is a subset of text classification, which is a well established
field, but the first papers about Bayesian spam filtering per se seem to have
been two given at the same conference in 1998, one by Pantel and Lin [2], and
another by a group from Microsoft Research [3].  
  
When I heard about this work I was a bit surprised. If people had been onto
Bayesian filtering four years ago, why wasn't everyone using it? When I read
the papers I found out why. Pantel and Lin's filter was the more effective of
the two, but it only caught 92% of spam, with 1.16% false positives.  
  
When I tried writing a Bayesian spam filter, it caught 99.5% of spam with less
than .03% false positives [4]. It's always alarming when two people trying the
same experiment get widely divergent results. It's especially alarming here
because those two sets of numbers might yield opposite conclusions. Different
users have different requirements, but I think for many people a filtering
rate of 92% with 1.16% false positives means that filtering is not an
acceptable solution, whereas 99.5% with less than .03% false positives means
that it is.  
  
So why did we get such different numbers? I haven't tried to reproduce Pantel
and Lin's results, but from reading the paper I see five things that probably
account for the difference.  
  
One is simply that they trained their filter on very little data: 160 spam and
466 nonspam mails. Filter performance should still be climbing with data sets
that small. So their numbers may not even be an accurate measure of the
performance of their algorithm, let alone of Bayesian spam filtering in
general.  
  
But I think the most important difference is probably that they ignored
message headers. To anyone who has worked on spam filters, this will seem a
perverse decision. And yet in the very first filters I tried writing, I
ignored the headers too. Why? Because I wanted to keep the problem neat. I
didn't know much about mail headers then, and they seemed to me full of random
stuff. There is a lesson here for filter writers: don't ignore data. You'd
think this lesson would be too obvious to mention, but I've had to learn it
several times.  
  
Third, Pantel and Lin stemmed the tokens, meaning they reduced e.g. both
``mailing'' and ``mailed'' to the root ``mail''. They may have felt they were
forced to do this by the small size of their corpus, but if so this is a kind
of premature optimization.  
  
Fourth, they calculated probabilities differently. They used all the tokens,
whereas I only use the 15 most significant. If you use all the tokens you'll
tend to miss longer spams, the type where someone tells you their life story
up to the point where they got rich from some multilevel marketing scheme. And
such an algorithm would be easy for spammers to spoof: just add a big chunk of
random text to counterbalance the spam terms.  
  
Finally, they didn't bias against false positives. I think any spam filtering
algorithm ought to have a convenient knob you can twist to decrease the false
positive rate at the expense of the filtering rate. I do this by counting the
occurrences of tokens in the nonspam corpus double.  
  
I don't think it's a good idea to treat spam filtering as a straight text
classification problem. You can use text classification techniques, but
solutions can and should reflect the fact that the text is email, and spam in
particular. Email is not just text; it has structure. Spam filtering is not
just classification, because false positives are so much worse than false
negatives that you should treat them as a different kind of error. And the
source of error is not just random variation, but a live human spammer working
actively to defeat your filter.  
  
**Tokens**  
  
Another project I heard about after the Slashdot article was Bill Yerazunis'
[CRM114](http://crm114.sourceforge.net) [5]. This is the counterexample to the
design principle I just mentioned. It's a straight text classifier, but such a
stunningly effective one that it manages to filter spam almost perfectly
without even knowing that's what it's doing.  
  
Once I understood how CRM114 worked, it seemed inevitable that I would
eventually have to move from filtering based on single words to an approach
like this. But first, I thought, I'll see how far I can get with single words.
And the answer is, surprisingly far.  
  
Mostly I've been working on smarter tokenization. On current spam, I've been
able to achieve filtering rates that approach CRM114's. These techniques are
mostly orthogonal to Bill's; an optimal solution might incorporate both.  
  
``A Plan for Spam'' uses a very simple definition of a token. Letters, digits,
dashes, apostrophes, and dollar signs are constituent characters, and
everything else is a token separator. I also ignored case.  
  
Now I have a more complicated definition of a token:

  1. Case is preserved.  
  

  2. Exclamation points are constituent characters.  
  

  3. Periods and commas are constituents if they occur between two digits. This lets me get ip addresses and prices intact.  
  

  4. A price range like $20-25 yields two tokens, $20 and $25.  
  

  5. Tokens that occur within the To, From, Subject, and Return-Path lines, or within urls, get marked accordingly. E.g. ``foo'' in the Subject line becomes ``Subject*foo''. (The asterisk could be any character you don't allow as a constituent.) 

Such measures increase the filter's vocabulary, which makes it more
discriminating. For example, in the current filter, ``free'' in the Subject
line has a spam probability of 98%, whereas the same token in the body has a
spam probability of only 65%.  
  
Here are some of the current probabilities [6]:  
  

    
    
    Subject*FREE      0.9999
    free!!            0.9999
    To*free           0.9998
    Subject*free      0.9782
    free!             0.9199
    Free              0.9198
    Url*free          0.9091
    FREE              0.8747
    From*free         0.7636
    free              0.6546
    

In the Plan for Spam filter, all these tokens would have had the same
probability, .7602. That filter recognized about 23,000 tokens. The current
one recognizes about 187,000.  
  
The disadvantage of having a larger universe of tokens is that there is more
chance of misses. Spreading your corpus out over more tokens has the same
effect as making it smaller. If you consider exclamation points as
constituents, for example, then you could end up not having a spam probability
for free with seven exclamation points, even though you know that free with
just two exclamation points has a probability of 99.99%.  
  
One solution to this is what I call degeneration. If you can't find an exact
match for a token, treat it as if it were a less specific version. I consider
terminal exclamation points, uppercase letters, and occurring in one of the
five marked contexts as making a token more specific. For example, if I don't
find a probability for ``Subject*free!'', I look for probabilities for
``Subject*free'', ``free!'', and ``free'', and take whichever one is farthest
from .5.  
  
Here are the alternatives [7] considered if the filter sees ``FREE!!!'' in the
Subject line and doesn't have a probability for it.  
  

    
    
    Subject*Free!!!
    Subject*free!!!
    Subject*FREE!
    Subject*Free!
    Subject*free!
    Subject*FREE
    Subject*Free
    Subject*free
    FREE!!!
    Free!!!
    free!!!
    FREE!
    Free!
    free!
    FREE
    Free
    free              
    

If you do this, be sure to consider versions with initial caps as well as all
uppercase and all lowercase. Spams tend to have more sentences in imperative
mood, and in those the first word is a verb. So verbs with initial caps have
higher spam probabilities than they would in all lowercase. In my filter, the
spam probability of ``Act'' is 98% and for ``act'' only 62%.  
  
If you increase your filter's vocabulary, you can end up counting the same
word multiple times, according to your old definition of ``same''. Logically,
they're not the same token anymore. But if this still bothers you, let me add
from experience that the words you seem to be counting multiple times tend to
be exactly the ones you'd want to.  
  
Another effect of a larger vocabulary is that when you look at an incoming
mail you find more interesting tokens, meaning those with probabilities far
from .5. I use the 15 most interesting to decide if mail is spam. But you can
run into a problem when you use a fixed number like this. If you find a lot of
maximally interesting tokens, the result can end up being decided by whatever
random factor determines the ordering of equally interesting tokens. One way
to deal with this is to treat some as more interesting than others.  
  
For example, the token ``dalco'' occurs 3 times in my spam corpus and never in
my legitimate corpus. The token ``Url*optmails'' (meaning ``optmails'' within
a url) occurs 1223 times. And yet, as I used to calculate probabilities for
tokens, both would have the same spam probability, the threshold of .99.  
  
That doesn't feel right. There are theoretical arguments for giving these two
tokens substantially different probabilities (Pantel and Lin do), but I
haven't tried that yet. It does seem at least that if we find more than 15
tokens that only occur in one corpus or the other, we ought to give priority
to the ones that occur a lot. So now there are two threshold values. For
tokens that occur only in the spam corpus, the probability is .9999 if they
occur more than 10 times and .9998 otherwise. Ditto at the other end of the
scale for tokens found only in the legitimate corpus.  
  
I may later scale token probabilities substantially, but this tiny amount of
scaling at least ensures that tokens get sorted the right way.  
  
Another possibility would be to consider not just 15 tokens, but all the
tokens over a certain threshold of interestingness. Steven Hauser does this in
his statistical spam filter [8]. If you use a threshold, make it very high, or
spammers could spoof you by packing messages with more innocent words.  
  
Finally, what should one do about html? I've tried the whole spectrum of
options, from ignoring it to parsing it all. Ignoring html is a bad idea,
because it's full of useful spam signs. But if you parse it all, your filter
might degenerate into a mere html recognizer. The most effective approach
seems to be the middle course, to notice some tokens but not others. I look at
a, img, and font tags, and ignore the rest. Links and images you should
certainly look at, because they contain urls.  
  
I could probably be smarter about dealing with html, but I don't think it's
worth putting a lot of time into this. Spams full of html are easy to filter.
The smarter spammers already avoid it. So performance in the future should not
depend much on how you deal with html.  
  
**Performance**  
  
Between December 10 2002 and January 10 2003 I got about 1750 spams. Of these,
4 got through. That's a filtering rate of about 99.75%.  
  
Two of the four spams I missed got through because they happened to use words
that occur often in my legitimate email.  
  
The third was one of those that exploit an insecure cgi script to send mail to
third parties. They're hard to filter based just on the content because the
headers are innocent and they're careful about the words they use. Even so I
can usually catch them. This one squeaked by with a probability of .88, just
under the threshold of .9.  
  
Of course, looking at multiple token sequences would catch it easily. ``Below
is the result of your feedback form'' is an instant giveaway.  
  
The fourth spam was what I call a spam-of-the-future, because this is what I
expect spam to evolve into: some completely neutral text followed by a url. In
this case it was was from someone saying they had finally finished their
homepage and would I go look at it. (The page was of course an ad for a porn
site.)  
  
If the spammers are careful about the headers and use a fresh url, there is
nothing in spam-of-the-future for filters to notice. We can of course counter
by sending a crawler to look at the page. But that might not be necessary. The
response rate for spam-of-the-future must be low, or everyone would be doing
it. If it's low enough, it [won't pay](wfks.html) for spammers to send it, and
we won't have to work too hard on filtering it.  
  
Now for the really shocking news: during that same one-month period I got
_three_ false positives.  
  
In a way it's a relief to get some false positives. When I wrote ``A Plan for
Spam'' I hadn't had any, and I didn't know what they'd be like. Now that I've
had a few, I'm relieved to find they're not as bad as I feared. False
positives yielded by statistical filters turn out to be mails that sound a lot
like spam, and these tend to be the ones you would least mind missing [9].  
  
Two of the false positives were newsletters from companies I've bought things
from. I never asked to receive them, so arguably they were spams, but I count
them as false positives because I hadn't been deleting them as spams before.
The reason the filters caught them was that both companies in January switched
to commercial email senders instead of sending the mails from their own
servers, and both the headers and the bodies became much spammier.  
  
The third false positive was a bad one, though. It was from someone in Egypt
and written in all uppercase. This was a direct result of making tokens case
sensitive; the Plan for Spam filter wouldn't have caught it.  
  
It's hard to say what the overall false positive rate is, because we're up in
the noise, statistically. Anyone who has worked on filters (at least,
effective filters) will be aware of this problem. With some emails it's hard
to say whether they're spam or not, and these are the ones you end up looking
at when you get filters really tight. For example, so far the filter has
caught two emails that were sent to my address because of a typo, and one sent
to me in the belief that I was someone else. Arguably, these are neither my
spam nor my nonspam mail.  
  
Another false positive was from a vice president at Virtumundo. I wrote to
them pretending to be a customer, and since the reply came back through
Virtumundo's mail servers it had the most incriminating headers imaginable.
Arguably this isn't a real false positive either, but a sort of Heisenberg
uncertainty effect: I only got it because I was writing about spam filtering.  
  
Not counting these, I've had a total of five false positives so far, out of
about 7740 legitimate emails, a rate of .06%. The other two were a notice that
something I bought was back-ordered, and a party reminder from Evite.  
  
I don't think this number can be trusted, partly because the sample is so
small, and partly because I think I can fix the filter not to catch some of
these.  
  
False positives seem to me a different kind of error from false negatives.
Filtering rate is a measure of performance. False positives I consider more
like bugs. I approach improving the filtering rate as optimization, and
decreasing false positives as debugging.  
  
So these five false positives are my bug list. For example, the mail from
Egypt got nailed because the uppercase text made it look to the filter like a
Nigerian spam. This really is kind of a bug. As with html, the email being all
uppercase is really conceptually _one_ feature, not one for each word. I need
to handle case in a more sophisticated way.  
  
So what to make of this .06%? Not much, I think. You could treat it as an
upper bound, bearing in mind the small sample size. But at this stage it is
more a measure of the bugs in my implementation than some intrinsic false
positive rate of Bayesian filtering.  
  
**Future**  
  
What next? Filtering is an optimization problem, and the key to optimization
is profiling. Don't try to guess where your code is slow, because you'll guess
wrong. _Look_ at where your code is slow, and fix that. In filtering, this
translates to: look at the spams you miss, and figure out what you could have
done to catch them.  
  
For example, spammers are now working aggressively to evade filters, and one
of the things they're doing is breaking up and misspelling words to prevent
filters from recognizing them. But working on this is not my first priority,
because I still have no trouble catching these spams [10].  
  
There are two kinds of spams I currently do have trouble with. One is the type
that pretends to be an email from a woman inviting you to go chat with her or
see her profile on a dating site. These get through because they're the one
type of sales pitch you can make without using sales talk. They use the same
vocabulary as ordinary email.  
  
The other kind of spams I have trouble filtering are those from companies in
e.g. Bulgaria offering contract programming services. These get through
because I'm a programmer too, and the spams are full of the same words as my
real mail.  
  
I'll probably focus on the personal ad type first. I think if I look closer
I'll be able to find statistical differences between these and my real mail.
The style of writing is certainly different, though it may take multiword
filtering to catch that. Also, I notice they tend to repeat the url, and
someone including a url in a legitimate mail wouldn't do that [11].  
  
The outsourcing type are going to be hard to catch. Even if you sent a crawler
to the site, you wouldn't find a smoking statistical gun. Maybe the only
answer is a central list of domains advertised in spams [12]. But there can't
be that many of this type of mail. If the only spams left were unsolicited
offers of contract programming services from Bulgaria, we could all probably
move on to working on something else.  
  
Will statistical filtering actually get us to that point? I don't know. Right
now, for me personally, spam is not a problem. But spammers haven't yet made a
serious effort to spoof statistical filters. What will happen when they do?  
  
I'm not optimistic about filters that work at the network level [13]. When
there is a static obstacle worth getting past, spammers are pretty efficient
at getting past it. There is already a company called Assurance Systems that
will run your mail through Spamassassin and tell you whether it will get
filtered out.  
  
Network-level filters won't be completely useless. They may be enough to kill
all the "opt-in" spam, meaning spam from companies like Virtumundo and
Equalamail who claim that they're really running opt-in lists. You can filter
those based just on the headers, no matter what they say in the body. But
anyone willing to falsify headers or use open relays, presumably including
most porn spammers, should be able to get some message past network-level
filters if they want to. (By no means the message they'd like to send though,
which is something.)  
  
The kind of filters I'm optimistic about are ones that calculate probabilities
based on each individual user's mail. These can be much more effective, not
only in avoiding false positives, but in filtering too: for example, finding
the recipient's email address base-64 encoded anywhere in a message is a very
good spam indicator.  
  
But the real advantage of individual filters is that they'll all be different.
If everyone's filters have different probabilities, it will make the spammers'
optimization loop, what programmers would call their edit-compile-test cycle,
appallingly slow. Instead of just tweaking a spam till it gets through a copy
of some filter they have on their desktop, they'll have to do a test mailing
for each tweak. It would be like programming in a language without an
interactive toplevel, and I wouldn't wish that on anyone.  
  
  
  
**Notes**  
  
[1] Paul Graham. ``A Plan for Spam.'' August 2002.
http://paulgraham.com/spam.html.  
  
Probabilities in this algorithm are calculated using a degenerate case of
Bayes' Rule. There are two simplifying assumptions: that the probabilities of
features (i.e. words) are independent, and that we know nothing about the
prior probability of an email being spam.  
  
The first assumption is widespread in text classification. Algorithms that use
it are called ``naive Bayesian.''  
  
The second assumption I made because the proportion of spam in my incoming
mail fluctuated so much from day to day (indeed, from hour to hour) that the
overall prior ratio seemed worthless as a predictor. If you assume that
P(spam) and P(nonspam) are both .5, they cancel out and you can remove them
from the formula.  
  
If you were doing Bayesian filtering in a situation where the ratio of spam to
nonspam was consistently very high or (especially) very low, you could
probably improve filter performance by incorporating prior probabilities. To
do this right you'd have to track ratios by time of day, because spam and
legitimate mail volume both have distinct daily patterns.  
  
[2] Patrick Pantel and Dekang Lin. ``SpamCop-- A Spam Classification &
Organization Program.'' Proceedings of AAAI-98 Workshop on Learning for Text
Categorization.  
  
[3] Mehran Sahami, Susan Dumais, David Heckerman and Eric Horvitz. ``A
Bayesian Approach to Filtering Junk E-Mail.'' Proceedings of AAAI-98 Workshop
on Learning for Text Categorization.  
  
[4] At the time I had zero false positives out of about 4,000 legitimate
emails. If the next legitimate email was a false positive, this would give us
.03%. These false positive rates are untrustworthy, as I explain later. I
quote a number here only to emphasize that whatever the false positive rate
is, it is less than 1.16%.  
  
[5] Bill Yerazunis. ``Sparse Binary Polynomial Hash Message Filtering and The
CRM114 Discriminator.'' Proceedings of 2003 Spam Conference.  
  
[6] In ``A Plan for Spam'' I used thresholds of .99 and .01. It seems
justifiable to use thresholds proportionate to the size of the corpora. Since
I now have on the order of 10,000 of each type of mail, I use .9999 and .0001.  
  
[7] There is a flaw here I should probably fix. Currently, when
``Subject*foo'' degenerates to just ``foo'', what that means is you're getting
the stats for occurrences of ``foo'' in the body or header lines other than
those I mark. What I should do is keep track of statistics for ``foo'' overall
as well as specific versions, and degenerate from ``Subject*foo'' not to
``foo'' but to ``Anywhere*foo''. Ditto for case: I should degenerate from
uppercase to any-case, not lowercase.  
  
It would probably be a win to do this with prices too, e.g. to degenerate from
``$129.99'' to ``$--9.99'', ``$--.99'', and ``$--''.  
  
You could also degenerate from words to their stems, but this would probably
only improve filtering rates early on when you had small corpora.  
  
[8] Steven Hauser. ``Statistical Spam Filter Works for Me.''
http://www.sofbot.com.  
  
[9] False positives are not all equal, and we should remember this when
comparing techniques for stopping spam. Whereas many of the false positives
caused by filters will be near-spams that you wouldn't mind missing, false
positives caused by blacklists, for example, will be just mail from people who
chose the wrong ISP. In both cases you catch mail that's near spam, but for
blacklists nearness is physical, and for filters it's textual.  
  
[10] If spammers get good enough at obscuring tokens for this to be a problem,
we can respond by simply removing whitespace, periods, commas, etc. and using
a dictionary to pick the words out of the resulting sequence. And of course
finding words this way that weren't visible in the original text would in
itself be evidence of spam.  
  
Picking out the words won't be trivial. It will require more than just
reconstructing word boundaries; spammers both add (``xHot nPorn cSite'') and
omit (``P#rn'') letters. Vision research may be useful here, since human
vision is the limit that such tricks will approach.  
  
[11] In general, spams are more repetitive than regular email. They want to
pound that message home. I currently don't allow duplicates in the top 15
tokens, because you could get a false positive if the sender happens to use
some bad word multiple times. (In my current filter, ``dick'' has a spam
probabilty of .9999, but it's also a name.) It seems we should at least notice
duplication though, so I may try allowing up to two of each token, as Brian
Burton does in SpamProbe.  
  
[12] This is what approaches like Brightmail's will degenerate into once
spammers are pushed into using mad-lib techniques to generate everything else
in the message.  
  
[13] It's sometimes argued that we should be working on filtering at the
network level, because it is more efficient. What people usually mean when
they say this is: we currently filter at the network level, and we don't want
to start over from scratch. But you can't dictate the problem to fit your
solution.  
  
Historically, scarce-resource arguments have been the losing side in debates
about software design. People only tend to use them to justify choices
(inaction in particular) made for other reasons.  
  
**Thanks** to Sarah Harlin, Trevor Blackwell, and Dan Giffin for reading
drafts of this paper, and to Dan again for most of the infrastructure that
this filter runs on.  
  
  
  
**Related:**  
  

