# Using Markov Random Field and Analytic Hierarchy Process to Account for Interdependent Criteria
## 3 MRF-Based AHP  
By incorporating discrete MRFs into the AHP, we aim to address the gap of accounting for interdependency between criteria. This integration ultimately enhances the decision-making process and leads to more accurate and reliable results. Step 1: Model the interdependencies between criteria using MRFs  
To begin with, we must establish the MRF's structure to illustrate the criteria's interconnections. An undirected graph can represent these interdependencies, where nodes denote the criteria and edges indicate direct connections between them. For example, we have an adjacency matrix of a graph with four criteria:  
\[\mathbf{A}= \begin{matrix}C_{1}\begin{bmatrix}0&1&1&0\\ 1&0&1&1\\ C_{3}&1&1&0&1\\ C_{4}&0&1&1&0\end{bmatrix}\end{matrix} \]  
The adjacency indicates that, for example, \(C_{1}\) is interdependent with \(C_{2}\) and \(C_{3}\). Step 2: Pairwise comparisons and local priorities  
Next, we need to obtain the local priorities of the criteria and alternatives based on pairwise comparisons. For this, we will use the traditional AHP approach. Decision-makers will provide pairwise comparison matrices for each criterion with respect to the goal and for each alternative with respect to each criterion. Step 3: Calculate joint probability distribution using MRFs  
With the MRF structure and local priorities in place, we can now calculate the joint probability distribution over all criteria using MRFs. This is completed by combining the local probability distributions (obtained from the pairwise comparisons) with the pairwise potentials between connected criteria, which capture the strength of the interdependencies. The joint probability distribution will provide the overall priorities of the criteria, accounting for their interdependencies. For example, if we have initial weights derived from the AHP as \(\mathbf{p}=[0.2,0.3,0.3,0.2]\), and the interdependencies between criteria can be quantified by expert opinion as:  
\[\mathbf{W}= \begin{matrix}C_{1}\begin{bmatrix}0&0.3&0.2&0\\ 0.3&0&0.1&0.4\\ C_{3}&0.2&0.1&0&0.3\\ C_{4}&0&0.4&0.3&0\end{bmatrix}\end{matrix} \]  
Note that the pairwise potentials matrix \(\mathbf{W}\) is derived to represent the strength of the relationships between criteria in the MRF framework. This matrix is a key component in calculating the joint probability distribution of criteria, accounting for their interdependencies. To obtain \(\mathbf{W}\), expert opinions or data-driven methods can be utilized to quantify the degree of dependency between criteria pairs. Each entry \(\mathbf{W}_{\bar{y}}\) in the matrix reflects the strength of the relationship between criteria \(i\) and \(j\). This process involves both subjective judgment and quantitative analysis to ensure that the pairwise potentials accurately represent the interdependencies within the decision-making context. In this paper, we simply assume that the potential matrix is quantified by the expert and the objective potential matrix could be explored by the future research. Then, we derive the joint probability distribution of each criterion as follows. Following our assumption above, let us denote the initial priorities of the criteria as \(\mathbf{P}=[p_{1},\,p_{2},\,p_{3},\,p_{4}]\) and the pairwise potentials matrix as \(\mathbf{W}\), where \(W_{\bar{y}}\) represents the strength of the relationship between criteria \(i\) and \(j\). The goal is to estimate the adjusted priorities \(\mathbf{P^{\prime}}=[p_{1}{}^{\prime},p_{2}{}^{\prime},p_{3}{}^{\prime},p_{4}{}^{ \prime}]\) through Gibbs sampling. The algorithm starts with the initial priority \(\mathbf{P}\) and proceeds as follows:  
1. For each criterion, \(i\), identify the incoming and outgoing pairwise potentials. Incoming edges are the rows in the \(\mathbf{W}\) matrix with a positive value in the \(i\)th column (\(W_{ij}>0,\forall j\)), and outgoing edges are the columns in the \(\mathbf{W}\) matrix with a positive value in the \(i\)th row (\(W_{ij}>0,\forall i\)). 2.