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What is tf * idf weighing?

It's the basic weighing scheme used in information retrieval. Tf stands for term frequency while idf is inverted document frequency. Term frequency is simply the number of times the term appears in a document, while document frequency is the number of documents in the database where the term appears.

Thus, the weight of the word for a document increases the more often it appears in the document and the less often it appears in other documents.

What are stop words?

Each document database is full of useless words. All the little words that appear in just about every document are completely useless for information retrieval purposes. Basically, their inverted document frequency is really low, so they never have much power in matching. Also, removing those words helps to make the index smaller and searching faster.