Dynamic Hybrid Random Fields for the Probabilistic Graphical Modeling of Sequential Data: Definitions, Algorithms, and an Application to Bioinformatics

Abstract

The paper introduces a dynamic extension of the hybrid random field (HRF), called dynamic HRF (D-HRF). The D-HRF is aimed at the probabilistic graphical modeling of arbitrary-length sequences of sets of (time-dependent) discrete random variables under Markov assumptions. Suitable maximum likelihood algorithms for learning the parameters and the structure of the D-HRF are presented. The D-HRF inherits the computational efficiency and the modeling capabilities of HRFs, subsuming both dynamic Bayesian networks and Markov random fields. The behavior of the D-HRF is first evaluated empirically on synthetic data drawn from probabilistic distributions having known form. Then, D-HRFs (combined with a recurrent autoencoder) are successfully applied to the prediction of the disulfide-bonding state of cysteines from the primary structure of proteins in the Protein Data Bank.

Appendices

A Formal Definition of HRF and Related Notions

The material presented in this Appendix and in the next one is drawn from [15]. HRFs are aimed at representing joint probability distributions underlying sets of random variables. The definition of HRF involves a few preliminary notions concerning directed and undirected graphs. First, let us define the concept of (directed) union of directed graphs:

Definition 1

If $\mathcal {G}_{1} = (\mathcal {V}_{1}, \mathcal {E}_{1})$ and $\mathcal {G}_{2} = (\mathcal {V}_{2}, \mathcal {E}_{2})$ are directed graphs, then the (directed) union of $\mathcal {G}_{1}$ and $\mathcal {G}_{2}$ (denoted by $\mathcal {G}_{1} \cup \mathcal {G}_{2}$) is the directed graph $\mathcal {G} = (\mathcal {V}, \mathcal {E})$ such that $\mathcal {V} = \mathcal {V}_{1} \cup \mathcal {V}_{2}$ and $\mathcal {E} = \mathcal {E}_{1} \cup \mathcal {E}_{2}$.

Given a set of directed graphs $\mathcal {G}_{1}, \ldots , \mathcal {G}_{n}$, the directed union $\mathcal {G} = \bigcup _{i=1}^{n} \mathcal {G}_{i}$ results from iterated application of the binary union operator. Similarly, we define the notion of (undirected) union for undirected graphs as follows:

Definition 2

If $\mathcal {G}_{1} = (\mathcal {V}_{1}, \mathcal {E}_{1})$ and $\mathcal {G}_{2} = (\mathcal {V}_{2}, \mathcal {E}_{2})$ are undirected graphs, then the (undirected) union of $\mathcal {G}_{1}$ and $\mathcal {G}_{2}$ (denoted by $\mathcal {G}_{1} \cup \mathcal {G}_{2}$) is the undirected graph $\mathcal {G} = (\mathcal {V}, \mathcal {E})$ such that $\mathcal {V} = \mathcal {V}_{1} \cup \mathcal {V}_{2}$ and $\mathcal {E} = \mathcal {E}_{1} \cup \mathcal {E}_{2}$.

Now, if $\mathcal {G} = (\mathcal {V}, \mathcal {E})$ is a directed graph, then we say that $\mathcal {G}^{*} = (\mathcal {V}^{*}, \mathcal {E}^{*})$ is the undirected version of $\mathcal {G}$ if $\mathcal {V}^{*} = \mathcal {V}$ and $\mathcal {E}^{*} = \{\{X_{i}, X_{j}\}: (X_{i}, X_{j}) \in \mathcal {E}\}$. Thus, we also define the undirected union for a pair of directed graphs:

Definition 3

If $\mathcal {G}_{1} = (\mathcal {V}_{1}, \mathcal {E}_{1})$ and $\mathcal {G}_{2} = (\mathcal {V}_{2}, \mathcal {E}_{2})$ are directed graphs, then the undirected union of $\mathcal {G}_{1}$ and $\mathcal {G}_{2}$ (denoted by $\mathcal {G}_{1} \Cup \mathcal {G}_{2}$) is the (undirected) graph $\mathcal {G}^{*} = \mathcal {G}_{1}^{*} \cup \mathcal {G}_{2}^{*}$ such that $\mathcal {G}_{1}^{*}$ and $\mathcal {G}_{2}^{*}$ are the undirected versions of $\mathcal {G}_{1}$ and $\mathcal {G}_{2}$ respectively.

Finally, a HRF can be defined as follows:

Definition 4

Let $\mathbf {X}$ be a set of random variables $X_{1}, \ldots , X_{n}$. A hybrid random field for $X_{1}, \ldots , X_{n}$ is a set of Bayesian networks $BN_{1}, \ldots , BN_{n}$ (with DAGs $\mathcal {G}_{1}, \ldots , \mathcal {G}_{n}$) such that:

• 1. Each $BN_{i}$ contains $X_{i}$ plus a subset $\mathcal {R}(X_{i})$ of $\mathbf {X} \setminus \{X_{i}\}$;

• 2. If $\mathcal {MB}_{i}(X_{i})$ denotes the Markov blanket [36] of $X_{i}$ in $BN_{i}$ (i.e. the set of the parents, the children, and the parents of the children of $X_{i}$ in $\mathcal {G}_{i}$) and $P(X_{i} \mid \mathcal {MB}_{i}(X_{i}))$ is the conditional distribution of $X_{i}$ given $\mathcal {MB}_{i}(X_{i})$, as derivable from $BN_{i}$, then at least one of the following conditions is satisfied:

  • (a) The directed graph $\mathcal {G} = \bigcup _{i=1}^{n} \mathcal {G}_{i}$ is acyclic and there is a Bayesian network $h_{\mathcal {G}}$ with DAG $\mathcal {G}$ such that, for each $X_{i}$ in $\mathbf {X}$, $P(X_{i} \mid \mathcal {MB}_{i}(X_{i})) = P(X_{i} \mid \mathcal {MB}_{\mathcal {G}}(X_{i}))$, where $\mathcal {MB}_{\mathcal {G}}(X_{i})$ is the Markov blanket of $X_{i}$ in $h_{\mathcal {G}}$ and $P(X_{i} \mid \mathcal {MB}_{\mathcal {G}}(X_{i}))$ is the conditional distribution of $X_{i}$ given $\mathcal {MB}_{\mathcal {G}}(X_{i})$, as entailed by $h_{\mathcal {G}}$;

  • (b) There is a Markov random field $h_{\mathcal {G}^{*}}$ with graph $\mathcal {G}^{*}$ such that $\mathcal {G}^{*} = \mathcal {G}_{1} \Cup \ldots \Cup \mathcal {G}_{n}$ and, for each $X_{i}$ in $\mathbf {X}$, $P(X_{i} \mid \mathcal {MB}_{i}(X_{i})) = P(X_{i} \mid \mathcal {MB}_{\mathcal {G}^{*}}(X_{i}))$, where $\mathcal {MB}_{\mathcal {G}^{*}}(X_{i})$ is the Markov blanket of $X_{i}$ in $h_{\mathcal {G}^{*}}$ and $P(X_{i} \mid \mathcal {MB}_{\mathcal {G}^{*}}(X_{i}))$ is the conditional distribution of $X_{i}$ given $\mathcal {MB}_{\mathcal {G}^{*}}(X_{i})$, as derived from $h_{\mathcal {G}^{*}}$.

The elements of $\mathcal {R}(X_{i})$ are called ‘relatives of $X_{i}$’. That is, the relatives of a node $X_{i}$ in a HRF are the nodes appearing in graph $\mathcal {G}_{i}$ (except for $X_{i}$ itself). An example of HRF over the variables $X_1, \ldots , X_4$ is shown in Fig. 10.

Fig. 10

The graphical components of a hybrid random field for the variables $X_{1}, \ldots , X_{4}$. Since each node $X_{i}$ has its own Bayesian network (where nodes in $\mathcal {MB}_{i}(X_{i})$ are shaded), there are four different DAGs ${\mathcal {G}}_1, \ldots , {\mathcal {G}}_4$. Relatives of $X_{i}$ that are not in $\mathcal {MB}_{i}(X_{i})$ are dashed

Two fundamental properties of HRFs stem from the following Modularity Theorem, whose proof is given in [15].

Theorem 1

Suppose that h is a hybrid random field for the random vector $\mathbf {X} = (X_{1}, \ldots , X_{n})$, and let h be made up by Bayesian networks $BN_{1}, \ldots , BN_{n}$ (with DAGs $\mathcal {G}_{1}, \ldots , \mathcal {G}_{n}$). Then, if each conditional distribution $P(X_{i} \mid \mathcal {MB}_{i}(X_{i}))$ is strictly positive (where $1 \le i \le n$), h has the following properties:

• 1. An ordered Gibbs sampler [22] applied to h, i.e. an ordered Gibbs sampler which is applied to the conditional distributions $P(X_{1} \mid \mathcal {MB}_{1}(X_{1})), \ldots , P(X_{n} \mid \mathcal {MB}_{n}(X_{n}))$, defines a joint probability distribution over $\mathbf {X}$ via its (unique) stationary distribution $P(\mathbf {X})$;

• 2. For each variable $X_{i}$ in $\mathbf {X}$, $P(\mathbf {X})$ is such that $P(X_{i} \mid \mathbf {X} \setminus \{X_{i}\}) = P(X_{i} \mid \mathcal {MB}_{i}(X_{i}))$.

The first condition in Theorem 1 is exploited in D-HRFs in order to express the overall likelihood of the D-HRF over an input sequence in terms of local, state-specific joint probability distributions (namely, the emission probabilities) modeled via HRFs. In turn, we refer to the second condition in Theorem 1 as the modularity property. Based on the modularity property, the set $\mathcal {MB}_{i}(X_{i})$ is a Markov blanket of $X_{i}$ in $\mathbf {X}$.

B Likelihood Versus Pseudo-Likelihood in HRFs

Let ${{\mathcal {X}}} = \{X_{1}, \ldots , X_{n}\}$ be any set of n discrete random variables of interest. In order to extract the joint probability $P({{\mathcal {X}}})$ from a HRF [15], Gibbs sampling techniques need to be used [15]. Unfortunately, Gibbs sampling can be computationally intractable [22]. Therefore, in practice we can follow in the footsteps of [3] and resort to the pseudo-likelihood function, defined as :

$$\begin{aligned} P^{*}({{\mathcal {X}}}) = \prod _{i=1}^{n} P(X_{i} \mid {{\mathcal {X}}} \setminus \{X_{i}\}) \end{aligned}$$

(29)

A discussion of the theoretical properties of the pseudo-likelihood is offered by [26]. Due to the modularity property, Eq. (29) can be rewritten as:

$$\begin{aligned} P^{*}({{\mathcal {X}}}) = \prod _{i=1}^{n} P(X_{i} \mid \mathcal {MB}_{i}(X_{i})) \end{aligned}$$

(30)

Therefore, in order to measure the pseudo-likelihood $P^{*}({{\mathcal {X}}})$ of the HRF given ${{\mathcal {X}}}$ we only need to be able to compute the conditional distribution of each node $X_{i}$ given the state $mb_{i}(X_{i})$ of $\mathcal {MB}_{i}(X_{i})$. A discussion of how this can be done in a simple and efficient way is given in [15].