Bayes is very simple, easily understood by a 6 year old, if explained properly. There are scary looking integral equations used to express it, but ultimately if you can count you can implement this.

Let's build a bayesian spam engine:

Start with a known-good email (not-spam) and a solicitation (spam). For each word in the not-spam document, count the number of times each word appears. That's your not-spam "corpus". Repeat for the known-spam document. Now you have a list of words+frequency (dictionary of counts). There will be many words which appear in both lists. This is normal.

Now, when you have an unclassified document, for each word in the document check both lists (spam & not-spam). In which list does this word appear more frequently? if the count for that word is higher in the spam list, then this is a "spam" word, therefore this document is +1 spam. If a not-spam word, then this document is +1 not-spam. Repeat for all words. At the end, which count (spam or not-spam) has the higher total? There's your answer. Spam or not-spam. Done.

Simple counting and comparison.

Seriously, that's the algorithm. It's that simple.

If the system miscategorizes a document, move it to the appropriate group yourself and count all the words again. (retrain). The system just "learned" from the mistake. It got "smarter".

You can easily add more categories, other than "spam" and "not spam" -- "questions from customers", "family & friends", etc. It works the same. The group with the highest total "wins" the classification.

extra credit: Now here's the interesting part. The corpus doesn't have to be words. It could be a switch set to "on" or "off." Joint angles on a robot. Ping time of a packet. Temperature from a thermistor.

And the classification doesn't need to be spam/not-spam, it could be "friend vs foe", upright/inverted, safe/not-safe, or anything else. The algorithm doesn't care, it's looking for probability of being in a classification group.

There you go. Have fun!