.1: Performance on synthetic data with d = 200, sp = 0.9, K = 1, 2, 3. . . . . . . 16 Table 4 .1: AUC performance on synthetic data with "band" graph type with d = 127. Table 4 .2: AUC performance on synthetic data with "cluster" graph type with d = 127. Table 4 .3: AUC performance on synthetic data with "random" graph type with d = 127. Table 4 .4: AUC performance on synthetic data with "scale-free" graph type with d = 127. Table 5 .1: AUC performance on synthetic data with "band" graph type with d = 127.
"K" indicates the number of components to generate the synthetic data. Table 5 .2: AUC performance on synthetic data with "cluster" graph type with d = 127. Table 5 .3: AUC performance on synthetic data with "random" graph type with d = 127. Table 5 .4: AUC performance on synthetic data with "scale-free" graph type with d = 127. Microbes are found almost everywhere on earth, including in environments deemed too extreme for other life forms, and they play critical roles in many biogeochemical processes [11, 31] .
Microbial communities are also found in association with higher life forms, including plants and animals; for instance, trillions of microbes live in or on the human body (almost as many human cells as there are in the body) [3, 30] and ongoing research continues to reveal the important roles that many of these microbes play in human health [27, 38, 57] . Microbial communities are typically structured and composed of members of different species. The microbes in a community do not exist in isolation, but interact with each other and also compete for the available carbon and energy sources. In a microbial community, the abundance level of a constituent member is determined by its interactions with other members of the community (resulting from the competition for available resources in an environment) and/or by its interaction with the host. Revealing the nature of these interactions and co-occurrences is important for understanding the functional roles of the community members, and has implications in many areas, including in the context of human health [38, 42] .
These interactions, along with resource availability and environmental parameters (like temperature, pH, and salinity) [16, 32, 58] , determine the taxonomic composition of the microbial community and the abundance levels of its constituents. Knowledge of these interactions is crucial for understanding the overall behaviour of the microbial community, and can be used to elucidate the biological mechanisms underlying microbe-associated disease progression and microbe-mediated processes (like biofilm formation).
The study of microbial communities has been greatly enabled with the advent of highthroughput next-generation DNA sequencing technologies [7, 26, 28] . With advances in highthroughput DNA sequencing, it is now possible to generate large volumes of sequence data (either targeting a taxonomic marker gene [33] or whole genome shotgun sequencing [39] ) at lower costs.
The taxonomic composition of a microbial community can be obtained by sequencing the DNA extracted from a biological sample that has been collected from the environment of interest. This is achieved either using a targeted approach, involving the sequencing of a taxonomic marker gene (for instance, the 16S ribosomal RNA gene, which is found in all bacteria [9, 33] ) or using a whole-genome shotgun sequencing approach [39] . Both approaches generate taxa counts that are compositional in nature, and that enable the estimation of the relative abundances of the constituent members of the community. Sequence data generated from a collection of n samples defined on d microbial taxa can be used to generate a sample-taxa matrix that contains counts reflecting the number of times a microbial taxon has been observed in a sample.
Microbial interactions can be modeled using a weighted graph (or network), where each node in the graph represents a taxon (or taxonomic group) and an (undirected) edge exists between two nodes if the corresponding taxa interact with, or influence, each other. The edge weight captures the strength of the interaction, with its sign reflecting whether the interaction is positive or negative.
This framework can be used to model a variety of microbial interactions, including competition and co-operation. While we do not consider it here, a directed graph can also be used to represent interactions, where the edge direction indicates the direction of influence (or causality). Microbial networks are typically constructed from sample-taxa count matrices. A sample-taxa count matrix is generated by sequencing multiple biological samples (n samples) collected from the environment of interest and identifying the counts of the observed taxa (d taxa) in each sample.
Microbial networks can be constructed using a variety of different approaches. To our knowledge, all of these methods assume that the sample-taxa matrix is associated with a single underlying stochastic process (that is, there is one underlying network topology and set of edge weights). However, this need not always be the case. In this dissertation, we consider an important extension to the network inference problem, whereby we develop a mixture modeling framework for inferring K microbial networks when the observed sample-taxa matrix is associated with K 2 underlying distributions. Another issue that we tackle in this dissertation is finding the proper K(number of components) in modeling microbial count data, by introducing a new practical framework.
We are motivated by large-scale human-associated and other environmental microbial community projects that are now possible due to cost-effective sequencing. For instance, human gut microbiome studies now routinely analyze large cohorts of individuals and generate microbial community data from several hundreds (to even thousands) of samples. An important research question in this area involves the definition of a "core" microbiome associated with a particular host phenotype [27, 38] . It is well understood that the gut microbiome composition is greatly influenced by many factors including diet and age, and thus it is not unreasonable to expect the associated microbial network interactions to also be different when these factors vary (for example, the gut microbial community interaction network in vegetarian hosts can be expected to be different compared to the network in non-vegetarian hosts). A similar situation also occurs in environmental studies where the microbial interactions are influenced greatly by the physical and chemical gradients of the environment. Often the collected metadata in these studies may not be comprehensive enough to discern these interactions in a supervised manner. Our proposed mixture frameworks offer a principled approach to identifying these multiple microbial interaction networks from a sample-taxa matrix.
Sample-taxa matrices can be used to study microbial associations [40] . These associations can be modeled using a weighted graph with d nodes in which the nodes represent microbial taxa, the edges represent pairwise associations, and edge weights represent the strength of the associations. Different methods have been proposed for inferring a single microbial network from a sample-taxa matrix, including correlation based approaches [15] , inference of the latent correlation structure after log-ratio transformation of the count data [43] [41] , and probabilistic graphical models [47] [45] .
Several methods have been proposed for constructing a single microbial network from an input sample-taxa matrix [24] . One approach involves using pairwise correlations (Pearson or Spearman) between taxa to define the edge weights in the graph. This is the most simple method that considers the pairwise correlation as similarity index to extract the interaction network. There are also other methods which calculate mutual information as similarity index or Kulback-Leiber criteria as a dissimilarity index to find the connections between the nodes in the interaction graph extracted from microbial data. However, the computation of these correlation networks directly from the observed count data can be misleading because of the compositional nature of these data [15] and causes spurious connection detection in the estimated network. Considering regular correlation as a criterion is not able to extract complex relations between the OTUs in the microbial communities. This issue is handles by multiple regression analysis methods that considers structural relation between the OTUs instead of pairwise [76] . Some methods use mutual information matrices to extract a specific structure such as the Chow-Liu tree approximation [77] . This method considers higher order interactions not just pairwise which gives the opportunity for better estimation of the interaction network. However this approach forces a tree structure to the network which is not a desired case in modeling the microbial interaction networks. 4 Furthermore, for a microbial network with d nodes, while there are d * (d − 1)/2 edge weights that need to be determined, the number of available samples n is often not large enough, with the result that the system of equations to determine all pairs of correlations is under-determined.
This later issue is typically handled by assuming that the network is sparse (that is, the number of edges is O(d)). Methods based on latent variable modelling have been proposed to infer correlation networks [12, 14] . These methods use log-ratio transformations of the original count data [5] to deal with the compositional nature of these data and subsequently infer the correlation matrix (i.e. edge weights) under the assumption of sparsity.
Microbial networks have also been constructed using a probabilistic graphical model framework [19] that enables the modeling of conditional dependencies associated with the interactions.
For instance, the assumption that the log-ratio transformed count data follow a Gaussian distribution, results in a Gaussian graphical model (GGM) framework. In this scenario, the graph structure represents the precision matrix (or inverse covariance matrix) of the underlying multivariate Gaussian distribution. This graph has the property that an edge exists between two nodes if the corresponding entry in the precision matrix is non-zero. A zero entry in the precision matrix indicates conditional independence between the two corresponding random variables. When the graph is assumed to be sparse, the GGM inference problem can be solved using sparse precision matrix estimation algorithms [13] . This approach has been used to construct microbial networks from sample-taxa matrices [21] [51] [52] .
An alternate approach to constructing a microbial network is to model the vector of observed taxa counts (in samples) using a multivariate distribution and to infer the parameters of 5 this distribution from the observed data using a maximum likelihood framework. Any candidate multivariate distribution for this approach will have to be flexible enough to capture the underlying covariance structure to model the network interactions (i.e., allow for both positive and negative covariances); this rules out distributions like the multinomial or the Dirichlet-Multinomial, which are popular choices for modeling microbial count data in certain situations [ [17, 22] ], but which cannot capture both types of interactions. The Multivariate Poisson Log-Normal (MPLN) distribution [6] can be used for modelling multivariate count data and its covariance structure can capture both positive and negative interactions. This distribution was used recently [10] to model counts in a sample-taxa matrix and infer an underlying microbial network using an assumption of sparsity.
Motivated by the observation that a sample-taxa matrix generated as part of a large cohort study involving multiple environmental or clinical parameters can be associated with more than one underlying microbial network, we proposed three novel mixture model frameworks [49] to infer K microbial networks from a sample-taxa matrix that is generated from n samples and d taxa.
Furthermore, we introduced a new practical framework to find the best number of components and model selection in modeling microbial data with a mixture modeling framework.
In our first approach, we modeled the taxa counts in a sample using a Multivariate Poisson Log-Normal (MPLN) (described in chapter 3) distribution [46, 73] . The mixture model framework consists of K MPLN distributions with different underlying parameters. We used a maximum likelihood setting to estimate the parameters, and presented an optimization algorithm based on the Minorization-Maximization (MM) principle [23] [18] , and involving gradient ascent and block updates. We extend this formulation based on an l 1 -penalty model and provide algorithms to infer K sparse networks. We evaluate the performance of our algorithms using both synthetic and real datasets. We also evaluate the performance of our method on compositional data obtained by subsampling from the true counts of the taxa. This evaluation models the real-world scenario, where 6 the sample-taxa matrices that we have access to, contain only relative abundances of the observed taxa.
The second algorithm (MixMCMC) (described in chapter 4) [50] which is introduced in this dissertation uses Markov Chain Monte Carlo (MCMC) sampling to estimate the latent parameters in the MPLN mixture model framework, and the third algorithm (MixGGM) (described in chapter 5) [50] specifically addresses the compositional nature of sequencing data (i.e., the counts in a sample-taxa matrix generated by sequencing reflect relative abundances and not absolute abundances of the taxa). For this approach, the count data are first transformed by applying the Centered Log-Ratio (CLR) transformation [46] . The CLR transformed matrix is assumed to follow a Multivariate Gaussian distribution, and the MixGGM algorithm uses the MM approach to estimate the parameters for a mixture of K Multivariate Gaussians. In all of these methods, the underlying precision matrices of the distributions associated with the mixture components correspond to the microbial networks that we care about. As it has been noted that microbial co-occurrence or interaction networks are often sparse, we extend the MixMCMC and MixGGM approaches to enable inference of sparse networks based on an l 1 -penalty model. Each of the three approaches (MixMPLN, MixMCMC, and MixGGM) are evaluated using synthetic datasets to assess recovery of the graphs that were used to create the data. We assess the performance of these methods on a comprehensive collection of graph types, consisting of band graphs, cluster graphs, scale-free graph and random graphs. MixMPLN and MixGGM are also applied to a real dataset. 
Prior to describing our mixture modeling framework, we describe a single MPLN distribution. In our discussions, we denote matrices using upper-case letters, column vectors using bold letters (upper-or lower-case), and scalar values using normal lower-case letters.
Consider an MPLN distribution with parameter set Θ = (µ, Σ), where µ represents its d-dimensional mean vector and Σ represents its d × d covariance matrix. Then, a d-dimensional count vector X = (x 1 , ..., x d ) T generated by this distribution has the following property
where P(c) denotes a Poisson distribution with mean c and N d (µ, Σ) denotes a d-dimensional multivariate normal distribution with mean µ and covariance Σ. An MPLN distribution thus has two layers, with the observed counts being generated by a mixture of independent poisson distributions whose (hidden) means follow a multivariate log normal distribution. If we use λ = (λ 1 , λ 2 , ..λ d ) to denote the latent (or hidden) variable representing the Poisson means, then the probability density function p(X|Θ) of the MPLN distribution is given by
Let X 1 , X 2 , ..X n denote n independent samples drawn from an MPLN distribution, where each X i is a d-dimensional count vector. We use X = [X 1 X 2 ..X n ] to denote the sample-taxa matrix generated from d taxa and n samples, and x ij to denote the count of the j th taxon in X i . We can estimate the parameter set Θ of this MPLN distribution using a likelihood framework by considering the d-dimensional latent variables λ 1 , λ 2 , .., and λ n associated with samples X 1 , X 2 , .., and X n respectively; let matrix Λ = [λ 1 λ 2 ..λ n ] and λ ij denote the j th entry in λ i . The log-likelihood function L(Θ|X, Λ) can be optimized using a simple iterative stepwise ascent (or conditional maximization) procedure to compute Θ and Λ. The estimated values of the parameters can be used to provide an approximation for p(X|Θ)
An analogous approach based on optimizing the log-posterior using an iterative conditional modes algorithm has been proposed previously [10] .
In our framework, we consider a mixture model involving K MPLN distributions. Let π 1 , ..., π K represent the mixing coefficients of the K components (where K l=1 π l = 1), and let p l and Θ l denote the l th component distribution and its parameter set. A d-dimensional sample vector X generated from this mixture has the following distribution:
For n independent samples X = [X 1 X 2 ..X n ], the probability distribution is given by
The general log-likelihood function is thus
where λ il is the d-dimensional latent variable associated with X i in component l. We use a maximum log-likelihood framework to estimate the parameters of the MixMPLN model from the observed data X by optimizing the function L(π 1 , ..., π K , Λ 1 , ..., Λ K , Θ 1 , ..., Θ K |X).
Optimizing the log-likelihood function using the MM principle
The MM principle is a general technique [18, 23] that has proven to be useful for tackling function instead of maximizing h(θ) directly; here, θ t denotes a fixed value of the parameter θ. The function g(θ|θ t ) is said to be a minorizer of h(θ) if:
Therefore, the first step in our MM approach is to find a minorizer function which has the required property. For this, we use the following observation that follows from the concavity property of the log function [23, 37] for m non-negative values β 1 , β 2 , ..β m :
Equation (6) can thus be lower-bounded based on this observation:
where, weight w t il is defined as follows:
We use the function on the right-hand side of equation 3.9 as the minorizer function for our problem.
Thus, we will define the new objective function (L) to be maximized as follows:
Steps of the MixMPLN optimization algorithm
We used a coordinate ascent approach in conjunction with a block update strategy to optimize L.
We present the details of parameter initialization and subsequent iterative updates below.
The n samples X 1 X 2 ..X n are partitioned into K clusters (components) using the K-means algorithm. Then, the samples assigned to a component are used to estimate the parameters of that component using the moments of an MPLN distribution [6] , given by the equations below; here, σ ij denotes the ij th entry in Σ.
Parameter estimation in iteration (t + 1)
The portion of function L (in equation 11) that is dependent on the π i 's can be optimized.
Since l π l = 1, we can optimize L 1 by introducing a Lagrange multiplier and identifying a stationary point of the subsequent Lagrangian [8] . This yields
where w t il is calculated using equation (10) .
Considering the part of the L function that is dependent on Λ and Θ, we have the following term to maximize:
Expanding p l using equation (3), we have:
where λ ijl is the j th entry in λ il . We solve for the parameters separately using the partial derivation method. Calculation of the partial derivative of L 2 with respect to λ ijl gives us:
where a, b denotes the inner product of vectors a and b, and Σ −1 .jl denotes the j th column of Σ −1 l .
We use the Newton-Raphson method to estimate λ t+1 ijl from this equation, using values from iteration t for Σ −1 , µ, and λ ij l (where j = j ).
Partial derivation with respect to µ l gives us:
Thus, µ l t+1 can be estimated from w t il and λ t+1 il .
And finally, partial derivation with respect to Σ −1 l results in :
We can solve this equation using matrix derivative rules to compute an estimate for the covariance matrix Σ l in iteration (t + 1) as:
Inferring sparse networks using an l 1 -penalty model
We extended the MixMPLN framework by incorporating an l 1 -norm penalty as follows:
where S denotes the empirical covariance matrix.
In our implementation of the extended MixMPLN framework, we calculated the empirical covariance matrix for each component using equation (20) in each iteration. We used the "glasso" and "huge" packages in R to solve the sparse precision matrix selection problem [13, 36] . We applied three different strategies using each of the two packages. In MixMPLN+glasso(cross validation), we used cross validation to determine the value of ρ (i.e. we picked that ρ value which gave the best log-likelihood value among the subsamples). In MixMPLN+glasso(fixed tuning parameter),
, as proposed in [10] . In this formulation, Σ −1 init is the estimated precision matrix after the initialization step of MixMPLN. In MixMPLN+glasso(iterative tuning parameter), we used the same formulation to initialize the ρ value, but then updated it in each iteration based on the new estimated precision matrix in that iteration. In MixMPLN+huge(StARS), we used the stability approach to regularization selection (StARS) method. This method selects the coefficient which results in the most edge stability in the final graph [21] . The tuning parameter selection method in MixMPLN+huge(fixed tuning parameter) and MixMPLN+huge(fixed tuning parameter) were the same as the corresponding implementations using glasso.
Synthetic sample-taxa count matrices were generated in order to assess the performance of the various MixMPLN algorithms. We evaluated the convergence properties of the algorithms as a function of increasing sample sizes. Since, in practice, sample-taxa count matrices generated from biological samples are compositional in nature, we also evaluated the effect of sampling from the true (or absolute) counts, and the subsequent application of data normalization, on network recovery and convergence. In addition, we also evaluated the accuracy in recovering sparse networks. Finally, we applied our mixture model framework to analyze a real dataset. component was a random d-length vector. For a sample-taxa matrix X generated this way, it is assumed that each entry in the matrix is the true (or absolute) abundance of taxon j in sample i. We refer to X as the orginal count matrix. To simulate compositional count data and sequencing depth differences between the biological samples, we generated a sampled data matrix XS from X by first normalizing each entry in a sample (by dividing by sample size) and subsequently scaling that . Finally, to study the effect of data normalization, we applied the trimmed mean of M-values (TMM) normalization procedure (from the "edgeR" package [29] ) to XS to generate the TMM normalized data matrix T S.
Real data: We also applied our mixture modeling framework to a sample-taxa count matrix produced by a recent microbiome study [35] that explored connections between gut microbiome 
Let A d×d denote the true precision matrix and B d×d an inferred (or predicted) precision matrix. For evaluating the performance of our algorithms on synthetic data, we used three different criteria to compare the inferred precision matrices with the original precision matrices that were used to generate the sample-taxa matrices.
• Relative difference between two matrices A and B defined as 1
• Frobenius norm of the difference between the partial correlations of matrices A and B.
• Area under the ROC (AUC): this measure was used to assess the recovery of the edges of the microbial network (i.e. identification of the zero entries in the precision matrix). The AUC was calculated by applying varying thresholds to the original and estimated precision matrices to define zero and non-zero entries. As any non-zero entry in the estimated precision matrix represents an edge in the graph, the specificity and sensitivity of detecting an edge at different thresholds can be computed and used to plot the ROC.
For the above measures, smaller values for relative difference and frobenius norm indicate increased proximity to the ground truth. Values closer to 1 for the AUC plots indicate increased accuracy in reconstructing the network topology. When K > 1, we first matched the predicted precision matrix (of a component) to the nearest true precision matrix from the set of K true precision matrices. We used the Frobenius norm measure for this. After pairing the true and predicted matrices, we report their mean value statistics.
We implemented the MixMPLN algorithms in R, and assessed their performance using the synthetic datasets. For our assessments, we generated sample-taxa count matrices X (and their corresponding XS and T S matrices) for the following four sets of parameters
For each dataset, each component mixing coefficient was 1/K. In addition, 5 replicates were generated for each parameter combination. In total, 465 datasets were generated and analyzed.
The datasets with sp = 0.9 were used to assess the performance of the MixMPLN algorithms in recovering sparse networks.
First we evaluated the ability of the MixMPLN algorithm to recover the true precision matrices with increasing sample size n. For this, we used the original count data, the sampled data, and the TMM normalized data. parameters. The figure also shows that the accuracy of recovery of the true precision matrices is not as high when using the sampled data (red curves/barchart) and the TMM normalized data (orange curves/barchart). In addition, from our analysis, it is not immediately evident that applying a TMM type data normalization is advantageous for the purpose of inferring the underlying covariance structure of the network.
We also evaluated the performance of MixMPLN and its l 1 -norm penalty variants in recovering sparse networks. Table 3 Since MixMPLN+glass(cross validation) performs better than the other approaches for n = 2d, we decided to apply this method to analyze the real dataset. We used the silhouette method from the "factoextra" R package [20] to compute the optimal number of components from this sample-taxa matrix. MCMC is a general framework that allows sampling from a target distribution, and can be used to estimate parameters when the target distribution is the posterior distribution. Second MC refers to Monte Carlo which is to find the approximate by averaging. The first MC is Mrakov Chain which is a mechanism to generate samples. A sequence X 0 , X 1 , ..., X n of random variables is a Markov Chain if, for any t, the conditional distribution of X t , given x t−1 , x t−2 , ..., x 0 is the same as the distribution of X t given x t−1 . In MCMC, we start navigating through the space and reject or accept the new sample until the convergence.
In our problem, the posterior distribution is P (λ l |X), the probability of the latent variable(λ l ) given the observed sample-taxa count matrix(X):
band random scale-free cluster We used the Rstan package [60] to implement the MixMCMC akgorithm. In iteration t + 1, the prior distribution of λ l to feed the MCMC is a Multivariate Gaussian distribution with mean µ t l and covariance matrix of Σ t l (where l represents the component number), and the likelihood function P (X|λ l ) for sampling λ l follows an MPLN distribution.
For each MCMC execution, we ran three chains in parallel [70] . TheR (between-chain variance relative to that of within-chain variance) [71] and n ef f (the effective sample size) [72] values were used to check convergence of these chains. While running the MCMC for each chain, we started with 1000 iterations and set half of them as warm-up iterations. After each running, convergence was assumed ifR was less than 1.1 and n ef f was higher than 100. In the case that the MCMC had not converged, we increased the number of iterations by 100 and ran the chains again. We repeated this process until convergence.
Aside from the mentioned difference (using the MCMC to estimate the latent variables of lambda), the other steps including applying CV to find the tuning parameter for the sparsity constraint in MixMCMC and MixMPLN are the same.
Ground truth datasets from real microbial communities containing information on microbial interactions, and that could be used to assess the performance of network inference methods, are currently lacking in this field. Thus, we generated realistic synthetic datasets using the Normal We used the huge package to generate three different graph types (band, cluster and scale-free) [68] ;
we also generated random graphs using our previous approach [49] . Each generated graph ( In our simulations, all generated graphs have 90% sparsity (i.e. only 10% of the possible edges are present). We used an HMP 16S rRNA dataset [67] to find the statistical features of real data, and thus d, the number of taxa, in our simulation is the same as the number of taxa in that dataset (d = 127).
Synthetic data corresponding to mixtures with K = 1, 2, and 3 components were generated, and used to assess the performance of our methods. For each evaluation, we generated three different sample sizes (d, 2d, and 3d) for each component, to reflect sample sizes often available in practice.
For each evaluation, five replicate datasets were generated, and the final performance numbers reported are the average values of the five replicates. 
We used the synthetic data generated from the different graph types for various values of K, d, and n to evaluate the methods with sparsity constraints . For each dataset, every method was run three times with different random starting points, and the solution with the largest log-likelihood value was selected for that method. All methods were assessed with respect to their ability to recover the 
), ..., log( x d g(X)
)] (5.1)
where g(X) = d d i=1 x i is the geometric mean.
Let matrix W denote the absolute abundances of the taxa in the samples. Suppose that Ω = Cov(log(W )) is the covariance of the log of the absolute count data that we are interested in estimating, and Γ = Cov(clr(X)) is the covariance of X after applying the CLR transformation to it. Then, these covariance matrices are related as follows [45] :
where G = I d − 1 d J, I d is a d × d identity matrix and J is a d × d matrix with all 1's. When d is large, G is close to the identity matrix, so an approximation of Ω by Γ is reasonable. For the MixGGM approach, we assume that the transformed data Z follows a Multivariate Gaussian distribution [46] .
MixGGM models the CLR transformed data Z by a mixture of Multivariate Gaussian distributions, which is defined by following equation:
Where p l (Z i |Φ l ) follows the normal distribution:
Therefore, the general log-likelihood function is:
where Φ l includes µ and Σ for the l th component. We also use the MM framework to maximize the log-likelihood function L and estimate the MixGGM parameters.
Minorizer function : The first step in applying the MM framework is to find a minorizer. We use equation (3.9) to find the minorizer for function in equation (5.5):
where, weight w t il is defined as follows:
Parameter estimation: The process for updating the mixing coefficients, the mean vector and the covariance matrix for each component in iteration t + 1 is similar to that employed for the MixMPLN algorithm [49] :
Applying sparsity constraint: The solution to the problem of inferring a sparse precision matrix for a Multivariate Gaussian model when we have the empirical precision matrix (S) and tuning parameter (ρ l ) for the l 1 -norm has been proposed previously (the graphical lasso) [44] . The graphical lasso formulation is as follows:
In the MixGGM approach, this constraint is applied separately for all K components. We used the huge package [55] in our implementation to solve the graphical lasso. In our previous work [49] , several strategies were evaluated to select the tuning parameter. Based on those results, we use Cross Validation (CV) strategy for parameter tuning. The CV is based on dividing the data into random subgroups, and selecting the parameter which results in the best likelihood value in these comparisons.
We used the synthetic data generated from the different graph types for various values of K, d, and n to evaluate the methods with sparsity constraints . For each dataset, every method was run three times with different random starting points, and the solution with the largest log-likelihood value was selected for that method. All methods were assessed with respect to their ability to recover the original graph topology, and thus, the area under the ROC curve (AUC) was used as the performance The performance results on band, cluster, random and scale-free graphs are given in Tables 5.1 ability of a method to recover the original graph is a function of the graph type. In general, the methods tend to be able to recover the topologies for band graphs and cluster graphs more accurately in comparison to topologies for scale-free graphs and random graphs (this is reflected in the comparatively higher AUC values for band and cluster graphs for each method). (c) MixGGM, which uses log-ratio transformation to deal with the compositionality of the data, generally tends to outperform MixMPLN on the different graph types.
• (Band graph) When K = 1, MixMPLN and MixGGM have comparable performances. For K > 1 MixGGM performs slightly better than MixMPLN (except for K = 3, n = 3d).
( • (Random graph) MixGGM performs better when K > 1 and n = 3d (although the difference range is 2% ). For the other cases, the two methods are comparable. (Table 5. 3)
• (Scale-free graph) When K = 1 and any value of n, MixGGM performs better. The other case which MixGGM works better is K = 3 and n = 3d where its AUC value is 3% higher than MixMPLN result. (Table 5 .4) (d) In addition, the average runtime of MixGGM is less than that of MixMPLN.
We applied MixGGM and MixMPLN to a genus-level sample-taxa matrix from 16S data generated from stool samples (HMP [74] ). The data consists of taxa counts derived from 319 stool samples and 95 genera. We set the number of components K = 1. Each method was run three times with different random starting points, and the model with the largest log-likelihood value was selected for that method. While we do not know the ground-truth network(s) for this dataset, it is possible to evaluate the consistency of the results generated by the two methods. 
In this chapter, we introduced MixGGM, to infer multiple networks from microbial count data.
We used synthetic datasets to evaluate these approaches with respect to recovering the original graph topologies, for different types of graphs. While the AUC values increase with increasing sample size of the input data, we observe that this is a function of the graph type. The methods tend to be able to recover the topologies for band graphs and cluster graphs more accurately in q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q comparison to topologies for scale-free graphs and random graphs. Another observation is that
MixGGM (which uses log-ratio transformation) generally tends to perform better compared to
MixMPLN. However, the MPLN distribution, which forms the basis of the MixMPLN framework, allows for the inclusion of biological and/or technical co-variates in the Poisson layer to control for confounding factors [47, 75] .
In the next chapter (chapter 6), we explores methods to estimate the number of components in a mixture model, and will introduce a novel and practical framework for model selection issue in modeling microbial data in a mixture framework.
So far, in all the proposed algorithms, the number of components(K) was assumed to be known.
This chapter's content is to find a framework to estimate the best number of components in modeling a sample-taxa matrix as a mixture model. As we are not aware of the original distribution which generated the data, we do not know the ground truth of our component generators and a technique is needed to find the best number of components. The best number of the components will be K which is considered as the input of the previous introduced algorithms to infer the microbial interactions.
There are many methods to find the optimal number of components some of which are not accurate enough or computationally expensive. One simple approach is to consider non-parametric methods for selecting the number of components such as calculating the Silhouete score [1] . This score checks how much the clusters are compact and well separated by calculating the mean distance between a sample and all other points in the same cluster and the mean distance between a sample and all other points in the next nearest cluster. The Silhouete score will be calculated for different number of components and the number of components with the higher Silhouete score is selected as the optimal one.
Another criterion to detect the best number of components in a mixture model comes from a Bayesian framework for model selection. The model should be run for different values for number of components and best model will be detected based on the decision criteria. This model selection criteria is trading off between the log-likelihood and model complexity in order to avoid over-fitting.
Among these types of methods we note here the Bayesian Information Criterion (BIC) [2] . The
Bayesian information criterion (BIC) is define as:
Where L is the likelihood function of the θ parameters given data. k is number of components and n is number of parameters in the model.
In finding the best number of components using BIC method, the best model will be the one that has the lowest BIC.
Reversible Jump MCMC (RJMCMC) [4] is another method which gives us the opportunity to choose from a collection of models, but it is computationally intense. Unlike other sampling algorithms, e.g. Gibbs sampling [64] or the generic Metropolis-Hastings algorithms [69] , this method explores different submodels and bridges between them by jumping across spaces of varying dimension to find the best generalized fit to the data.
All the methods which are mentioned so far need to fit the data to the model several times using different number of components to find the best model. Thus, finding the optimal number of the component associated with the data can be computationally expensive. This fact is the reason that we are interested in the Variational Inference(VI) [78] algorithm which can be used to estimate the number of components by running the algorithm on the data once instead of running it several times for different models.
VI optimizes the marginal log-likelihood function associated with the data instead of the regular log-likelihood function and provides approximate solutions to the inference problem. It gives us the opportunity to consider a prior distribution for a parameter in the model. In other words, each point parameter (called hidden variable) will be replaced with a distribution (called variational posterior).
Optimizing the marginal log-likelihood function, tries to make the variational posteriors as close as possible to the true posteriors. This approach runs the mixture model on a large fixed number of components when the only parameters are mixing coefficients and everything else is a hidden variable. Therefore, mixing coefficients corresponding to unwanted components will go to zero after the convergence [80] .
In the following equations, Q(θ) includes all the posteriors for the hidden variables of θ and only π is the parameter including the mixing coefficients to be optimized. Following equations [65, 78] show how we calculate new objective function in VI.
ln(P (D|π)) = ln P (D, θ|π)dθ (6.1)
Where D is data, θ includes all the hidden variables and π is the parameter which includes all the mixing coefficients.
Equation 6 .1 can be written as following equation:
Where the lower bound for equation 6.2 can be written as:
Based on previous equation, we can define the Evidence Lower Bound (Elbo) function which is our marginal log-likelihood [78] .
Elbo(θ) = Q(θ)ln P (D, θ|π) Q(θ) )dθ (6.4) In the previous equation, Elbo function which is the lower bound for the marginal log-likelihood function is shown. The Elbo function is our objective function to maximise in VI for model selection.
One way to solve the explained optimization problem is to find the updating formulas for the unknown parameters. By considering a specific situation that general posterior distribution has a conjugate form which is indicated in equation(6.5), the optimal posterior distributions are then given by equation(6.6) [78] [53] [54] :
optimal posterior for each parameter is calculated as [78] :
Knowing the following fact, we can easily obtain the updating formulas for finding the optimal parameters.
Another way to solve the variational Inference is to use Variational Auto Encoder(VAE)(depicted in Figure 6 .1). VAE is used to reconstruct the input data, by setting the loss to minimise the distance between the input and reconstructed data in output layer. By defining a proper loss and considering the KL divergence between the prior and posterior distribution of the parameters we can get to the optimal variables for the variational inference problem. Which can be written as [81] :
Knowing the definition of KL divergence, we can get to a proper Elbo function for VAE in modeling the VI [81] :
In the next section, we applied the tensorflow implementation for the Gaussian mixture model and extended it to the Poisson log normal distribution.
Implementing the VI for model selection in tensorflow framework
We have used the tensorflow framework to optimize the mentioned Elbo function by setting proper prior distribution for mixture of Gaussian distribution:
For the Gaussian mixture model the prior distributions for the parameters are described by the following equations:
Where k is number of components, µ k , δ k are mean and standard deviation corresponding to the k th component.
The only parameters in the model are the mixing coefficients which are forced to sum up to one.
Following equations show the prior distribution that we selected for the mean and the inverse of variances:
In the preprocessing stage, Principle Component Analysis(PCA) is applied to the normalized data.
We selected number of components by setting 90% cut-off threshold on cumulative variance. After PCA another normalization is applied to the data to make the convergence in the optimization process faster.
One main disadvantage of using tensorflow in solving the VI for model selection is that for any new dataset, all the hyper-parameters of the optimization should be specialized in order to get to the steady state level in the loss function value. These hyper-parameters include learning rate, number of iterations, batch size and initial points for parameters inside the distributions. However we can get to the optimal number of components in this method, the mentioned disadvantage makes it impractical. 
In the next step, we extended our implementation for the variational inference in tensorflow for mixture of poisson log normal distribution by setting the proper distributions as following equations:
Here data for each component is coming from a Poisson Log-Normal distribution, where λ is latent variable , i number of sample and k number of component.
The only parameters in the model are the mixing coefficients which are forced to sum up to one.
Prior for hidden variables are defines the same as previous simulation: As it is shown in Figure 6 .1, Figure 6 .2 and Figure 6 .3, if we get to the steady state of the loss function, we will find the optimal number of the components. The issue is tuning the hyper-parameters for each new input data to get to the steady state in the loss function. This issue makes the VI with tensorflow an impractical framework and persuades us to find a practical framework to solve the VI for model selection.
The mentioned reason motivated us to derive the formula directly as will be explained in the next sections.
As it is mentioned in the previous section, using tensorflow to solve the VI for model selection is not practical. This issue motivates us to find a closed form solution for the problem. We have included the formula derivation for the model selection using the Variational Inference in Gaussian mixture model in this section [80] . As we discussed previously to estimate the number of components using the VI, we optimise the marginal log-likelihood instead of regular log-likelihood function. Here we consider the posterior distribution of variables as multiplication of posterior for each variable, thus we are able to use equation 6.6 to find the optimal posterior functions [80] .
P (D, µ, T, S) = P (D|µ, T, S)P (S|π)P (µ))P (T ) (6.10)
Where D is data, µ, T are mean, precision matrices for all components. S is latent variable. s in = 1
when i th sample is coming from n th component, and s in = 0 vice versa. π is the only parameter in the model which includes mixing coefficients.
Where N is number of samples, M is number of components. Using latent variable S, we can re write the equation 6.11 as following equation:
Prior distributions are shown in following equations. Drichlete distribution for S, Normal distribution for mean vectors and Wishart distribution for Precision matrices :
N (0, βI) (6.14)
Next equations shows posterior distributions for hidden variables: 
x n < S in > (6.23) 
calculating the expectations---------------------------- P (D|λ) ∼ P oisson(e λ ) (6.36)
Where N is number of samples, M is number of components. Using latent variable S, we can re write the equation 6.11 as following equation:
Prior distributions are shown in following equations. Drichlete distribution for S, Normal distribution for mean vectors and Wishart distribution for Precision matrices :
Next equations shows posterior distributions for hidden variables:
The difference between the formulation here and the GMM is that here in addition to previous hidden variables we have a latent variable of λ. Therefore we need to marginalize the equations based on the λ too.
We are able to use equation 6.6 to derive closed form updating equations for all the parameters but 
calculating the expectations----------------------------
calculating the Elbo function--------------------------- To evaluate our algorithm, we generated sample-taxa count matrices X for the following four sets In addition, 10 replicates were generated for each parameter combination. In total, 20 datasets were generated for each specific value of K.
In the preprocessing stage, Principle Component Analysis(PCA) is applied to the normalized data.
We selected number of components by setting 90% cut-off threshold on cumulative variance. After PCA another normalization is applied to the data to make the convergence in the optimization process faster.
Initial value for v degrees of freedom is selected the same as detected number of components in PCA. larger values of v (for fixed scale matrix V) results in smaller values in Precision Matrix.
For scale matrix V as diagonal matrix, larger values on diagonal tend to make Precision Matrix to have smaller entries; and vice versa. Initial matrix for V was selected a d × d diagonal matrix with diagonal entries equal to 1e-1. For beta, initial small value of 1e-6 is selected. In this chapter, we presented a practical and efficient framework to find the number of components (K) in modeling a sample-taxa matrices with a mixture model. Previous methods such as BIC are not computationally efficient as they need to run the training dataset on several different models to find the best one, while VI gets the best model in one training run. Furthermore they are not as precise as VI. We have also showed that implementing VI for model selection in tensorflow is not practical as needs the hyper parameters to be tuned for each new dataset. We have shown that by driving a closed form formulation for VI in model selection, we will have a precise, practical and computationally efficient method compared to the previous ones.
In this chapter, we presented a practical and efficient framework to find the number of components (K) in modeling a sample-taxa matrices with a mixture model. Previous methods such as BIC are not computationally efficient as they need to run the training dataset on several different models to find the best one, while VI gets the best model in one training run. Furthermore they are not as precise as VI. We have also showed that implementing VI for model selection in tensorflow is not 68 practical as needs the hyper parameters to be tuned for each new dataset. We have shown that by driving a closed form formulation for VI in model selection, we will have a precise, practical and computationally efficient method compared to the previous ones.
This dissertation addressed the scenario when the sample-taxa matrix is associated with K microbial networks and considers the computational problem of inferring K microbial networks from a given sample-taxa matrix.
In chapter one, we described the motivation behind defining the mentioned computational problem.
A literature review on previous methods to infer one single microbial network from a microbial data was included in the next chapter.
In chapter three, we presented MixMPLN, a mixture model framework and network inference algorithms, to analyze sample-taxa matrices that are associated with K microbial networks.
In chapter four, We introduced MixMCMC to infer multiple networks from microbial count data. It is based on the MM principle applied in a mixture model framework and build off of the MixM-PLN framework that we introduced previously [59] . We used synthetic datasets to evaluate these approaches with respect to recovering the original graph topologies, for different types of graphs.
While the AUC values increase with increasing sample size of the input data, we observe that this is a function of the graph type. The methods tend to be able to recover the topologies for band graphs and cluster graphs more accurately in comparison to topologies for scale-free graphs and random graphs. However, MixMCMC is significantly slower.
In chapter five, we introduced MixGGM, to infer multiple networks from microbial count data.
We used synthetic datasets to evaluate these approaches with respect to recovering the original graph topologies, for different types of graphs. While the AUC values increase with increasing sample size of the input data, we observe that this is a function of the graph type. The methods tend to be able to recover the topologies for band graphs and cluster graphs more accurately in comparison to topologies for scale-free graphs and random graphs. Another observation is that
MixGGM (which uses log-ratio transformation) generally tends to perform better compared to MixMPLN. However, the MPLN distribution, which forms the basis of the MixMPLN framework, allows for the inclusion of biological and/or technical co-variates in the Poisson layer to control for confounding factors [47, 75] .
In the next chapter (chapter 6), we presented a practical and efficient framework to find the number of components (K) in modeling a sample-taxa matrices with a mixture model. Previous methods such as BIC are not computationally efficient as they need to run the training dataset on several different models to find the best one, while VI gets the best model in one training run. Furthermore they are not as precise as VI. We have also showed that implementing VI for model selection in tensorflow is not practical as needs the hyper parameters to be tuned for each new dataset. We have shown that by driving a closed form formulation for VI in model selection, we will have a precise, practical and computationally efficient method compared to the previous ones.
The contributions of this dissertation include 1) new frameworks to generate synthetic sample-taxa count data; 2)novel methods to combine mixture modeling with probabilistic graphical models to infer multiple interaction/association networks from microbial count data; 3) dealing with the compositionality aspect of microbial count data;4) extensive experiments on real and synthetic data; 5)new methods for model selection to infer correct value of K.