# Convergence of a Belief Propagation Algorithm for Biological Networks

Authors: Sang-Mok Choo; Young-Hee Kim
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

Constructing network models of biological systems is important for effective understanding and control of the biological systems. For the construction of biological networks, a stochastic approach for link weights has been recently developed by using experimental data and belief propagation on a factor graph. The link weights were variable nodes of the factor graph and determined from their marginal probability mass functions which were approximated by using an iterative scheme. However, there is no convergence analysis of the iterative scheme. In this paper, at first, we present a detailed explanation of the complicated multistep process step by step with a network of small size and artificial experimental data, and then we show a sufficient condition for the convergence of the iterative scheme. Numerical examples are given to illustrate the whole process and to verify our result.


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## Body

## 1. Introduction

Systems of mathematical equations have been used for modeling interactions of genes or proteins in biological systems. The modeling consists of two parts: one is to construct a network of nodes and links, where nodes usually represent genes or proteins and links denote interactions of nodes. The second is to build a set of mathematical formulas for governing the network dynamics, which might be expressed as a system of differential equations or Boolean logic functions [1, 2]. Modeling of a biological network has been focused on the construction of a directed network for a given biological system and its mathematical formulas, with which biological phenomena and mechanisms have been explained and controlled [3] and, furthermore, modeling of a biological network is a useful tool for understanding and explaining the distribution of the biological communities too [4]. However, there are few researches on how to determine an optimal set of nodes and links which are key components of networks for biological phenomena.There could be multiple networks for explaining a given biological system. So, it would be meaningful to use a probabilistic modeling approach which could yield multiple networks for the biological system. One approach was recently developed in [5, 6] for the construction of network models of a melanoma cell line (SK-MEL-133), the most serious skin cancer, in which each link weight was considered as a discrete random variable and its marginal probability mass function (PMF) was constructed by using a factor graph and a belief propagation (BP) algorithm. The marginal PMFs were used to choose multiple weights of high-probability values and then to construct multiple networks for the biological system. The initial step of the approach was to determine a prior knowledge network (PKN) for melanoma and its mathematical formula structure. Nodes were chosen for representing some drugs (drug nodes) for melanoma or proteins (measured nodes) which take important roles in melanoma. The drug nodes had outgoing links to each measured node without incoming links. Each measured node had an outgoing link to each other measured node and an incoming link from each other measured node. Each link weight in the mathematical formulas was not determined in the initial step. Next, each protein concentration was measured at steady state before and after treatment of the drugs, which produced experimental data for determining the link weights. Then, the joint PMF of link weights was defined as the Boltzmann/Gibbs form [7] for the factorization of the joint PMF. The form contained a cost function of the experimental data and the simulated values of measured nodes at steady state, where the simulated values were not constants for undetermined link weights. So, the factorization of the joint PMF directly led to the factor graph which is a bipartite graph consisting of variable nodes (link weights in the network) and factor nodes (factors in the factorization). Finally, the marginal PMFs were approximated with an iterative scheme constructed from belief (message) propagation on the factor graph.In general, computation time could be high to directly compute marginal distributions from a joint probability distribution. So, factor graphs and BP have been applied to infer marginals in diverse problems [8–12]. It is known that a marginal obtained from applying BP becomes the true marginal for an acyclic factor graph and BP could be successfully applied to a cyclic graph [13–18]. There have been many researches on conditions for the convergence of beliefs, which can be applied for computing marginals from a Gaussian joint probability density function (PDF) [19, 20]. However, the convergence conditions cannot be applied to the convergence of beliefs (messages) in [5, 6], since the joint random variable is discrete. And the authors [5, 6] did not analyze the convergence. As far as we know, there is no framework for the inference of marginal PMFs based on both BP and perturbation data except the approach in [5, 6]. For the use of the stochastic approach [5] in the construction of networks based on BP as well as perturbation data, a sufficient condition for the convergence of messages is necessary. Under such a sufficient condition marginal PMFs can be identified and, as a result, multiple network models for the given biological system can be constructed.In this paper, with our simple network, we present a detailed explanation of the long and intricate process of constructing a system of equations with messages from variable nodes to factor nodes, and vice versa for the approximation of marginal PMFs. Since approximate marginal PMFs are defined with the solution of the system of equations, we identify recursive relations for the solution of the system of equations and construct iterative schemes for solving the recursive relations. Finally we show a sufficient condition for the convergence of the schemes by using a Banach fixed-point theorem (see, e.g., [21, 22]). Numerical examples are given to illustrate the process in Section 4 and to verify our results.


## 2. Preliminaries

For a clear explanation of the convergence of the iterative schemes mentioned in the Introduction section, we make a simple network of three nodes and four links (see Figure1(a)), which is assumed to be a PKN. Since the simple network has key components necessary for determining approximate marginal PMFs, the simple network can be extended to any network, including the biological network in [5]. So, our convergence analysis in the Results section can be applied for any network. The network has two measured nodes (x1,x2) and one drug node (x3). We consider two treatments as the 1st and 2nd perturbations. Symbols xiυ(υ=1,2,i=1,2,3) in Figure 1(b) denote log2xiυ,after/xiυ,before, where xiυ,before and xiυ,after are the concentrations of xi at steady state before and after the υth perturbation, respectively. Since each drug node has an outgoing link to each measured node and no incoming link to any node, drug node x3 has two outgoing links to x1 and x2 with weights w1,3 and w2,3, respectively. Each measured node has an outgoing link to each other measured node and incoming links from each other nodes, and so, measured node x1 has one outgoing link to x2 with weight w2,1 and two incoming links from x2 and x3 with weights w1,2 and w1,3, respectively. Similarly, measured node x2 has one outgoing link to x1 with weight w1,2 and two incoming links from x1 and x3 with weights w2,1 and w2,3, respectively.Figure 1
Given information and our goal. (a) A prior knowledge network of two measured nodesx1 and x2, and one drug node x3. The symbol wi,j denotes the link weight from xi to xj, which is a discrete random variable. (b) Experimental data. The 1st and 2nd perturbations mean the 1st and 2nd experiments. Symbol xiυ(i=1,2,3,υ=1,2) denotes the log⁡2-ratio concentration of xi at steady state before and after the υth perturbation. (c) Differential equation model for the υth perturbation. Symbol xi,υ(t) denotes the relative change of xi at time t in the υth perturbation. (d) Construction of approximate marginal PMFs from a joint PMF. Set w∣w=-1,0,1 denotes the domain of each marginal PMF.
(a) (b) (c) (d)As in [5], the dynamics of the given situation is modeled with the mathematical formulas as in Figure 1(c). Simulated values xiυ,s at steady state under the υth perturbation are assumed to be the form(1)xiυ,s=ϕ∑j=1,j≠i3wi,jxjυi=1,2,x3υ,s=x3υ,where ϕ(x)=tanh⁡x. From now on, weights wi,j are considered as discrete random variables with three values w (-1, 0 and 1) and their PMFs are denoted as follows: for w∈-1,0,1(2)Pw1,2=w,Pw1,3=w,Pw2,1=w,Pw2,3=w,which are approximately calculated in the Results section by using a factor graph and BP based on the factor graph. In [3], discretization over the 3 weight values led to a high rate of convergence and did not capture appropriate uncertainty in a probability distribution and then 11 weight values were used. However, our goals in this paper are to find a convergence condition for messages and to explain the complicated multistep process in [3] step by step by using a prior knowledge network of small size. So, the three weight values are enough for our goals. For the extension of our results to networks of large size and real perturbation data, some notations and mathematical expressions used in our paper might be rather cumbersome in later sections.


## 3. Results

In this section, we present a system of equations for each marginal PMF, iterative schemes for the solutions and a sufficient condition for the convergence of the iterative schemes.


### 3.1. System of Equations for Marginal PMFs

To define a joint PMF of link weights based on the experimental data we consider the sum of squared errors between simulated valuesxiυ,s and experimental values xiυ(3)∑i=13∑υ=12xiυ,s-xiυ2as well as the penalty(4)∑i=13∑j=1,j≠i3δwi,j,δwi,j=w=0w=01w≠0,which reflects the property that the total number of links tends not to be large. Note that w3,j=0(j=1,2) due to no incoming link to drug node x3. Then the cost function is defined as(5)Cost=β∑i=13∑υ=12xiυ,s-xiυ2+λ∑i=12∑j=1,j≠i3δwi,j,where the error and penalty are weighted by positive constants β and λ, respectively. Using (1), the cost is written as(6)Cost=β∑i=12∑υ=12ϕ∑j=1,j≠i3wi,jxjυ-xiυ2+λ∑i=12∑j=1,j≠i3δwi,j.Posteriori probability (joint PMF) PW=Pw1,2,w1,3,w2,1,w2,3 is defined as(7)PW=1Zexp⁡-Cost,where Z is the constant ensuring that the sum of the probabilities over all model configurations is equal to one. Let Z=Z1Z2, where Z1 and Z2 are the constants ensuring that the sums of probabilities Pw1,2,w1,3 and Pw2,1,w2,3 over all model configurations are equal to one, respectively. Substituting the cost (6) into PW gives (8)PW=1Zexp⁡∑i=12-β∑υ=12ϕ∑j=1,j≠i3wi,jxjυ-xiυ2-λ∑j=1,j≠i3δwi,j=∏i=121Ziexp⁡-λ∑j=1,j≠i3δwi,j∏υ=12exp⁡-βϕ∑j=1,j≠i3wi,jxjυ-xiυ2≡Pw1,2,w1,3Pw2,1,w2,3,which implies that Pw1,2,w1,3 and Pw2,1,w2,3 can be independently determined by using a similar approach. So, we consider Pw1,2,w1,3 to show the approximation of marginal PMF Pw1,2:(9)Pw1,2=∑w1,3Pw1,2,w1,3=1Z1∑w1,3e-λ∑j=23δw1,j∏υ=12e-βϕ∑j=23w1,jxjυ-x1υ2.It is not efficient to calculate the exact marginal with the enumeration in the cases where the numbers of nodes in a network or experimental perturbations become large. Therefore, the process to approximate the marginal is presented step by step by using a factor graph and BP on the factor graph.Step 1 (introduction of a factor graph and BP on the factor graph).
Using the factorization in (9), factor nodes F1υ(υ=1,2) are defined as(10)F1υw1,2,w1,3=exp⁡-βϕ∑j=23w1,jxjυ-x1υ2and then marginal P(w1,2) in (9) can be written as follows:(11)Pw1,2=1Z1∑w1,3e-λ∑j=23δw1,j∏υ=12F1υw1,2,w1,3,which yields the factor graph of two variable nodes (w1,2,w1,3) and two factor nodes (F11,F12) as in Figure 2. Following BP on the factor graph, message Pυ(w1,j) from variable node w1,j to factor node F1υ(w1,2,w1,3) is defined as (12)Pυw1,j=1Z1,jυe-λδw1,j∏μ=1,μ≠υ2ρμw1,jυ=1,2,j=2,3,where Z1,jυ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one and ρμ(w1,j) is the message from F1μ(w1,2,w1,3) to w1,j defined as(13)ρμw1,j=∑w1,2,w1,3-w1,jF1μw1,2,w1,3∏2≤l≤3,l≠j3Pμw1,l.Using message ρυw1,j, marginal PMF P(w1,j) in (11) can be approximated as(14)Pw1,j=1Z1,je-λδw1,j∏υ=12ρυw1,j,where Z1,j is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one.Figure 2
Two representations of the factor graph: bipartite graph (a) and cyclic graph (b). The factor graph contains a cycle. The connection of factor nodeF11 to variable nodes w1,2 and w1,3 denotes the dependence of F11 on w1,2 and w1,3 since F11 is a function of w1,2 and w1,3.
(a) (b)Step 2 (approximation of the summation (13)).
The process of the approximation used in [5] can be divided into two parts: the first is to change multiple summations into a single summation by introducing a new random variable. The next is to change the single summation into an integral.
Step 2-1. Since ρυ in (13) is a function of w1,j, all random variables in F1υw1,2,…,w1,n, which are in (13), can be divided into two type of random variables: one is w1,j and the other is s1,jυ=∑ξ≠1,jnw1,ξxξυ(n=3), which becomes the linear combination of random variables w1,ξ in the cases where the number n of nodes becomes large. Using the definitions of s1,jυ and F1υw1,2,…,w1,n in (10), we can write F1υ in (13) as(15)F1υs1,jυ,w1,j=exp⁡-βϕs1,jυ+w1,jxjυ-x1υ2,which is a function of random variables s1,jυ and w1,j. Then (13) becomes(16)ρυw1,j=∑w1,ξ1<ξ≤n,ξ≠jF1υs1,jυ,w1,j∏2≤l≤n,l≠jPυw1,l.Letting(17)∑1<ξ≤n,ξ≠jw~1,ξxξυ∣w~1,ξ∈-1,0,1,1<ξ≤n,ξ≠j=s~1,j,kυ∣1≤k≤mfor some positive integer m, message ρυ in (16) becomes(18)ρυw1,j=∑k=1m∑w1,ξ=w~1,ξ1<ξ≤n,ξ≠j∑1<ξ≤n,ξ≠jw~1,ξxξυ=s~1,j,kυF1υs1,jυ,w1,j∏2≤l≤n,l≠jPυw1,l=∑k=1mF1υs~1,j,kυ,w1,j∑w1,ξ=w~1,ξ1<ξ≤n,ξ≠j∑1<ξ≤n,ξ≠jw~1,ξxξυ=s~1,j,kυ∏2≤l≤n,l≠jPυw1,l.Note that(19)∑k=1m∑w1,ξ=w~1,ξ1<ξ≤n,ξ≠j∑1<ξ≤n,ξ≠jw~1,ξxξυ=s~1,j,kυ∏2≤l≤n,l≠jPυw1,l=∑w1,ξ1<ξ≤n,ξ≠j∏2≤l≤n,l≠jPυw1,l=∏2≤l≤n,l≠j∑w1,lPυw1,l=1,which implies that the following can be a PMF of s1,jυ(20)Psυs1,jυ=s~1,j,kυ=∑w1,ξ=w~1,ξ1<ξ≤n,ξ≠j∑1<ξ≤n,ξ≠jw~1,ξxξυ=s~1,j,kυ∏2≤l≤n,l≠jPυw1,l1≤k≤m.So, message ρυ in (18) can be written as a single summation(21)ρυw1,j=∑s1,jυF1υs1,jυ,w1,jPsυs1,jυ.
Step 2-2. Note that s1,jυ is a sum of random variables w1,ξ(1<ξ≤n,ξ≠j), which are assumed to be independent. Even though s1,jυ is not identically distributed, the authors [5] invoked the central limit theorem to approximate the PMF of s1,jυ as a Gaussian with reference to [23], where there was no explicit justification for the application of this theorem. Since sums of independent random variables converge in distribution to the standard normal as long as some condition (e.g., the Lindeberg Condition [24]) is satisfied, we think that such a condition might be implicitly assumed in [24]. So, the approximate PMF of s1,jυ becomes (22)Psυs1,jυ=12πΔ1,jυexp⁡-s1,jυ-s1,jυ¯22Δ1,jυ,where average s1,jυ¯ and variance Δ1,jυ of s1,jυ are defined as(23)s1,jυ¯=Es1,jυ=E∑l≠1,j3w1,lxlυ=∑l≠1,j3w1,l¯xlυ,(24)Δ1,jυ=Vs1,jυ=V∑l≠1,j3w1,lxlυ=∑l≠1,j3Vw1,lxlυ2=∑l≠1,j3w1,l2¯-w1,l¯2xlυ2.Symbols w1,l¯ and w1,l2¯ denote the averages of w1,l and (w1,l)2, respectively:(25)w1,l¯=∑wwPυw1,l=w,w1,l2¯=∑ww2Pυw1,l=wand then the sum over configurations w1,l∣2≤l≤3,l≠j in (19) is approximated with a Gaussian integration:(26)ρυw1,j≈∫-∞∞F1υs1,jυ,w1,jPsυs1,jυds1,jυ.Step 3 (approximation of the improper integral (26)).
For the calculation of the improper integral (26), error ϕs1,jυ+w1,jxjυ-xjυ is linearized with respect to the maximization of the fitness in (15). Note that the equality(27)ϕs1,jυ+w1,jxjυ-x1υ=0,can be written as(28)ϕ-1x1υ-w1,jxjυ-s1,jυ=0,under the assumption that(29)experimental  datax1υare  contained  in  the  codomain  ofϕ.Then error ϕs1,jυ+w1,jxjυ-x1υ in (15) is approximated by(30)ϕ-1x1υ-w1,jxjυ-s1,jυ,which yields (31)F1υs1,jυ,w1,j=exp-βϕs1,jυ+w1,jxjυ-x1υ2≈exp⁡-βϕ-1x1υ-w1,jxjυ-s1,jυ2.Hence we have (32)ρυw1,j≈∫-∞∞F1υs1,jυ,w1,jPsυs1,jυds1,jυ≈∫-∞∞exp⁡-βϕ-1x1υ-w1,jxjυ-s1,jυ2·12πΔ1,jυexp⁡-s1,jυ-s1,jυ¯22Δ1,jυds1,jυ=11+2βΔ1,jυ1/2exp⁡-βϕ-1x1υ-w1,jxjυ-s1,jυ¯21+2βΔ1,jυ,where the last equality is obtained both by the identity (33)-βϕ-1x1υ-w1,jxjυ-s1,jυ2-s1,jυ-s1,jυ¯22Δ1,jυ=-121+2βΔ1,jυs1,jυ-s1,jυ¯+βϕ-1x1υ-w1,jxjυ2Δ1,jυ/1+2βΔ1,jυ2Δ1,jυ-βϕ-1x1υ-w1,jxjυ-s1,jυ¯21+2βΔ1,jυand the property of a PDF (the integral of a PDF over its domain is equal to 1). Therefore adopting the final approximation of ρυ in [5], approximate marginal PMF P(w1,j=w) in (14) can be obtained by solving the system of equations(34)ρυw1,j=w=exp⁡-βϕ-1x1υ-wxjυ-s1,jυ¯21+2βΔ1,jυ,(35)Pυw1,j=w=1Z1,jυe-λδw1,j=wρ3-υw1,j=w,where υ=1,2, j=2,3, w=-1,0,1, s1,jυ¯ and Δ1,jυ are in (23)–(25).Remark 1.
Since the approximation (34) was used in [5] and our goal in this paper is to find a convergence condition for the algorithm in [5], we also use (34) instead of(36)ρυw1,j=11+2βΔ1,jυ1/2exp⁡-βϕ-1x1υ-w1,jxjυ-s1,jυ¯21+2βΔ1,jυ.Even when using (36) instead of (34), the convergence analysis in the later subsections could be applied (we do not show it in this paper).Remark 2.
Similarly approximate marginal PMFP(w2,j) can be obtained by replacing subscripts 1 and j=2,3, related with node x1, in w1,j, s1,jυ¯, Δ1,jυ, and w1,l in (34), (35), and (23)–(25) with 2 and j=1,3, related with node x2: for υ=1,2, j=1,3:(37)Pw2,j=1Z2,je-λδw2,j∏υ=12ρυw2,j,(38)ρυw2,j=exp⁡-βϕ-1x2υ-w2,jxjυ-s2,jυ¯21+2βΔ2,jυ,(39)Pυw2,j=1Z2,jυe-λδw2,jρ3-υw2,j,where Z2,jυ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one and (40)s2,jυ¯=∑l≠2,j3∑wwPυw2,l=wxlυ,(41)Δ2,jυ=∑l≠2,j3∑ww2Pυw2,l=w-∑wwPυw2,l=w2xlυ2.


### 3.2. Iterative Schemes for Solving the System of Equations

In this subsection, we construct sequences{ρ1,j,nυw} and {P1,j,nυ(w)}(υ=1,2,j=2,3,w=-1,0,1) by using (34) and (35). In the next subsection we present a sufficient condition for the convergence of the two sequences, which implies that the two limits satisfy (34) and (35). So, the limit of {ρ1,j,nυw} becomes value ρυw1,j=w, leading to the construction of approximate marginal PMF P(w1,j) in (14). We assume that(42)initial  termsρ1,j,0υwofρ1,j,nυware  given  as  positive  numbersand initial terms of {P1,j,nυw} are defined as(43)P1,j,0υw=1Z1,j,0υe-λδwρ1,j,03-υw,where Z1,j,0υ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one. The 1st iterations ρ1,j,1υw and P1,j,1υw are defined similarly to (34) and (35) and so we need to define s1,j,0υ¯ and Δ1,j,0υ by using (23) and (24): (44)s1,j,0υ¯=∑l≠1,j3w1,l,0¯xlυ=∑l≠1,j3∑w∈-1,0,1wP1,l,0υwxlυ=∑l≠1,j3-P1,l,0υ-1+P1,l,0υ1xlυ=-P1,3,0υ-1+P1,3,0υ1x3υj=2-P1,2,0υ-1+P1,2,0υ1x2υj=3=e-λZ1,5-j,0υ-ρ1,5-j,03-υ-1+ρ1,5-j,03-υ1x5-jυ,and (45)Δ1,j,0υ=∑l≠1,j3w1,l,02¯-w1,l,0¯2xlυ2=∑l≠1,j3∑ww2P1,l,0υw-∑wwP1,l,0υw2xlυ2=∑l≠1,j3P1,l,0υ-1+P1,l,0υ1--P1,l,0υ-1+P1,l,0υ12xlυ2=e-λZ1,5-j,0υρ1,5-j,03-υ-1+ρ1,5-j,03-υ1-e-λZ1,5-j,0υ-ρ1,5-j,03-υ-1+ρ1,5-j,03-υ12x5-jυ2.So, the 1st iteration is defined as(46)ρ1,j,1υw=exp⁡-βϕ-1x1υ-wxjυ-s1,j,0υ¯21+2βΔ1,j,0υ,P1,j,1υw=1Z1,j,1υe-λδwρ1,j,13-υw,where Z1,j,1υ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one. Similarly the (n+1)th iteration is defined as(47)ρ1,j,n+1υw=Φ1,j,wυρ1,5-j,n3-υ-1,ρ1,5-j,n3-υ1n≥0,(48)P1,j,n+1υw=1Z1,j,n+1υe-λδwρ1,j,n+13-υw,for υ=1,2, j=2,3 and w=-1,0,1. Here Z1,j,n+1υ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one and(49)Φ1,j,wυρ1,5-j,n3-υ-1,ρ1,5-j,n3-υ1=exp⁡-βϕ-1x1υ-wxjυ-e-λ/Z1,5-j,nυ-ρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ1x5-jυ21+2βe-λ/Z1,5-j,nυρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ1-e-λ/Z1,5-j,nυ-ρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ12x5-jυ2.Therefore the desired iterative schemes consist of (42), (43), (47), (48), and (49) under the assumption (29).Remark 3.
SinceP1,j,n+1υw1,j defined in (48) is a PMF due to (47) and (49), we have(50)0<e-λZ1,5-j,nυρ1,5-j,n3-υ-1<1,0<e-λZ1,5-j,nυρ1,5-j,n3-υ1<1,which gives the positivity of the following term in (49)(51)ρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ1-e-λZ1,5-j,nυ-ρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ12>0.So, we obtain a lower bound of ρ1,j,n+1υw(52)ρ1,j,n+1υw≥exp⁡-βϕ-1x1υ+xjυ+2x5-jυ2.Therefore we obtain an upper bound(53)e-λZ1,j,nυ=e-λe-λρ1,j,nυ-1+ρ1,j,nυ0+e-λρ1,j,nυ1≤e-λUj,β,λυ,Uj,β,λυ=11+2e-λexpβϕ-1x1υ+xjυ+2x5-jυ2,which is used in the proof of Lemma 8.Remark 4.
Note that sequence{ρ1,j,nυw} in the recursive relation (47) contains no P1,j,nυ(w). Then, in the next subsection, we present a sufficient condition for the convergence of sequence {ρ1,j,nυw} without using {P1,j,nυ(w)} and, as a result, the limit of {ρ1,j,nυw} is used to define the message ρυw1,j=w.Remark 5.
Similarly, to define messagesρυw2,j=w(υ=1,2,j=1,3,w=-1,0,1), we use the following sequences:(54)ρ2,j,n+1υw=Φ2,j,wυρ2,4-j,n3-υ-1,ρ2,4-j,n3-υ1n≥0,(55)P2,j,n+1υw=1Z2,j,n+1υe-λδwρ2,j,n+13-υw,where Z2,j,n+1υ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one and(56)Φ2,j,wυρ2,4-j,n3-υ-1,ρ2,4-j,n3-υ1=exp⁡-βϕ-1x2υ-wxjυ-e-λ/Z2,4-j,nυ-ρ2,4-j,n3-υ-1+ρ2,4-j,n3-υ1x4-jυ21+2βe-λ/Z2,4-j,nυρ2,4-j,n3-υ-1+ρ2,4-j,n3-υ1-e-λ/Z2,4-j,nυ-ρ2,4-j,n3-υ-1+ρ2,4-j,n3-υ12x4-jυ2.


### 3.3. A Sufficient Condition for the Convergence of the Iterative Schemes

Let(57)X1n=X1,1n,X1,2n,X1,3n,X1,4n,X1,5n,X1,6n,X1,7n,X1,8n,X1,9n,X1,10n,X1,11n,X1,12n=ρ1,2,n1-1,ρ1,2,n10,ρ1,2,n11,ρ1,3,n1-1,ρ1,3,n10,ρ1,3,n11,ρ1,2,n2-1,ρ1,2,n20,ρ1,2,n21,ρ1,3,n2-1,ρ1,3,n20,ρ1,3,n21∈R12and(58)Φ1=Φ1,1,Φ1,2,Φ1,3,Φ1,4,Φ1,5,Φ1,6,Φ1,7,Φ1,8,Φ1,9,Φ1,10,Φ1,11,Φ1,12=Φ1,2,-11,Φ1,2,01,Φ1,2,11,Φ1,3,-11,Φ1,3,01,Φ1,3,11,Φ1,2,-12,Φ1,2,02,Φ1,2,12,Φ1,3,-12,Φ1,3,02,Φ1,3,12,where the subscript 1 of X1 and Φ1 represents node x1. The (n+1)th iteration in (47) is written as(59)X1n+1=Φ1X1n.It follows from (47), (57), and (58) that Φ1,θ(1≤θ≤12) are functions of two independent variables: for X∈R12(60)Φ1X=Φ1,1X10,X12,Φ1,2X10,X12,Φ1,3X10,X12,Φ1,4X7,X9,Φ1,5X7,X9,Φ1,6X7,X9,Φ1,7X4,X6,Φ1,8X4,X6,Φ1,9X4,X6,Φ1,10X1,X3,Φ1,11X1,X3,Φ1,12X1,X3.We use Banach fixed-point theorem [21] for the convergence of the sequence (59) to prove Theorem 9, which is our main result.Theorem 6.
LetD be a closed subset of Rm for a positive integer m. If a function Ψ:D→D satisfies that for a constant k∈(0,1) and all x,y in D(61)Ψx-Ψy≤kx-y,then there exists a unique fixed point x∗∈D such that Ψ(x∗)=x∗, which is the limit of sequence x(n+1)=Ψ(x(n))(n≥0) for any x(0)∈D.For the application of Theorem6 to the proof of our main result we need the following lemmas.Lemma 7.
Assume that experimental datax1υ are contained in the codomain of ϕ. Assume that positive constants β and λ satisfy inequalities 1-2βe-λUj,β,λυ2xjυ2>0 for υ=1,2,j=2,3 and Uj,β,λυ defined in (53). Then Φ1 defined in (49) and (57)–(60) becomes a function from domain [0,1]12 to codomain [0,1]12.Proof.
It follows from (49) and (57)–(60) that for (θ,w)=(1,-1),(2,0),(3,1)(62)Φ1,θX=Φ1,2,w1X10,X12,X=X1,…,X12∈0,112and(63)Φ1,2,w1X10,X12=exp⁡-βϕ-1x11-wx21-e-λ/Z1,3,n1-X10+X12x3121+2βe-λ/Z1,3,n1X10+X12-e-λ/Z1,3,n1-X10+X122x312.We have that for β>0, λ>0 and {X10,X12}⊂[0,1](64)1+2βe-λZ1,3,n1X10+X12-e-λZ1,3,n1-X10+X122x312≥1-2βe-λU3,β,λ12x312>0,and for β>0(65)-βϕ-1x11-wx21-e-λZ1,3,n1-X10+X12x312≤0,and so, we obtain(66)0<Φ1,θX≤11≤θ≤3.Similarly, under the same condition, we can obtain(67)0<Φ1,θX≤14≤θ≤12,which are desired results.Lemma 8.
Assume that experimental datax1υ are contained in the codomain of ϕ. Assume that positive constants β and λ satisfy inequalities 1-2βe-λUj,β,λυ2xjυ2>0 for υ=1,2,j=2,3 and Uj,β,λυ defined in (53). Then for Φ1 defined in (49) and (57)–(60) and {X,Y}⊂[0,1]12(68)Φ1X-Φ1Y≤6max⁡Cj,β,λυ∣υ=1,2,j=2,3βe-λX-Y,where(69)Cj,β,λυ=2ϕ-1x1υ+xjυ+e-λU5-j,β,λυx5-jυU5-j,β,λυx5-jυ×1+2βe-λU5-j,β,λυ2+e-λU5-j,β,λυx5-jυ2+ϕ-1x1υ+xjυ+e-λU5-j,β,λυx5-jυ2×2βU5-j,β,λυ1+2e-λU5-j,β,λυx5-jυ2×1-2βe-λU5-j,β,λυ2x5-jυ2-2.Proof.
Using (62) and (63), we have that for {X,Y}⊂[0,1]12(70)Φ1,1X-Φ1,1Y=Φ1,2,-11X10,X12-Φ1,2,-11Y10,Y12(71)=exp⁡fX10,X12-exp⁡fY10,Y12,where X10,X12,Y10,Y12⊂[0,1] and function f:[0,1]2→[0,1]2 is defined as follows: for ζ,η∈[0,1](72)fζ,η=-βϕ-1x11--1x21-e-λ/Z1,3,n1-ζ+ηx3121+2βe-λ/Z1,3,n1ζ+η-e-λ/Z1,3,n1-ζ+η2x312.Due to the mean value theorem there exists a constant c in (0,1) such that(73)exp⁡fX10,X12-exp⁡fY10,Y12=∇exp⁡f1-cX10,X12+cY10,Y12·X10-Y10,X12-Y12.Note that(74)∂∂ζexp⁡fζ,η≤∂∂ζfζ,η,where(75)∂∂ζfζ,η=-β2ϕ-1x11+x21-e-λ/Z1,3,n1-ζ+ηx31e-λ/Z1,3,n1x31×1+2βe-λ/Z1,3,n1ζ+η-e-λ/Z1,3,n1-ζ+η2x312-ϕ-1x11+x21-e-λ/Z1,3,n1-ζ+ηx312×2βe-λ/Z1,3,n11-2e-λ/Z1,3,n1-ζ+η-1x3121+2βe-λ/Z1,3,n1ζ+η-e-λ/Z1,3,n1-ζ+η2x3122.Using {ζ,η}⊂[0,1], we have(76)∂∂ζfζ,η≤β2ϕ-1x11+x21+e-λ/Z1,3,n1x31e-λ/Z1,3,n1x31×1+2βe-λ/Z1,3,n12+e-λ/Z1,3,n1x312+ϕ-1x11+x21+e-λ/Z1,3,n1x312×2βe-λ/Z1,3,n11+2e-λ/Z1,3,n1x3121+2βe-λ/Z1,3,n1ζ+η-e-λ/Z1,3,n1-ζ+η2x3122.Using the given condition in this lemma and (53), we have the positive lower bound(77)1+2βe-λZ1,3,n1ζ+η-e-λZ1,3,n1-ζ+η2x312≥1-2βe-λU3,β,λ12x312>0,which gives that(78)∂∂ζfζ,η≤βe-λ2ϕ-1x11+x21+e-λU3,β,λ1x31U3,β,λ1x31×1+2βe-λU3,β,λ12+e-λU3,β,λ1x312+ϕ-1x11+x21+e-λU3,β,λ1x312×2βU3,β,λ11+2e-λU3,β,λ1x3121-2βe-λU3,β,λ12x3122=βe-λC2,β,λ1.Similarly,(79)∂∂ηfζ,η≤βe-λC2,β,λ1.Substituting (73), (74), (78), and (79) into (70), we have(80)Φ1,1X-Φ1,1Y=Φ1,2,-11X-Φ1,2,-11Y=exp⁡fX10,X12-exp⁡fY10,Y12≤C2,β,λ1βe-λX10-Y10+X12-Y12.Following the calculations for obtaining (80), we can obtain that for the 1st perturbation(81)Φ1,θX-Φ1,θY≤C2,β,λ1βe-λX10-Y10+X12-Y12θ=2,3,Φ1,θX-Φ1,θY≤C3,β,λ1βe-λX7-Y7+X9-Y9θ=4,5,6,where C3,β,λ1 is defined in Lemma 8.
Similarly we can have that for the2nd perturbation(82)Φ1,θX-Φ1,θY≤C2,β,λ2βe-λX4-Y4+X6-Y6θ=7,8,9,Φ1,θX-Φ1,θY≤C3,β,λ2βe-λX1-Y1+X3-Y3θ=10,11,12,where Cj,β,λ2(j=2,3) are defined in Lemma 8. Therefore, it follows from (80)–(82) that (83)Φ1X-Φ1Y2=∑θ=112Φ1,θX-Φ1,θY2≤6maxυ=1,2j=2,3Cj,β,λυ2β2e-2λX-Y2,which gives the desired result.Using Theorem6 and Lemmas 7 and 8, we can obtain our main result.Theorem 9.
Assume that the experimental datax1υ(υ=1,2) are contained in the codomain of ϕ. Suppose that positive constants β and λ satisfy(84)1-2βe-λUj,β,λ12xjυ2>0and(85)6max⁡Cj,β,λυ∣υ=1,2,j=2,3βe-λ<1,for υ=1,2,j=2,3, Uj,β,λυ defined in (53) and Cj,β,λυ defined in Lemma 8. Then sequence X1n+1=Φ1X1n(n≥0) converges for any X1(0)∈[0,1]12.Proof.
LetD be the closed subset [0,1]12 of R12 and(86)k=6max⁡Cj,β,λυ∣υ=1,2,j=2,3βe-λ.Then Lemmas 7 and 8 give that Φ1:D→D and(87)Φ1X-Φ1Y≤kX-Y.Therefore we can apply Theorem 6 and so we obtain a unique fixed point X1∗∈D such that Φ1(X1∗)=X1∗, which implies the convergence of sequence X1n+1=Φ1X1n(n≥0) for any X1(0)∈[0,1]12 with limit X1∗.Remark 10.
We define(88)ρ1w1,2=-1,ρ1w1,2=0,ρ1w1,2=1,ρ1w1,3=-1,ρ1w1,3=0,ρ1w1,3=1,ρ2w1,2=-1,ρ2w1,2=0,ρ2w1,2=1,ρ2w1,3=-1,ρ2w1,3=0,ρ2w1,3=1=ρ1,2,-11,ρ1,2,01,ρ1,2,11,ρ1,3,-11,ρ1,3,01,ρ1,3,11,ρ1,2,-12,ρ1,2,02,ρ1,2,12,ρ1,3,-12,ρ1,3,02,ρ1,3,12,where the vector in the right side denotes limit X1∗ in Theorem 9. Since X1n in (57)–(59) converges, {P1,j,nυw} in (48) also converges to limit e-λδwρ1,j,w3-υ. Therefore, ρυw1,j in (88) and Pυw1,j=e-λδw1,jρ1,j,w1,j3-υ satisfy the system of equations in (34) and (35), which give approximate marginal PMFs Pw1,j=1/Z1,je-λδw1,j∏υ=12ρυw1,j in (14) as follows: (89)Pw1,j=w=1Z1,je-λδw1,j=w∏υ=12ρ1,j,wυj=2,3,w=-1,0,1.Remark 11.
When using the sequences in (54)–(56) instead of the sequences in (47)–(49), we can obtain the limits of the sequences in (54)–(56) under conditions similar to those in Theorem 9: assume that experimental data x2υ(υ=1,2) are contained in the codomain of ϕ. Let Uj,β,λυ=1/1+2e-λexp⁡βϕ-1(x2υ)+xjυ+2x4-jυ2(j=1,3). Suppose that positive constants β and λ satisfy that 1-2βe-λUj,β,λ12xjυ2>0 and(90)6max⁡Cj,β,λυ∣υ=1,2,j=1,3βe-λ<1,where(91)Cj,β,λυ=2ϕ-1x2υ+xjυ+e-λU4-j,β,λυx4-jυU4-j,β,λυx4-jυ×1+2βe-λU4-j,β,λυ2+e-λU4-j,β,λυx4-jυ2+ϕ-1x2υ+xjυ+e-λU4-j,β,λυx4-jυ2×2βU4-j,β,λυ1+2e-λU4-j,β,λυx4-jυ2×1-2βe-λU4-j,β,λ12x4-jυ2-2.We can define the messages as in Remark 4(92)ρ1w2,1=-1,ρ1w2,1=0,ρ1w2,1=1,ρ1w2,3=-1,ρ1w2,3=0,ρ1w2,3=1,ρ2w2,1=-1,ρ2w2,1=0,ρ2w2,1=1,ρ2w2,3=-1,ρ2w2,3=0,ρ2w2,3=1=ρ2,1,-11,ρ2,1,01,ρ2,1,11,ρ2,3,-11,ρ2,3,01,ρ2,3,11,ρ2,1,-12,ρ2,1,02,ρ2,1,12,ρ2,3,-12,ρ2,3,02,ρ2,3,12and then obtain the approximate marginal PMFs(93)Pw2,j=w=1Z2,je-λδw2,j=w∏υ=12ρ2,j,wυj=1,3,w=-1,0,1.


## 3.1. System of Equations for Marginal PMFs

To define a joint PMF of link weights based on the experimental data we consider the sum of squared errors between simulated valuesxiυ,s and experimental values xiυ(3)∑i=13∑υ=12xiυ,s-xiυ2as well as the penalty(4)∑i=13∑j=1,j≠i3δwi,j,δwi,j=w=0w=01w≠0,which reflects the property that the total number of links tends not to be large. Note that w3,j=0(j=1,2) due to no incoming link to drug node x3. Then the cost function is defined as(5)Cost=β∑i=13∑υ=12xiυ,s-xiυ2+λ∑i=12∑j=1,j≠i3δwi,j,where the error and penalty are weighted by positive constants β and λ, respectively. Using (1), the cost is written as(6)Cost=β∑i=12∑υ=12ϕ∑j=1,j≠i3wi,jxjυ-xiυ2+λ∑i=12∑j=1,j≠i3δwi,j.Posteriori probability (joint PMF) PW=Pw1,2,w1,3,w2,1,w2,3 is defined as(7)PW=1Zexp⁡-Cost,where Z is the constant ensuring that the sum of the probabilities over all model configurations is equal to one. Let Z=Z1Z2, where Z1 and Z2 are the constants ensuring that the sums of probabilities Pw1,2,w1,3 and Pw2,1,w2,3 over all model configurations are equal to one, respectively. Substituting the cost (6) into PW gives (8)PW=1Zexp⁡∑i=12-β∑υ=12ϕ∑j=1,j≠i3wi,jxjυ-xiυ2-λ∑j=1,j≠i3δwi,j=∏i=121Ziexp⁡-λ∑j=1,j≠i3δwi,j∏υ=12exp⁡-βϕ∑j=1,j≠i3wi,jxjυ-xiυ2≡Pw1,2,w1,3Pw2,1,w2,3,which implies that Pw1,2,w1,3 and Pw2,1,w2,3 can be independently determined by using a similar approach. So, we consider Pw1,2,w1,3 to show the approximation of marginal PMF Pw1,2:(9)Pw1,2=∑w1,3Pw1,2,w1,3=1Z1∑w1,3e-λ∑j=23δw1,j∏υ=12e-βϕ∑j=23w1,jxjυ-x1υ2.It is not efficient to calculate the exact marginal with the enumeration in the cases where the numbers of nodes in a network or experimental perturbations become large. Therefore, the process to approximate the marginal is presented step by step by using a factor graph and BP on the factor graph.Step 1 (introduction of a factor graph and BP on the factor graph).
Using the factorization in (9), factor nodes F1υ(υ=1,2) are defined as(10)F1υw1,2,w1,3=exp⁡-βϕ∑j=23w1,jxjυ-x1υ2and then marginal P(w1,2) in (9) can be written as follows:(11)Pw1,2=1Z1∑w1,3e-λ∑j=23δw1,j∏υ=12F1υw1,2,w1,3,which yields the factor graph of two variable nodes (w1,2,w1,3) and two factor nodes (F11,F12) as in Figure 2. Following BP on the factor graph, message Pυ(w1,j) from variable node w1,j to factor node F1υ(w1,2,w1,3) is defined as (12)Pυw1,j=1Z1,jυe-λδw1,j∏μ=1,μ≠υ2ρμw1,jυ=1,2,j=2,3,where Z1,jυ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one and ρμ(w1,j) is the message from F1μ(w1,2,w1,3) to w1,j defined as(13)ρμw1,j=∑w1,2,w1,3-w1,jF1μw1,2,w1,3∏2≤l≤3,l≠j3Pμw1,l.Using message ρυw1,j, marginal PMF P(w1,j) in (11) can be approximated as(14)Pw1,j=1Z1,je-λδw1,j∏υ=12ρυw1,j,where Z1,j is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one.Figure 2
Two representations of the factor graph: bipartite graph (a) and cyclic graph (b). The factor graph contains a cycle. The connection of factor nodeF11 to variable nodes w1,2 and w1,3 denotes the dependence of F11 on w1,2 and w1,3 since F11 is a function of w1,2 and w1,3.
(a) (b)Step 2 (approximation of the summation (13)).
The process of the approximation used in [5] can be divided into two parts: the first is to change multiple summations into a single summation by introducing a new random variable. The next is to change the single summation into an integral.
Step 2-1. Since ρυ in (13) is a function of w1,j, all random variables in F1υw1,2,…,w1,n, which are in (13), can be divided into two type of random variables: one is w1,j and the other is s1,jυ=∑ξ≠1,jnw1,ξxξυ(n=3), which becomes the linear combination of random variables w1,ξ in the cases where the number n of nodes becomes large. Using the definitions of s1,jυ and F1υw1,2,…,w1,n in (10), we can write F1υ in (13) as(15)F1υs1,jυ,w1,j=exp⁡-βϕs1,jυ+w1,jxjυ-x1υ2,which is a function of random variables s1,jυ and w1,j. Then (13) becomes(16)ρυw1,j=∑w1,ξ1<ξ≤n,ξ≠jF1υs1,jυ,w1,j∏2≤l≤n,l≠jPυw1,l.Letting(17)∑1<ξ≤n,ξ≠jw~1,ξxξυ∣w~1,ξ∈-1,0,1,1<ξ≤n,ξ≠j=s~1,j,kυ∣1≤k≤mfor some positive integer m, message ρυ in (16) becomes(18)ρυw1,j=∑k=1m∑w1,ξ=w~1,ξ1<ξ≤n,ξ≠j∑1<ξ≤n,ξ≠jw~1,ξxξυ=s~1,j,kυF1υs1,jυ,w1,j∏2≤l≤n,l≠jPυw1,l=∑k=1mF1υs~1,j,kυ,w1,j∑w1,ξ=w~1,ξ1<ξ≤n,ξ≠j∑1<ξ≤n,ξ≠jw~1,ξxξυ=s~1,j,kυ∏2≤l≤n,l≠jPυw1,l.Note that(19)∑k=1m∑w1,ξ=w~1,ξ1<ξ≤n,ξ≠j∑1<ξ≤n,ξ≠jw~1,ξxξυ=s~1,j,kυ∏2≤l≤n,l≠jPυw1,l=∑w1,ξ1<ξ≤n,ξ≠j∏2≤l≤n,l≠jPυw1,l=∏2≤l≤n,l≠j∑w1,lPυw1,l=1,which implies that the following can be a PMF of s1,jυ(20)Psυs1,jυ=s~1,j,kυ=∑w1,ξ=w~1,ξ1<ξ≤n,ξ≠j∑1<ξ≤n,ξ≠jw~1,ξxξυ=s~1,j,kυ∏2≤l≤n,l≠jPυw1,l1≤k≤m.So, message ρυ in (18) can be written as a single summation(21)ρυw1,j=∑s1,jυF1υs1,jυ,w1,jPsυs1,jυ.
Step 2-2. Note that s1,jυ is a sum of random variables w1,ξ(1<ξ≤n,ξ≠j), which are assumed to be independent. Even though s1,jυ is not identically distributed, the authors [5] invoked the central limit theorem to approximate the PMF of s1,jυ as a Gaussian with reference to [23], where there was no explicit justification for the application of this theorem. Since sums of independent random variables converge in distribution to the standard normal as long as some condition (e.g., the Lindeberg Condition [24]) is satisfied, we think that such a condition might be implicitly assumed in [24]. So, the approximate PMF of s1,jυ becomes (22)Psυs1,jυ=12πΔ1,jυexp⁡-s1,jυ-s1,jυ¯22Δ1,jυ,where average s1,jυ¯ and variance Δ1,jυ of s1,jυ are defined as(23)s1,jυ¯=Es1,jυ=E∑l≠1,j3w1,lxlυ=∑l≠1,j3w1,l¯xlυ,(24)Δ1,jυ=Vs1,jυ=V∑l≠1,j3w1,lxlυ=∑l≠1,j3Vw1,lxlυ2=∑l≠1,j3w1,l2¯-w1,l¯2xlυ2.Symbols w1,l¯ and w1,l2¯ denote the averages of w1,l and (w1,l)2, respectively:(25)w1,l¯=∑wwPυw1,l=w,w1,l2¯=∑ww2Pυw1,l=wand then the sum over configurations w1,l∣2≤l≤3,l≠j in (19) is approximated with a Gaussian integration:(26)ρυw1,j≈∫-∞∞F1υs1,jυ,w1,jPsυs1,jυds1,jυ.Step 3 (approximation of the improper integral (26)).
For the calculation of the improper integral (26), error ϕs1,jυ+w1,jxjυ-xjυ is linearized with respect to the maximization of the fitness in (15). Note that the equality(27)ϕs1,jυ+w1,jxjυ-x1υ=0,can be written as(28)ϕ-1x1υ-w1,jxjυ-s1,jυ=0,under the assumption that(29)experimental  datax1υare  contained  in  the  codomain  ofϕ.Then error ϕs1,jυ+w1,jxjυ-x1υ in (15) is approximated by(30)ϕ-1x1υ-w1,jxjυ-s1,jυ,which yields (31)F1υs1,jυ,w1,j=exp-βϕs1,jυ+w1,jxjυ-x1υ2≈exp⁡-βϕ-1x1υ-w1,jxjυ-s1,jυ2.Hence we have (32)ρυw1,j≈∫-∞∞F1υs1,jυ,w1,jPsυs1,jυds1,jυ≈∫-∞∞exp⁡-βϕ-1x1υ-w1,jxjυ-s1,jυ2·12πΔ1,jυexp⁡-s1,jυ-s1,jυ¯22Δ1,jυds1,jυ=11+2βΔ1,jυ1/2exp⁡-βϕ-1x1υ-w1,jxjυ-s1,jυ¯21+2βΔ1,jυ,where the last equality is obtained both by the identity (33)-βϕ-1x1υ-w1,jxjυ-s1,jυ2-s1,jυ-s1,jυ¯22Δ1,jυ=-121+2βΔ1,jυs1,jυ-s1,jυ¯+βϕ-1x1υ-w1,jxjυ2Δ1,jυ/1+2βΔ1,jυ2Δ1,jυ-βϕ-1x1υ-w1,jxjυ-s1,jυ¯21+2βΔ1,jυand the property of a PDF (the integral of a PDF over its domain is equal to 1). Therefore adopting the final approximation of ρυ in [5], approximate marginal PMF P(w1,j=w) in (14) can be obtained by solving the system of equations(34)ρυw1,j=w=exp⁡-βϕ-1x1υ-wxjυ-s1,jυ¯21+2βΔ1,jυ,(35)Pυw1,j=w=1Z1,jυe-λδw1,j=wρ3-υw1,j=w,where υ=1,2, j=2,3, w=-1,0,1, s1,jυ¯ and Δ1,jυ are in (23)–(25).Remark 1.
Since the approximation (34) was used in [5] and our goal in this paper is to find a convergence condition for the algorithm in [5], we also use (34) instead of(36)ρυw1,j=11+2βΔ1,jυ1/2exp⁡-βϕ-1x1υ-w1,jxjυ-s1,jυ¯21+2βΔ1,jυ.Even when using (36) instead of (34), the convergence analysis in the later subsections could be applied (we do not show it in this paper).Remark 2.
Similarly approximate marginal PMFP(w2,j) can be obtained by replacing subscripts 1 and j=2,3, related with node x1, in w1,j, s1,jυ¯, Δ1,jυ, and w1,l in (34), (35), and (23)–(25) with 2 and j=1,3, related with node x2: for υ=1,2, j=1,3:(37)Pw2,j=1Z2,je-λδw2,j∏υ=12ρυw2,j,(38)ρυw2,j=exp⁡-βϕ-1x2υ-w2,jxjυ-s2,jυ¯21+2βΔ2,jυ,(39)Pυw2,j=1Z2,jυe-λδw2,jρ3-υw2,j,where Z2,jυ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one and (40)s2,jυ¯=∑l≠2,j3∑wwPυw2,l=wxlυ,(41)Δ2,jυ=∑l≠2,j3∑ww2Pυw2,l=w-∑wwPυw2,l=w2xlυ2.


## 3.2. Iterative Schemes for Solving the System of Equations

In this subsection, we construct sequences{ρ1,j,nυw} and {P1,j,nυ(w)}(υ=1,2,j=2,3,w=-1,0,1) by using (34) and (35). In the next subsection we present a sufficient condition for the convergence of the two sequences, which implies that the two limits satisfy (34) and (35). So, the limit of {ρ1,j,nυw} becomes value ρυw1,j=w, leading to the construction of approximate marginal PMF P(w1,j) in (14). We assume that(42)initial  termsρ1,j,0υwofρ1,j,nυware  given  as  positive  numbersand initial terms of {P1,j,nυw} are defined as(43)P1,j,0υw=1Z1,j,0υe-λδwρ1,j,03-υw,where Z1,j,0υ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one. The 1st iterations ρ1,j,1υw and P1,j,1υw are defined similarly to (34) and (35) and so we need to define s1,j,0υ¯ and Δ1,j,0υ by using (23) and (24): (44)s1,j,0υ¯=∑l≠1,j3w1,l,0¯xlυ=∑l≠1,j3∑w∈-1,0,1wP1,l,0υwxlυ=∑l≠1,j3-P1,l,0υ-1+P1,l,0υ1xlυ=-P1,3,0υ-1+P1,3,0υ1x3υj=2-P1,2,0υ-1+P1,2,0υ1x2υj=3=e-λZ1,5-j,0υ-ρ1,5-j,03-υ-1+ρ1,5-j,03-υ1x5-jυ,and (45)Δ1,j,0υ=∑l≠1,j3w1,l,02¯-w1,l,0¯2xlυ2=∑l≠1,j3∑ww2P1,l,0υw-∑wwP1,l,0υw2xlυ2=∑l≠1,j3P1,l,0υ-1+P1,l,0υ1--P1,l,0υ-1+P1,l,0υ12xlυ2=e-λZ1,5-j,0υρ1,5-j,03-υ-1+ρ1,5-j,03-υ1-e-λZ1,5-j,0υ-ρ1,5-j,03-υ-1+ρ1,5-j,03-υ12x5-jυ2.So, the 1st iteration is defined as(46)ρ1,j,1υw=exp⁡-βϕ-1x1υ-wxjυ-s1,j,0υ¯21+2βΔ1,j,0υ,P1,j,1υw=1Z1,j,1υe-λδwρ1,j,13-υw,where Z1,j,1υ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one. Similarly the (n+1)th iteration is defined as(47)ρ1,j,n+1υw=Φ1,j,wυρ1,5-j,n3-υ-1,ρ1,5-j,n3-υ1n≥0,(48)P1,j,n+1υw=1Z1,j,n+1υe-λδwρ1,j,n+13-υw,for υ=1,2, j=2,3 and w=-1,0,1. Here Z1,j,n+1υ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one and(49)Φ1,j,wυρ1,5-j,n3-υ-1,ρ1,5-j,n3-υ1=exp⁡-βϕ-1x1υ-wxjυ-e-λ/Z1,5-j,nυ-ρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ1x5-jυ21+2βe-λ/Z1,5-j,nυρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ1-e-λ/Z1,5-j,nυ-ρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ12x5-jυ2.Therefore the desired iterative schemes consist of (42), (43), (47), (48), and (49) under the assumption (29).Remark 3.
SinceP1,j,n+1υw1,j defined in (48) is a PMF due to (47) and (49), we have(50)0<e-λZ1,5-j,nυρ1,5-j,n3-υ-1<1,0<e-λZ1,5-j,nυρ1,5-j,n3-υ1<1,which gives the positivity of the following term in (49)(51)ρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ1-e-λZ1,5-j,nυ-ρ1,5-j,n3-υ-1+ρ1,5-j,n3-υ12>0.So, we obtain a lower bound of ρ1,j,n+1υw(52)ρ1,j,n+1υw≥exp⁡-βϕ-1x1υ+xjυ+2x5-jυ2.Therefore we obtain an upper bound(53)e-λZ1,j,nυ=e-λe-λρ1,j,nυ-1+ρ1,j,nυ0+e-λρ1,j,nυ1≤e-λUj,β,λυ,Uj,β,λυ=11+2e-λexpβϕ-1x1υ+xjυ+2x5-jυ2,which is used in the proof of Lemma 8.Remark 4.
Note that sequence{ρ1,j,nυw} in the recursive relation (47) contains no P1,j,nυ(w). Then, in the next subsection, we present a sufficient condition for the convergence of sequence {ρ1,j,nυw} without using {P1,j,nυ(w)} and, as a result, the limit of {ρ1,j,nυw} is used to define the message ρυw1,j=w.Remark 5.
Similarly, to define messagesρυw2,j=w(υ=1,2,j=1,3,w=-1,0,1), we use the following sequences:(54)ρ2,j,n+1υw=Φ2,j,wυρ2,4-j,n3-υ-1,ρ2,4-j,n3-υ1n≥0,(55)P2,j,n+1υw=1Z2,j,n+1υe-λδwρ2,j,n+13-υw,where Z2,j,n+1υ is the normalization constant ensuring that the sum of the probabilities over all model configurations is equal to one and(56)Φ2,j,wυρ2,4-j,n3-υ-1,ρ2,4-j,n3-υ1=exp⁡-βϕ-1x2υ-wxjυ-e-λ/Z2,4-j,nυ-ρ2,4-j,n3-υ-1+ρ2,4-j,n3-υ1x4-jυ21+2βe-λ/Z2,4-j,nυρ2,4-j,n3-υ-1+ρ2,4-j,n3-υ1-e-λ/Z2,4-j,nυ-ρ2,4-j,n3-υ-1+ρ2,4-j,n3-υ12x4-jυ2.


## 3.3. A Sufficient Condition for the Convergence of the Iterative Schemes

Let(57)X1n=X1,1n,X1,2n,X1,3n,X1,4n,X1,5n,X1,6n,X1,7n,X1,8n,X1,9n,X1,10n,X1,11n,X1,12n=ρ1,2,n1-1,ρ1,2,n10,ρ1,2,n11,ρ1,3,n1-1,ρ1,3,n10,ρ1,3,n11,ρ1,2,n2-1,ρ1,2,n20,ρ1,2,n21,ρ1,3,n2-1,ρ1,3,n20,ρ1,3,n21∈R12and(58)Φ1=Φ1,1,Φ1,2,Φ1,3,Φ1,4,Φ1,5,Φ1,6,Φ1,7,Φ1,8,Φ1,9,Φ1,10,Φ1,11,Φ1,12=Φ1,2,-11,Φ1,2,01,Φ1,2,11,Φ1,3,-11,Φ1,3,01,Φ1,3,11,Φ1,2,-12,Φ1,2,02,Φ1,2,12,Φ1,3,-12,Φ1,3,02,Φ1,3,12,where the subscript 1 of X1 and Φ1 represents node x1. The (n+1)th iteration in (47) is written as(59)X1n+1=Φ1X1n.It follows from (47), (57), and (58) that Φ1,θ(1≤θ≤12) are functions of two independent variables: for X∈R12(60)Φ1X=Φ1,1X10,X12,Φ1,2X10,X12,Φ1,3X10,X12,Φ1,4X7,X9,Φ1,5X7,X9,Φ1,6X7,X9,Φ1,7X4,X6,Φ1,8X4,X6,Φ1,9X4,X6,Φ1,10X1,X3,Φ1,11X1,X3,Φ1,12X1,X3.We use Banach fixed-point theorem [21] for the convergence of the sequence (59) to prove Theorem 9, which is our main result.Theorem 6.
LetD be a closed subset of Rm for a positive integer m. If a function Ψ:D→D satisfies that for a constant k∈(0,1) and all x,y in D(61)Ψx-Ψy≤kx-y,then there exists a unique fixed point x∗∈D such that Ψ(x∗)=x∗, which is the limit of sequence x(n+1)=Ψ(x(n))(n≥0) for any x(0)∈D.For the application of Theorem6 to the proof of our main result we need the following lemmas.Lemma 7.
Assume that experimental datax1υ are contained in the codomain of ϕ. Assume that positive constants β and λ satisfy inequalities 1-2βe-λUj,β,λυ2xjυ2>0 for υ=1,2,j=2,3 and Uj,β,λυ defined in (53). Then Φ1 defined in (49) and (57)–(60) becomes a function from domain [0,1]12 to codomain [0,1]12.Proof.
It follows from (49) and (57)–(60) that for (θ,w)=(1,-1),(2,0),(3,1)(62)Φ1,θX=Φ1,2,w1X10,X12,X=X1,…,X12∈0,112and(63)Φ1,2,w1X10,X12=exp⁡-βϕ-1x11-wx21-e-λ/Z1,3,n1-X10+X12x3121+2βe-λ/Z1,3,n1X10+X12-e-λ/Z1,3,n1-X10+X122x312.We have that for β>0, λ>0 and {X10,X12}⊂[0,1](64)1+2βe-λZ1,3,n1X10+X12-e-λZ1,3,n1-X10+X122x312≥1-2βe-λU3,β,λ12x312>0,and for β>0(65)-βϕ-1x11-wx21-e-λZ1,3,n1-X10+X12x312≤0,and so, we obtain(66)0<Φ1,θX≤11≤θ≤3.Similarly, under the same condition, we can obtain(67)0<Φ1,θX≤14≤θ≤12,which are desired results.Lemma 8.
Assume that experimental datax1υ are contained in the codomain of ϕ. Assume that positive constants β and λ satisfy inequalities 1-2βe-λUj,β,λυ2xjυ2>0 for υ=1,2,j=2,3 and Uj,β,λυ defined in (53). Then for Φ1 defined in (49) and (57)–(60) and {X,Y}⊂[0,1]12(68)Φ1X-Φ1Y≤6max⁡Cj,β,λυ∣υ=1,2,j=2,3βe-λX-Y,where(69)Cj,β,λυ=2ϕ-1x1υ+xjυ+e-λU5-j,β,λυx5-jυU5-j,β,λυx5-jυ×1+2βe-λU5-j,β,λυ2+e-λU5-j,β,λυx5-jυ2+ϕ-1x1υ+xjυ+e-λU5-j,β,λυx5-jυ2×2βU5-j,β,λυ1+2e-λU5-j,β,λυx5-jυ2×1-2βe-λU5-j,β,λυ2x5-jυ2-2.Proof.
Using (62) and (63), we have that for {X,Y}⊂[0,1]12(70)Φ1,1X-Φ1,1Y=Φ1,2,-11X10,X12-Φ1,2,-11Y10,Y12(71)=exp⁡fX10,X12-exp⁡fY10,Y12,where X10,X12,Y10,Y12⊂[0,1] and function f:[0,1]2→[0,1]2 is defined as follows: for ζ,η∈[0,1](72)fζ,η=-βϕ-1x11--1x21-e-λ/Z1,3,n1-ζ+ηx3121+2βe-λ/Z1,3,n1ζ+η-e-λ/Z1,3,n1-ζ+η2x312.Due to the mean value theorem there exists a constant c in (0,1) such that(73)exp⁡fX10,X12-exp⁡fY10,Y12=∇exp⁡f1-cX10,X12+cY10,Y12·X10-Y10,X12-Y12.Note that(74)∂∂ζexp⁡fζ,η≤∂∂ζfζ,η,where(75)∂∂ζfζ,η=-β2ϕ-1x11+x21-e-λ/Z1,3,n1-ζ+ηx31e-λ/Z1,3,n1x31×1+2βe-λ/Z1,3,n1ζ+η-e-λ/Z1,3,n1-ζ+η2x312-ϕ-1x11+x21-e-λ/Z1,3,n1-ζ+ηx312×2βe-λ/Z1,3,n11-2e-λ/Z1,3,n1-ζ+η-1x3121+2βe-λ/Z1,3,n1ζ+η-e-λ/Z1,3,n1-ζ+η2x3122.Using {ζ,η}⊂[0,1], we have(76)∂∂ζfζ,η≤β2ϕ-1x11+x21+e-λ/Z1,3,n1x31e-λ/Z1,3,n1x31×1+2βe-λ/Z1,3,n12+e-λ/Z1,3,n1x312+ϕ-1x11+x21+e-λ/Z1,3,n1x312×2βe-λ/Z1,3,n11+2e-λ/Z1,3,n1x3121+2βe-λ/Z1,3,n1ζ+η-e-λ/Z1,3,n1-ζ+η2x3122.Using the given condition in this lemma and (53), we have the positive lower bound(77)1+2βe-λZ1,3,n1ζ+η-e-λZ1,3,n1-ζ+η2x312≥1-2βe-λU3,β,λ12x312>0,which gives that(78)∂∂ζfζ,η≤βe-λ2ϕ-1x11+x21+e-λU3,β,λ1x31U3,β,λ1x31×1+2βe-λU3,β,λ12+e-λU3,β,λ1x312+ϕ-1x11+x21+e-λU3,β,λ1x312×2βU3,β,λ11+2e-λU3,β,λ1x3121-2βe-λU3,β,λ12x3122=βe-λC2,β,λ1.Similarly,(79)∂∂ηfζ,η≤βe-λC2,β,λ1.Substituting (73), (74), (78), and (79) into (70), we have(80)Φ1,1X-Φ1,1Y=Φ1,2,-11X-Φ1,2,-11Y=exp⁡fX10,X12-exp⁡fY10,Y12≤C2,β,λ1βe-λX10-Y10+X12-Y12.Following the calculations for obtaining (80), we can obtain that for the 1st perturbation(81)Φ1,θX-Φ1,θY≤C2,β,λ1βe-λX10-Y10+X12-Y12θ=2,3,Φ1,θX-Φ1,θY≤C3,β,λ1βe-λX7-Y7+X9-Y9θ=4,5,6,where C3,β,λ1 is defined in Lemma 8.
Similarly we can have that for the2nd perturbation(82)Φ1,θX-Φ1,θY≤C2,β,λ2βe-λX4-Y4+X6-Y6θ=7,8,9,Φ1,θX-Φ1,θY≤C3,β,λ2βe-λX1-Y1+X3-Y3θ=10,11,12,where Cj,β,λ2(j=2,3) are defined in Lemma 8. Therefore, it follows from (80)–(82) that (83)Φ1X-Φ1Y2=∑θ=112Φ1,θX-Φ1,θY2≤6maxυ=1,2j=2,3Cj,β,λυ2β2e-2λX-Y2,which gives the desired result.Using Theorem6 and Lemmas 7 and 8, we can obtain our main result.Theorem 9.
Assume that the experimental datax1υ(υ=1,2) are contained in the codomain of ϕ. Suppose that positive constants β and λ satisfy(84)1-2βe-λUj,β,λ12xjυ2>0and(85)6max⁡Cj,β,λυ∣υ=1,2,j=2,3βe-λ<1,for υ=1,2,j=2,3, Uj,β,λυ defined in (53) and Cj,β,λυ defined in Lemma 8. Then sequence X1n+1=Φ1X1n(n≥0) converges for any X1(0)∈[0,1]12.Proof.
LetD be the closed subset [0,1]12 of R12 and(86)k=6max⁡Cj,β,λυ∣υ=1,2,j=2,3βe-λ.Then Lemmas 7 and 8 give that Φ1:D→D and(87)Φ1X-Φ1Y≤kX-Y.Therefore we can apply Theorem 6 and so we obtain a unique fixed point X1∗∈D such that Φ1(X1∗)=X1∗, which implies the convergence of sequence X1n+1=Φ1X1n(n≥0) for any X1(0)∈[0,1]12 with limit X1∗.Remark 10.
We define(88)ρ1w1,2=-1,ρ1w1,2=0,ρ1w1,2=1,ρ1w1,3=-1,ρ1w1,3=0,ρ1w1,3=1,ρ2w1,2=-1,ρ2w1,2=0,ρ2w1,2=1,ρ2w1,3=-1,ρ2w1,3=0,ρ2w1,3=1=ρ1,2,-11,ρ1,2,01,ρ1,2,11,ρ1,3,-11,ρ1,3,01,ρ1,3,11,ρ1,2,-12,ρ1,2,02,ρ1,2,12,ρ1,3,-12,ρ1,3,02,ρ1,3,12,where the vector in the right side denotes limit X1∗ in Theorem 9. Since X1n in (57)–(59) converges, {P1,j,nυw} in (48) also converges to limit e-λδwρ1,j,w3-υ. Therefore, ρυw1,j in (88) and Pυw1,j=e-λδw1,jρ1,j,w1,j3-υ satisfy the system of equations in (34) and (35), which give approximate marginal PMFs Pw1,j=1/Z1,je-λδw1,j∏υ=12ρυw1,j in (14) as follows: (89)Pw1,j=w=1Z1,je-λδw1,j=w∏υ=12ρ1,j,wυj=2,3,w=-1,0,1.Remark 11.
When using the sequences in (54)–(56) instead of the sequences in (47)–(49), we can obtain the limits of the sequences in (54)–(56) under conditions similar to those in Theorem 9: assume that experimental data x2υ(υ=1,2) are contained in the codomain of ϕ. Let Uj,β,λυ=1/1+2e-λexp⁡βϕ-1(x2υ)+xjυ+2x4-jυ2(j=1,3). Suppose that positive constants β and λ satisfy that 1-2βe-λUj,β,λ12xjυ2>0 and(90)6max⁡Cj,β,λυ∣υ=1,2,j=1,3βe-λ<1,where(91)Cj,β,λυ=2ϕ-1x2υ+xjυ+e-λU4-j,β,λυx4-jυU4-j,β,λυx4-jυ×1+2βe-λU4-j,β,λυ2+e-λU4-j,β,λυx4-jυ2+ϕ-1x2υ+xjυ+e-λU4-j,β,λυx4-jυ2×2βU4-j,β,λυ1+2e-λU4-j,β,λυx4-jυ2×1-2βe-λU4-j,β,λ12x4-jυ2-2.We can define the messages as in Remark 4(92)ρ1w2,1=-1,ρ1w2,1=0,ρ1w2,1=1,ρ1w2,3=-1,ρ1w2,3=0,ρ1w2,3=1,ρ2w2,1=-1,ρ2w2,1=0,ρ2w2,1=1,ρ2w2,3=-1,ρ2w2,3=0,ρ2w2,3=1=ρ2,1,-11,ρ2,1,01,ρ2,1,11,ρ2,3,-11,ρ2,3,01,ρ2,3,11,ρ2,1,-12,ρ2,1,02,ρ2,1,12,ρ2,3,-12,ρ2,3,02,ρ2,3,12and then obtain the approximate marginal PMFs(93)Pw2,j=w=1Z2,je-λδw2,j=w∏υ=12ρ2,j,wυj=1,3,w=-1,0,1.


## 4. Numerical Examples

In this section, we verify our main result and show its application.Step 1 (generation of artificial experimental data).
We randomly generate 100 sets of artificial experimental data, where each set consists of the six values ofxjυ(υ=1,2,j=1,2,3) as in Figure 3(a). Every randomly generated experimental data has a value contained in the open interval (-1,1), the codomain of ϕ(x)=tanh⁡x, and so the condition on the experimental data in Theorem 9 is satisfied.Figure 3
Experimental data and cost parameters. (a) Each of 100 color matrices denotes the values of 6 artificial experimental dataxiυ(υ=1,2,i=1,2,3). Values of the x- and y-axis in each color matrix denote the values of i and υ, respectively, where the color in the entry (i,υ) denotes the value of xiυ. (b) A value on the x-axis denotes an experimental data set in Figure 3(a). The data set at entry (i,j) in Figure 3(a) is represented as the 10(i-1)+jth data set in Figure 3(b). There are few values of β and λ greater than 0.1 and we do not show them here.
(a)
100 sets of 6 experimental data(xυi,υ=1,2,i=1,2,3) (b)
Cost parameters(β,λ)Step 2 (determination of cost parameters).
We find a pair of positive cost parameters(β,λ) as in Figure 3(b) for each experimental data set, which satisfy the condition on (β,λ) in Theorem 9.Step 3 (generation of values of initial terms in sequences).
We randomly generate initial values of 24 termsρi,j,0υ(w)(υ=1,2,i=1,2,j≠i,1≤j≤3,w=-1,0,1) for each experimental data set, which satisfy the condition on initial values in Theorem 9 (see Figure 4).Figure 4
Initial values for the sequences. Due to the recursive relations in (47), (48), (54), and (55), it is enough to independently consider the convergences of the two types of the sequences ρ1,j,nυ(w)(υ=1,2,j=2,3,w=-1,0,1) and ρ2,j,nυ(w)(υ=1,2,j=1,3,w=-1,0,1). (a) Each set contains values of 12 initial terms ρ1,j,0υ(w)(υ=1,2,j=2,3,w=-1,0,1) for each experimental data set. (b) Each set contains values of 12 initial terms ρ2,j,0υ(w)(υ=1,2,j=1,3,w=-1,0,1) for each experimental data set.
(a)
100 sets of 12 initial values(ρ1,j,0υw,υ=1,2,j=2,3,w=-1,0,1) (b)
100 sets of 12 initial values(ρ2,j,0υw,υ=1,2,j=1,3,w=-1,0,1)Step 4 (convergence of the sequences).
We obtain the convergence of all sequencesρi,j,nυ(w)(υ=1,2,i=1,2,j≠i,1≤j≤3,w=-1,0,1) for each experimental data set in Figure 3(a), (β,λ) in Figure 3(b), and initial values in Figure 4, where every sequence converges within 14 iterations (see Figure 5). Due to the recursive relations in (47), (48), (49), (53), (54), (55), and (56), we can divide all sequences into four groups ({ρ1,2,n1(w),ρ1,3,n2(w)}, {ρ1,3,n1(w),ρ1,2,n2(w)}, {ρ2,1,n1(w),ρ2,3,n2(w)}, and {ρ2,3,n1(w),ρ2,1,n2(w)}) and show their convergences as in Figure 5.Figure 5
Iteration number for the convergence. (a) The system of recursive relations forρ1,2,nυ(w) and ρ1,3,nυ(w)(υ=1,2,w=-1,0,1) converges within 10 iterations for tolerance 10-10. Symbols and data in Figures 5(b), 5(c), and 5(d) have similar meanings as in Figure 5(a).
(a) (b) (c) (d)Step 5 (approximation of marginal PMFs).
Substituting the limits (Figure6(c)) of the sequences for the 1st experimental data set (Figure 6(a)) and its corresponding cost parameters (β,λ) (Figure 6(a)) into (89) and (93), we obtain the approximate marginal PMFs P(w1,2=w), P(w1,3=w), P(w2,1=w), and P(w2,3=w) as in Figure 6(d), where the maximum value of each PMF is marked as red.Figure 6
Approximate marginal PMFs. (a) The experimental data and cost parameters(β,λ) are the 61th data set and its corresponding values of (β,λ) in Figures 3(a) and 3(b), respectively. (b) 12 initial values in the left and right denote the values in the 61th initial data set in Figures 4(a) and 4(b), respectively. (c) 12 limit values in the top and bottom are obtained for Figures 6(a) and 6(b). (d) The approximate marginal PMFs are obtained from (89) and (93). The red numbers denote the maximum values of the marginal PMFs. The first network and its mathematical model in the right denote those in Figures 1(a) and 1(c) with link weights of maximum values. Replacing w12=1 in the first network with w12=0 gives the second network and its mathematical model.
(a) (b) (c) (d)Step 6 (construction of multiple networks and their mathematical formulas from the approximate marginal PMFs).
We can use the maximum values of four link weights to make a network and its corresponding mathematical formulas (Figure6(d), first grey box), which could be considered as the most possible network model of the artificial biological system. In addition, since value P(w2,1=0) is almost equal to value P(w2,1=1), we can also consider the network with w2,1=0 and its mathematical formula as in the second grey box in Figure 6(d). We present two networks and their mathematical formulas in the two grey boxes to show that there could be multiple networks for explaining the artificial biological system by using the approach in [5]. Similarly, in cases where a given PKN has more nodes and discrete values of link weights, we could obtain multiple networks and their corresponding mathematical formulas, where probability P(wi,j=w) is high.


## 5. Conclusions and Future Work

In this paper, using a simple network with artificial experimental data, we have presented a detailed explanation of the whole process for determining approximate marginal PMFs of link weights in a prior knowledge network based on BP on the factor graph. For the factorization of the joint PMF of link weights, the Boltzmann/Gibbs form has been used. And we have shown a sufficient condition for the convergence of beliefs, which leads to determining the approximate marginal PMFs. Since our simple PKN and experimental data have key components necessary for the application of BP in determining approximate marginal PMFs, our results can be extended to any PKN with perturbation data, including the PKN in [5], which is our future study.



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