# Query Likelihood Model - [[Language Models|Unigram]] language model is defined as $ P\left(t \mid M_{d}\right)=\frac{t f(t, d)}{d l(d)} $ - A document is a multinomial distribution over words. - If some vocabulary terms do not appear in document $d$, then $P\left(t \mid M_{d}\right)=0$ - This is addressed by [[Language Models#Laplace add 1 smoothing]] How do we match these two distr ibutions? It is given by Query Likelihood Model. - Likelihood of a document given a query $ P(d \mid q)=\frac{P(q \mid d) P(d)}{P(q)} $ - The prior distribution over queries $P(q)$ does not affect matching for a particular query, so $ P(d \mid q) \stackrel{\operatorname{rank}}{=} P(q \mid d) P(d) $ - Usually, the prior distribution over documents $P(d)$ is assumed to be uniform ($M_d$ is the model of the document) $ P(d \mid q) \stackrel{\text { rank }}{=} P(q \mid d)=P\left(q \mid M_{d}\right) $ - "Bag of words" assumption: terms are independent. Therefore, $ P\left(q \mid M_{d}\right)=\prod_{t \in q} P\left(t \mid M_{d}\right)=\prod_{t \in q} \frac{t f(t, d)}{d l(d)} $ --- ## References 1. IR1 Course 2021, UvA 2. https://course.ccs.neu.edu/cs6200sp15/slides/m03.s06%20-%20query%20likelihood%20retrieval.pdf