Consider ranking of documents using a Language Modeling$-$based IR algorithm.

$P(q,M_d)=pro{d}_{\text{d}\text{i}\text{s}\text{t}\text{i}\text{n}\text{c}\text{t}\text{t}\text{e}\text{r}\text{n}\text{t}in?}P(t,M_d{)}^{tht}$

where $t{f}_{t,q}$ is the term frequency $-$ number of occurrences of $t$ in query

We estimate the parameters $P(t,M_d)$ using Maximum Likelihood Estimate $($MLE$)$ as:

$(M_d|)=\frac{tf{t}_{t}d}{d}$

where

$|d|$ is the length of document $d$

$t{f}_{t,d}$ is the term frequency $-$ number of occurrences of $t$ in documen d

To avoid problem with zero probabilities, we smooth the estimates. First, we define:

where

$M_c$ is the collection model.

$c{f}_t$ is the number of occurrences of $t$ in the collection.

$T={\displaystyle \underset{t}{\overset{?}{}}}c{f}_t$ is the total number of tokens in the collection.

We use $(M_c|)$ to smooth $P(t|d)$ using Jelinek$-$Mercer smoothing as:

$(M_d(M_c|)|)$

Consider the following documents and $d_2)$ and query $(q)$ :

$d_1=$ epistemological considerations should also address learning design.

$d_2=$ epistemological considerations such as what is being measured.

$q$ : epistemological design

Rank $d_1$ and $d_2$ with respect to $q$ using Jelinek$-$Mercer smoothing. Use $=0\mathrm{.}75.$