# A Study of a Gain Based Approach for Query Aspects in Recall Oriented Tasks
## 2 A GAIN-Based Approach  
In [PERSON] et al. [21], define the intent-aware gain metric and the requirements that it should satisfy. They identify the following properties: importance, documents relevant to a central aspect of the search topic produce higher gain than those relevant to a peripheral one; relevance, highly relevant documents produce higher gain than partially relevant ones; novelty, documents relevant to an unexplored aspect produce higher gain than those relevant to a fully explored aspect.  
The set of aspects \(A_{t}\) of a topic \(t\) is estimated through the process described in [22]: first, a set of subtopics \(S_{t}\) is mined given a topic \(t\); then, the subtopics are grouped into a set of clusters \(C_{t}\). These clusters are regarded as the \"facets\" (We use _facets_ instead of _aspects_ to not repeat the same term that will be use to identify the most representative subtopic.) of \(t\). The most representative subtopic \(s\) is chosen from each cluster as formulation of the topic aspect \(a\) using the formula \(a=argmax_{s\in C_{t}}\text{Imp}_{t}(s)\), where the importance of a subtopic \(s\) is defined as:  
\[\text{Imp}_{t}(s)=\sum_{d\in D_{s}^{N}\cap d\in D_{t}^{N}}\frac{1}{\text{Rank }_{t}(d)} \tag{1}\]  
\(D_{s}^{N}\) and \(D_{t}^{N}\) denote the sets of the top \(N\) retrieved documents for a subtopic \(s\) and the topic \(t\), respectively, and \(\text{Rank}_{t}(d)\) is the rank of the document \(d\) in the ranked list for \(t\).  
It is crucial to stress that the definition of _importance_, and the following definition of _gain_, derives from the assumption that there is a known \"reference\" topic \(t\) that describes completely the information need. For such topic \(t\) the retrieved documents can be different compared to the ones observed for a query which represents just one aspect \(a\) of the topic.  
In Figure 1, we show an example of a number of subtopics found for a topic \(t\) and grouped into three clusters, each one with a representative aspect.  
The Intent-Aware Gain is defined for a set of documents \(D\) as:  
\[\text{Gain-IA}_{t}(D)=\sum_{a\in A_{t}}P(a|t)\cdot\text{Gain}_{t,a}(D) \tag{2}\]  
which is a sort of expected value of the gain across the different aspects. \(P(a|t)\) is the probability that an aspect \(a\) is important to the topic \(t\), and \(\text{Gain}_{t,a}(D)\) is the gain that can be obtained by the aspect \(a\) from the documents D. The importance probability for an aspect of a topic is computed as:  
\[P(a|t)=\frac{\text{Imp}_{t}(a)}{\sum_{a^{\prime}\in A_{t}}\text{Imp}_{t}(a^{ \prime})} \tag{3}\]  
while the gain which measures how the documents \(D\) retrieved for a query contribute to increment the information relative to a specific aspect of the topic is:  
\[\text{Gain}_{t,a}(D)=\left[1-\prod_{d\in D}(1-\text{Rel}_{t,a}(d))\right] \tag{4}\]  
This last part that is required to compute the Intent-Aware Gain contains the term \(\text{Rel}_{t,a}(d)\) which is the relevance degree of a document \(d\) with respect to an aspect \(a\), estimated as follows:  
\[\text{Rel}_{t,a}(d)=\frac{\sum_{s\in C_{a}}\text{Imp}_{t}(s)\cdot\text{Rel}_ {s}(d)}{\sum_{s\in C_{a}}\text{Imp}_{t}(s)} \tag{5}\]  
where \(C_{a}\in C_{t}\) is the cluster of subtopics belonging to the aspect \(a\), and \(Rel_{s}(d)\) is the relevance degree of a document \(d\) to a subtopic \(s\) estimated as \(\text{Rel}_{s}(d)=1/\sqrt{\text{Rank}_{s}(d)}\).