Several established R packages can fit and visualize networks (R-Core-Team, 2024), e.g., network<\/em> (Butts, 2008), statnet<\/em> (Handcock et al., 2008), and igraph<\/em> (Csardi & Nepusz, 2006). In psychology, network psychometrics has its specific packages like bootnet<\/em> to estimate networks and their stability (Epskamp et al.,<\/a> 2018), mgm<\/em> for mixed graphical models (Haslbeck & Waldorp, 2015), IsingFit<\/em> for Ising models (van Borkulo et al., 2015), and qgraph<\/em> to visualize networks (Epskamp et al., 2012). These packages include a variety of widely used analysis tools, but they implement the frequentist analysis of networks. Other packages have emerged that allow for the Bayesian analysis of graphical models, in particular, bgms<\/em> (Marsman & Haslbeck, 2023), BDgraph<\/em> (Mohammadi & Wit, 2019), and BGGM<\/em> Williams and Mulder (2020b). Unfortunately, these packages and their papers are mostly aimed at researchers with advanced statistical knowledge and programming experience. At present, the use of Bayesian network analysis is not accessible to applied researchers. To bring the benefits of Bayesian analysis of graphical models to all researchers, we present the user-friendly package easybgm<\/em>. The package performs the Bayesian estimation of graphical models. It is specifically aimed at users with little experience of Bayesian inference or programming experience. easybgm <\/em>consolidates the three existing packages for Bayesian graphical modeling in the social sciences (i.e., bgms<\/em>, BDgraph<\/em>, and BGGM<\/em>) and combines their functionality to model all different data types.<\/p>\n\n\n\n The remainder of this paper is organized as follows. First, we introduce the methodological background of Bayesian graphical modeling. This also allows us to define the notation used throughout the paper; key terms are also defined in the glossary at the end of the paper. Readers new to the material can find a more extensive and accessible introduction to the Bayesian analysis of networks<\/a> in Huth et al. (2023). Next, we describe the workflow of the package. We introduce how the package obtains the model fit and give an overview of the different prior specifications. We discuss the output of the package and introduce the suite for visualizing the network results. The functionality of easybgm <\/em>is demonstrated using a sample dataset on women and mathematics and with an additional example on a self-worth scale.<\/p>\n\n\n\n Graphical models are a subset of networks in which the network graph encodes probabilistic relationships between variables (i.e., links between nodes are not observed but must be estimated). In graphical models, the nodes are random variables and their pairwise relationships\u2013the edges\u2013are conditional dependencies (Lauritzen & Wermuth, 1989). We focus here on undirected graphical models in which the edges do not imply a direction.<\/p>\n\n\n\n In practice, we are interested in two aspects of the network: the underlying network structure \\(\\mathcal{S}\\) and the matrix of pairwise association strengths \u0398<\/strong>. First, the network structure \\(\\mathcal{S}\\) is a configuration of unweighted, undirected edges, where the presence of an edge indicates conditional dependence, and the absence of an edge indicates conditional independence. The number of possible structures grows exponentially as the number of variables increases. Since further inference is based on the structure estimate, it becomes critical to adequately quantify the uncertainty associated with the structure. Second, given a structure, we are interested in the strength and nature of the present edges which are encoded in \u0398<\/strong>, the matrix of edge weights. A positive value of \u03b8ij <\/sub><\/em>indicates a positive association between the variables i <\/em>and j<\/em>, and negative values indicate a negative association of the variables. A higher absolute \u03b8ij <\/sub><\/em>indicates a stronger association. The uncertainty of the partial association estimate is of interest, especially when drawing inferences about the strength of association and differences between network edges.<\/p>\n\n\n\n The Bayesian framework allows us to quantify the uncertainty of the network\u2019s structure and its parameters. To make use of the Bayesian approach we must first specify how likely certain parameter values and network structures are before we have seen the data. A Bayesian does this by specifying a prior distribution, a probability distribution that describes, for each value of the parameter, our belief that the respective value will produce the data (van de Schoot et al., 2015; Wagenmakers et al., 2018). In many cases, we have little expectation about the value of the parameter and specify a default, diffuse prior<\/a>. We discuss the (default) specification of prior distributions below. After observing the data, Bayes\u2019 rule is used to update the prior into a posterior distribution. The posterior is a probability distribution that describes, for each value of the parameter, our belief that the respective value produced the data we observed. This updated distribution contains everything we know about the structure of the network and its parameters, what we expected before seeing the data, and what we learned from the data. Formally, we can express this cycle of knowledge updating as<\/p>\n\n\n\n \\(\\begin{equation}\\label{Eq:Bayes1} \\underbrace{p(\\boldsymbol{\\Theta}\\text{, }\\mathcal{S} \\mid \\text{data})}_{\\text{joint posterior}} \\propto \\underbrace{p(\\text{data} \\mid \\boldsymbol{\\Theta} \\text{, }\\mathcal{S})}_{\\text{likelihood}} \\times \\underbrace{p(\\boldsymbol{\\Theta}\\mid\\mathcal{S})}_{\\substack{\\text{prior on the} \\\\ \\text{parameters}}} \\times \\underbrace{p(\\mathcal{S}).}_{\\substack{\\text{prior on the} \\\\ \\text{graph structure}}} \\end{equation}\\tag{1}\\)<\/p>\n\n\n\n The posterior distribution can be used to describe our network and the uncertainty associated with its estimation.[1]<\/a> Next, we elaborate on how to quantify the uncertainty of the structure (i.e., how likely are edges to be included in the network) and the parameter estimates (i.e., how certain are the edge weights).<\/p>\n\n\n\n In network psychometrics, we are oftentimes primarily interested in the structure of the network, i.e., which edges are present and which are absent. For example, to assess the link between node i <\/em>and j<\/em>, we are interested in the two hypotheses<\/p>\n\n\n\n We can use the posterior distribution of the structures (i.e., \\(p(\\mathcal{S} \\mid \\text{data})\\)) to determine how likely both hypotheses are. As such, we obtain the posterior probability for edge inclusion and exclusion. To obtain the posterior probability of edge presence between node \\(i\\) and \\(j\\) (i.e., \\(\\mathcal{H}^{(ij)}_1\\)), we sum the posterior probabilities of all structures \\(\\mathcal{S}^\\prime\\) that include the edge between the variables<\/p>\n\n\n\n \\(p(\\mathcal{H}^{(ij)}_1 \\mid \\text{data}) = \\sum_{\\mathcal{S}^\\prime \\in \\mathcal{S}^{(i-j)}} p(\\mathcal{S}^\\prime \\mid \\text{data})\\)<\/p>\n\n\n\n and for H0<\/sub>(ij<\/em>)<\/sup> that there is no edge between the two variables by summing over all structures where the edge is absent. This technique is known as Bayesian model-averaging (BMA; e.g., Hinne et al., 2020; Hoeting et al., 1999). In BMA, each possible graph structure is treated as a separate statistical model and we aggregate over all possible models rather than choosing a single most likely one. Two of the three packages implemented in easybgm<\/em> use BMA (Marsman et al., 2023; Mohammadi & Wit, 2015). Since BMA can be computationally and analytically challenging, some approaches use the posterior distribution of a single structure (e.g., BGGM<\/em>; Williams & Mulder, 2020b). Here, Equation (1) reduces \\(\\mathcal{S}_s\\) and and \u0398s<\/sub><\/em><\/strong> <\/em>for one particular structure s<\/em>.[2]<\/a><\/p>\n\n\n\n With the posterior edge inclusion probabilities at hand, we can test the two hypotheses against each other. In the Bayesian framework, the hypothesis testing tool is the Bayes Factor (Kass & Raftery, 1995; Wagenmakers et al., 2016). A Bayes factor is a measure used to compare how well two competing models or hypotheses explain the observed data, indicating which explanation is more strongly supported by the data. Concerning networks, the Bayes factor is used to compare the evidence for edge presence against the evidence for edge absence. The Bayes factor can be obtained in the following way<\/p>\n\n\n\n \\(\\begin{equation}\\label{BF} \\text{BF}_{10} = \\underbrace{\\frac{p(\\mathcal{H}^{(ij)}_1\\mid \\text{data})}{p(\\mathcal{H}^{(ij)}_0 \\mid \\text{data})}}_{\\substack{\\text{posterior}\\\\\\text{inclusion odds}}} \\bigg\/ \\underbrace{\\frac{p(\\mathcal{H}^{(ij)}_1)}{p(\\mathcal{H}^{(ij)}_0)}}_{\\substack{\\text{prior}\\\\\\text{inclusion odds}}}. \\end{equation}\\tag{2}\\)<\/p>\n\n\n\n We call this the inclusion Bayes factor (Huth et al., 2023; Marsman & Haslbeck, 2023; Marsman et al., 2022; Sekulovski, Keetelaar, Huth, Wagenmakers et al., 2023), which is interpreted as the strength of evidence in the data for edge inclusion (i.e., conditional dependence). Conversely, we can take the inverse, \\(\\text{BF}_{01}= 1\/\\text{BF}_{10}\\), to quantify the evidence for edge exclusion (i.e., conditional independence). For example, a \\(\\text{BF}_{10} = 5\\) would mean that the data are 5 times more likely under a network structure that includes the edge between variables \\(i\\) and \\(j\\) than under a network structure that excludes that edge. Bayes factors greater than \\(1\\) indicate some evidence for edge inclusion; a \\(BF_{10} > 3\\) indicates weak and a \\(BF_{10} > 10\\) indicates strong evidence for edge inclusion (Jeffreys, 1961). Conversely, a \\(BF_{10} < 1\\) indicates evidence for edge exclusion, where a \\(BF_{10} < \\frac{1}{3}\\) denotes weak and a \\(BF_{10} < \\frac{1}{10}\\) strong evidence for edge exclusion (Kass & Raftery, 1995). We suggest to consider a \\(\\text{BF}_{10} \\geq 10\\) as evidence for edge inclusion and conversely a \\(\\text{BF}_{10} \\leq \\frac{1}{10}\\) as evidence for edge exclusion (e.g., Huth et al., 2023). All \\(\\text{BF}_{10}\\) <\/sub>in between are considered inconclusive edges. Researchers may choose their cutoffs, depending on the substantive research question.<\/p>\n\n\n\n To quantify parameter uncertainty, we can use the posterior distribution of the partial association parameters \\(p(\\boldsymbol{\\Theta} \\mid \\mathcal{S}\\text{, }\\text{data})\\) and obtain posterior standard deviations and credible intervals for each partial association parameter \u03b8ij<\/sub><\/em>. The credible interval is the Bayesian equivalent of the frequentist confidence interval and is a range of values within which \u03b8ij<\/sub><\/em> is believed to lie with x<\/em>% probability, i.e., covers x<\/em>% of the posterior distribution. A special type of credible interval, the highest posterior density interval, is the shortest interval that covers x<\/em>% of the posterior distribution.<\/p>\n\n\n\n Figure 1<\/strong><\/p>\n\n\n\n Workflowof the easybgm Package<\/em><\/p>\n\n\n\nMethodological Background<\/h2>\n\n\n\n
Structure Uncertainty – Edge Presence or Absence<\/h3>\n\n\n\n
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Parameter Uncertainty<\/h3>\n\n\n\n
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