# A Focused Event Crawler with Temporal Intent
## 3 Mathematical Proof and Identification of Temporal Intent  
### Mathematical Proof of Temporal Intent Using Bayes Formula  
In general, there will be little web information about the event before its occurrence. This means that compared to the total amount of web information about the event, its number may be negligible. It can therefore be concluded that the web information is irrelevant to the event if it is published before the event starts. After the event, there will be a series of reports on the web. Even after the event is over, there will be new web information about the event for a period of time. For example, on the anniversary of an important event, there will be some new web information on the occasion of thisspecial event. Therefore, the web information may or may not be relevant to the event if the publication time of the web information is equal to or greater than the start time of the event. From a probability perspective, the above conclusions can be expressed as \"given an event, the probability of web information published earlier than its start time is 0, and the probability of web information published on or later than its start time can be 0 or 1\". Assuming that the start time of the event T is \(t_{s}\) and the publication time of a random web page is \(t\), the above conclusion can be formally expressed as Equation (1) according to the definition of conditional probability. \[\Pr(t/T)= \begin{cases}0&t<t_{s}\\ x&t\geq t_{s}\end{cases}  \tag{1}\]  
where \(Pr(t/T)\) is the probability that a time \(t\) is relevant to an event T using the distribution of relevant web pages. The value \(x\) is an arbitrary number between 0 and 1. \(Pr(t/T)\) can be estimated by Bayes Formula, as shown in Equation (2). \[Pr(t/T)=\frac{Pr(T/t)\times Pr(t)}{Pr(T)}=\frac{Pr(T/t)\times Pr(t)}{\sum_{t1 \in Time(WP)}Pr(T/t1)\times Pr(t1)} \tag{2}\]  
where \(Pr(t)\) represents the probability that time \(t\) contains a web page (relevant to T or not), \(Pr(T)\) is the prior probability of finding a web page relevant to event T, and serves as a normalizing factor. A variable of \(WP\) represents a collection of web pages and \(Time(WP)\) is a collection of publication times of WP. \(Pr(T/t)\) is the probability that the web page published in t is relevant to the event T. For example, a random web page selected from the web pages published on 3 November 2013 has a higher probability of being relevant to the event \"Typhoon Haiyan\" than a random web page selected from the web pages published in November 2014. Suppose the web page collection \(WP\) contains the number of \(Num(WP)\) web pages and the number of \(Num(WP,t)\) web pages published at time \(t\), \(\Pr(t)\) can be estimated by Equation (3). \[Pr(t)=\frac{Num(WP,t)}{Num(WP)} \tag{3}\]  
Suppose the collection of web pages related to the given event T is \(R_{T}\). It contains the number of \(Num(R_{T})\) web pages and the number of \(Num(R_{T},t)\) web pages published at time \(t\). Therefore, \(Pr(T/t)\) can be estimated using the distribution of the web pages in \(R_{T}\) over time with Equation (4). \[Pr(T/t)=\frac{Num(R_{T},t)}{Num(WP,t)} \tag{4}\]  
Equations (3) and (4) are then combined with (2) to obtain Equation (5). \[Pr(t/T)=\frac{Num(R_{T},t)}{Num(R_{T})} \tag{5}\]  
In general, we do not know the ground truth \(Num(R_{T})\) and \(Num(R_{T},t)\), but when time is infinitely large, the value of \(Num(R_{T})\) is fixed and invariable. Therefore, the probability \(Pr(t/T)\) is proportional to the value of \(Num(R_{T},t)\). According to this, it is possible to draw two important roles of the temporal intent in the focused crawler. The first role is that the start time of an event can be utilized to filter out irrelevant web pages. As discussed in the first paragraph of this section, when the web information is published earlier than the start time of the event (\(t<t_{s}\)), the value of \(Num(R_{T},t)\) is approximately equal to 0.