$[5$pts$]$ What are the differences between standard vector space tf$-$idf weighting and the BIM probabilistic retrieval model $($in the case where no document relevance information is available$)?$

$[5$pts$]$ How might a language model be used in a spelling correction system? In particular, consider the case of context$-$sensitive spelling correction, and correcting incorrect usages of words, such as their in Are you their? $($See Section $3\mathrm{.}5$ in the textbook for pointers to some literature on this topic.$)$

$[5$pts$]$ Programming: For this last programming assignment, you are to select $2$ web search engines, and compare them in a brief analysis on a basis of your choosing. You may choose the types of test queries to execute on them and compare the results in a quantitative and qualitative manner. Your submission should be $1-2$ paragraphs in length, about a page will suffice.