21.2: Structure Inference

Key Questions in Structure Inteference

How to choose network models? A number of models exist for representing networks, a key problem is choosing between them based on data and predicted dynamics.

How to choose learning methods? Two broad methods exist for learning networks. Unsupervised methods attempt to infer relationships for unalabeled datapoints and will be described in sections to come. Supervised methods take a subset of network edges known to be regulatory, and learn a classifier to predict new ones.²

How to incorporate data? A variety of data sources can be used to learn and build networks including Motifs, ChIP binding assays, and expression. Data sources are always expanding; expanding availability of data is at the heart of the current revolution in analyzing biological networks.

Abstract Mathematical Representations for Networks

Think of a network as a function, a black box. Regulatory networks for example, take input expressions of regulators and spit out output expression of targets. Models differ in choosing the nature of functions and assigning meaning to nodes and edges.

Boolean Network This model discretizes node expression levels and interactions. Functions represented by edges are logic gates.

Differential Equation Model These models capture network dynamics. Expression rate changes are func- tion of expression levels and rates of change of regulators. For these it can be very difficult to estimate parameters. Where do you find data for systems out of equilibrium?

Probabilistic Graphical Model These systems model networks as a joint probability distribution over random variables. Edges represent conditional dependencies. Probabilistic graphical models (PGMs) are focused on in the lecture.

Probabilistic Graphical Models

Probabilistic graphical models (PGMs) are trainable and able to deal with noise and thus they are good Bayesian Network Directed graphical technique.³ In PGMs, nodes can be transcription factors or genes and they are modeled by random variables. If you know the joint distribution over these random variables, you can build the network as a PGMs. Since this graph structure is a compact representation of the network, we can work with it easily and accomplish learning tasks. Examples of PGMS include:

Bayesian Network Directed graphical technique. Every node is either a parent or a child. Parents fully determine the state of children but their states may not be available to the experimenter. The network structure describes the full joint probablility distribution of the network as a product of individual distributions for the nodes. By breaking up the network into local potentials, computational complexity is drastically reduced.

Dynamic Bayesian Network Directed graphical technique. Static bayesian networks do not allow cyclic dependencies but we can try to model them with bayesian networks allowing arbitrary dependencies between nodes at di↵erent time points. Thus cyclic dependencies are allowed as the network progresses through time and the network joint probability itself can be described as a joint over all times.

Markov Random Field Undirected graphical technique. Models potentials in terms of cliques. Allows modelling of general graphs including cyclic ones with higher order than pairwise dependencies.

Factor Graph Undirected graphical technique. Factor graphs introduce “factor” nodes specifying interac- tion potentials along edges. Factor nodes can also be introduced to model higher order potentials than pairwise.

It is easiest to learn networks for Bayesian models. Markov random fields and factor graphs require determination of a tricky partition function. To encode network structure, it is only necessary to assign random variables to TFs and genes and then model the joint probability distribution.

Bayesian networks provide compact representations of JPD

The main strength of Bayesian networks comes from the simplicity of their decomposition into parents and children. Because the networks are directed, the full joint probability distribution decomposes into a product of conditional distributions, one for each node in the network.⁴

Network Inference From Expression Data

Using expression data and prior knowledge, the goal of network inference is to produce a network graph. Graphs will be undirected or directed. Regulatory networks for example will often be directed while expression nets for example will be undirected.

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²Supervised methods will not be addressed today.
³These are Dr. Roys models of choice for dealing with biological nets.