# Patent Data Analysis of Artificial Intelligence Using Bayesian Interval Estimation
## 3 Bayesian Prediction Interval Estimation for Patent Data Analysis  
The frequentist estimates the model parameters using only the observed data. But, the Bayesian adds the prior belief in data to the observed data for the model parameter estimating [17]. The main difference between frequentist and Bayesian is the use of prior belief in data for statistical inference such as estimation, testing, modeling, etc. [10]. Because patent data analysis is an analysis of technology, more explanatory results can be expected when the expert's knowledge is reflected in data analysis. So, we consider the Bayesian approach for patent data analysis. Basically, the Bayesian approach is based on Bayesian rules as follows [18]. \[P(\theta|x)=\frac{P(\theta)P(x|\theta)}{P(x)} \tag{1}\]  
where \(x\) and \(y\) are variables representing the patent keywords. That is, [PERSON] rule is dependent on the conditional probability. In addition, \(P(x)=\sum_{\theta}P(\theta)P(x|\theta)\). In Bayesian inference, \(P(\theta|x)\) and \(P(\theta)\) are posterior and prior respectively. Also, \(P(x|\theta)\) is likelihood function based on observed data. The \(P(x)\) does not depend on \(\theta\) so we can omit it as follow [10]. \[P(\theta|x)\propto P(\theta)P(x|\theta) \tag{2}\]  
This is core formula of Bayesian inference. In this paper, the \(\theta\) represents the keyword extracted from patent documents, and we are finally interested in \(P(\theta|x)\) called posterior density. So, Bayesian inference reflects not only the collected data but also the domain knowledge to which the data belongs, through posterior distribution as follow. \[Posterior\propto Prior\times Likelihood\]  
This feature of Bayesian statistics is useful for analyzing patent data. This is because patent data represents technology and technology requires the experience of domain experts. We apply Bayesian inference to patent data analysis for technology analysis as shown in Table 1.  
In this paper, we combine the domain expert knowledge with collected patent data for technology patent analysis. Using this expression based on prior, likelihood and posterior, we compute Bayesian confidence interval for each keyword (\(\theta\)). The \(100(1-\alpha)\%\) Bayesian confidence interval \(C_{\alpha}\) of keyword \(\theta\) is defined as follows [10,17]. \[P(\theta\in C_{\alpha}|x)=(1-\alpha) \tag{3}\]  
The \(\theta\) represents the frequency of the keyword having posterior distribution and is defined as follows [10,17]. \[\int_{\theta\in C_{\alpha}}P(\theta|x)d\theta=1-\alpha \tag{4}\]  
In this case, we find the largest \(C_{\alpha}\) that satisfies  
\[P(\theta\in C_{\alpha}|x)\leq 1-\alpha \tag{5}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Bayesian Component** & **Density** & **Patent Technology Data** \\ \hline Prior & \(P(\theta)\) & Domain knowledge of target technology \\ Likelihood & \(P(x|\theta)\) & Collected patent documents \\ Posterior & \(P(\theta|x)\) & Combine domain knowledge with patent data \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bayesian inference and patent technology analysis. In general, Bayesian confidence intervals are not unique, because we can take many intervals satisfied with the proper probability coverage for a given posterior density. In this paper, \(x_{i}\) represents \(i\)th keyword in the patent-keyword matrix. Let \(n\) keywords data follows a normal distribution as follows. \[x_{1},\ x_{2},\ \ldots,\ x_{n}\ \sim\ N\big{(}\theta,\sigma^{2}\big{)} \tag{6}\]  
where \(\sigma^{2}\) is known, and the parameter \(\theta\) is distributed as \(N\big{(}\mu,v^{2}\big{)}\). Then the posterior distribution of \(\theta\) is shown as follow [10; 17]. \[N\Bigg{(}\Bigg{(}1-\frac{\sigma^{2}}{\sigma^{2}+nv^{2}}\Bigg{)}\overline{x}+ \Bigg{(}\frac{\sigma^{2}}{\sigma^{2}+nv^{2}}\Bigg{)}\mu,\frac{\sigma^{2}}{n} \Bigg{(}1-\frac{\sigma^{2}}{\sigma^{2}+nv^{2}}\Bigg{)}\Bigg{)} \tag{7}\]  
So, we can take \(100(1-\alpha)\%\) Bayesian credible interval as follow [10; 13].