# Construction Method of Probabilistic Boolean Networks Based on Imperfect Information
## 3 Construction Problem and Its Solution Method  
Using the matrix-based representation, the problem of finding a PBN is reduced to the problem of finding matrices. In this section, we formulate the construction problem of PBNs and propose its solution method. First, let \(s_{i,p}^{k+1}\), \(s_{i,p}^{k}\), \(p=1,2,\ldots,\overline{p}\) denote the sample mean of the states, which are measured multiple times at a certain time interval \(k\) and \(k+1\) under a certain pattern \(p\) of the experiment, where \(\overline{p}\) is the number of patterns. Each experiment can be characterized by the initial state. However, in an experiment of biological systems, there is the case where the initial state cannot be set precisely. In this paper, we suppose that the initial state for each pattern may be unknown. Next, we give the construction problem of a PBN as follows. **Problem 1**. : _For a PBN, suppose that the index set \(\mathcal{N}_{i}\), \(i=1,2,\ldots,n\) and the sample mean \(s_{i,p}^{k+1}\), \(s_{i,p}^{k}\), \(i=1,2,\ldots,n\), \(p=1,2,\ldots,\overline{p}\) are given. Then, find the probabilities \(c_{l}^{(i)}\), \(l=1,2,\ldots,q(i)\) and the Boolean functions \(f_{l}^{(i)}\), \(l=1,2,\ldots,q(i)\) satisfying \(E[x_{i}(k+1)]=s_{i,p}^{k+1}\) and \(E[x_{i}(k)]=s_{i,p}^{k}\) for the pair of \(s_{i,p}^{k+1}\) and \(s_{i,p}^{k}\)._  
We remark here that in this problem, candidates of Boolean functions (\(f_{l}^{(i)}\)), those selection probabilities (\(c_{l}^{(i)}\)), and the number of candidates of Boolean functions (\(q(i)\)) are not given. Problem 1 can be equivalently rewritten as the following problem using the matrix-based representation. **Problem 2**. : _For a PBN, suppose that the index set \(\mathcal{N}_{i}\), \(i=1,2,\ldots,n\) and the sample mean \(s^{k+1}_{i,p},s^{k}_{i,p}\), \(i=1,2,\ldots,n\), \(p=1,2,\ldots,\overline{p}\) are given. Then, find the probabilities \(c^{(i)}_{l}\), \(l=1,2,\ldots,q(i)\) and the matrices \(\tilde{A}^{(i)}_{l}\), \(l=1,2,\ldots,q(i)\) satisfying the following condition:_  
\[\widetilde{s}^{k+1}_{i,p}=\sum_{l=1}^{q(i)}\left(c^{(i)}_{l}\tilde{A}^{(i)}_{ l}\right)\left(\bigotimes_{j\in\mathcal{N}_{i}}\widetilde{s}^{k}_{i,p}\right), \tag{5}\]  
_where \(\widetilde{s}^{k}_{i,p}:=[\;1-s^{k}_{i,p}\;\;s^{k}_{i,p}\;]^{\top}\)._  
It is difficult to directly reduce Problem 2 to a certain optimization problem. Here, we propose a solution method consisting of two steps. In the proposed method, first, the matrices including both the selection probabilities and the matrices expressing the Boolean functions are calculated by solving a linear programming (LP) problem. Next, the solution of Problem 2 is derived from the matrices obtained. First, defining \(\tilde{B}^{(i)}:=\sum_{l=1}^{q(i)}c^{(i)}_{l}\tilde{A}^{(i)}_{l}\), (5) can be rewritten as  
\[\widetilde{s}^{k+1}_{i,p}=\tilde{B}^{(i)}\left(\bigotimes_{j\in\mathcal{N}_{i }}\widetilde{s}^{k}_{i,p}\right), \tag{6}\]  
where each element of \(\tilde{B}^{(i)}\) must take on a real variable on \([0,1]\). In addition, the sum of each column must be equal to 1. Here, consider the following problem. **Problem 3**. : _For a PBN, suppose that the index set \(\mathcal{N}_{i}\), \(i=1,2,\ldots,n\) and the sample mean \(s^{k+1}_{i,p},s^{k}_{i,p}\), \(i=1,2,\ldots,n\), \(p=1,2,\ldots,\overline{p}\) are given. Then, find the matrices \(\tilde{B}^{(i)}\), \(i=1,2,\ldots,n\) satisfying (6)._  
By a simple calculation, Problem 3 can be reduced to \(n\) LP problems with no objective function. Using \(\tilde{B}^{(i)}\) obtained by solving Problem 3, we can obtain the probabilities \(c^{(i)}_{l}\), \(l=1,2,\ldots,q(i)\), the matrices \(\tilde{A}^{(i)}_{l}\), \(l=1,2,\ldots,q(i)\), and \(q(i)\). See e.g., [27] for the further details. Now we consider the case where the candidate of Boolean functions is given by only one based on theoretical biology and past experiments. In this case, it is desirable that the probability that the Boolean function given in advance is chosen is higher than other selection probabilities.