Bayesian Networks
What are Bayesian Networks?
Bayesian Networks – also known as Bayes network, Bayes model, belief network, and decision network, is a graph-based model representing a set of variables and their dependencies
Bayesian networks are a type of probabilistic graphical model that uses Bayesian inference for probability computations. Bayesian networks aim to model conditional dependence, and therefore causation, by representing conditional dependence by edges in a directed graph. Through these relationships, one can efficiently conduct inference on the random variables in the graph through the use of factors.
Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that anyone of several possible known causes was the contributing factor. An example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.
Efficient algorithms can perform inference and learn in Bayesian networks. Bayesian networks that model sequences of variables (e.g. speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams.
Directed acyclic graphs
Bayesian networks are directed acyclic graphs (DAGs) whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters, or hypotheses. Edges represent conditional dependencies; nodes that are not connected (no path connects one node to another) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node’s parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if {\displaystyle m} parent nodes represent {\displaystyle m} Boolean variables, then the probability function could be represented by a table of {\displaystyle 2^{m}} entries, one entry for each of the {\displaystyle 2^{m}} possible parent combinations. Similar ideas may be applied to undirected, and possibly cyclic, graphs such as Markov networks.
Bayesian networks perform three main inference tasks:
- Inferring unobserved variables
Because a Bayesian network is a complete model for its variables and their relationships, it can be used to answer probabilistic queries about them. For example, the network can be used to update knowledge of the state of a subset of variables when other variables (the evidence variables) are observed. This process of computing the posterior distribution of variables given evidence is called probabilistic inference. The posterior gives a universal sufficient statistic for detection applications when choosing values for the variable subset that minimize some expected loss function, for instance, the probability of decision error. A Bayesian network can thus be considered a mechanism for automatically applying Bayes’ theorem to complex problems.
The most common exact inference methods are variable elimination, which eliminates (by integration or summation) the non-observed non-query variables one by one by distributing the sum over the product; clique tree propagation, which caches the computation so that many variables can be queried at one time and new evidence can be propagated quickly; and recursive conditioning and AND/OR search, which allow for a space-time tradeoff and match the efficiency of variable elimination when enough space is used. All of these methods have complexity that is exponential in the network’s treewidth. The most common approximate inference algorithms are importance sampling, stochastic MCMC simulation, mini-bucket elimination, loopy belief propagation, generalized belief propagation, and variational methods.
2. Parameter learning
In order to fully specify the Bayesian network and thus fully represent the joint probability distribution, it is necessary to specify for each node X the probability distribution for X conditional upon X’s parents. The distribution of X conditional upon its parents may have any form. It is common to work with discrete or Gaussian distributions since that simplifies calculations. Sometimes only constraints on distribution are known; one can then use the principle of maximum entropy to determine a single distribution, the one with the greatest entropy given the constraints. (Analogously, in the specific context of a dynamic Bayesian network, the conditional distribution for the hidden state’s temporal evolution is commonly specified to maximize the entropy rate of the implied stochastic process.)
Often these conditional distributions include parameters that are unknown and must be estimated from data, e.g., via the maximum likelihood approach. Direct maximization of the likelihood (or of the posterior probability) is often complex given unobserved variables. A classical approach to this problem is the expectation-maximization algorithm, which alternates computing expected values of the unobserved variables conditional on observed data, with maximizing the complete likelihood (or posterior) assuming that previously computed expected values are correct. Under mild regularity conditions, this process converges on maximum likelihood (or maximum posterior) values for parameters.
A more fully Bayesian approach to parameters is to treat them as additional unobserved variables and to compute a full posterior distribution over all nodes conditional upon observed data, then to integrate out the parameters. This approach can be expensive and lead to large dimension models, making classical parameter-setting approaches more tractable.
3. Structure learning
In the simplest case, a Bayesian network is specified by an expert and is then used to perform inference. Other applications, the task of defining the network is too complex for humans. In this case, the network structure and the parameters of the local distributions must be learned from data.
Automatically learning the graph structure of a Bayesian network (BN) is a challenge pursued within machine learning. The basic idea goes back to a recovery algorithm developed by Rebane and Pearl and rests on the distinction between the three possible patterns allowed in a 3-node DAG:
Junction patterns | |
Pattern | Model |
Chain | {\displaystyle X\rightarrow Y\rightarrow Z} |
Fork | {\displaystyle X\leftarrow Y\rightarrow Z} |
Collider | {\displaystyle X\rightarrow Y\leftarrow Z} |
The first 2 represent the same dependencies ({\displaystyle X} and {\displaystyle Z} are independent given {\displaystyle Y}) and are, therefore, indistinguishable. The collider, however, can be uniquely identified, since {\displaystyle X} and {\displaystyle Z} are marginally independent and all other pairs are dependent. Thus, while the skeletons (the graphs stripped of arrows) of these three triplets are identical, the directionality of the arrows is partially identifiable. The same distinction applies when {\displaystyle X} and {\displaystyle Z} have common parents, except that one must first condition on those parents. Algorithms have been developed to systematically determine the skeleton of the underlying graph and, then, orient all arrows whose directionality is dictated by the conditional independence observed.
An alternative method of structural learning uses an optimization-based search. It requires a scoring function and a search strategy. A common scoring function is the posterior probability of the structure given the training data, like the BIC or the BDeu. The time requirement of an exhaustive search returning a structure that maximizes the score is superexponential in the number of variables. A local search strategy makes incremental changes aimed at improving the score of the structure. A global search algorithm like Markov chain Monte Carlo can avoid getting trapped in local minima. Friedman et al. discuss using mutual information between variables and finding a structure that maximizes this. They do this by restricting the parent candidate set to k nodes and exhaustively searching therein.
A particularly fast method for exact BN learning is to cast the problem as an optimization problem and solve it using integer programming. Acyclicity constraints are added to the integer program (IP) during solving in the form of cutting planes. Such a method can handle problems with up to 100 variables.
In order to deal with problems with thousands of variables, a different approach is necessary. One is to first sample one order, and then find the optimal BN structure with respect to that ordering. This implies working on the search space of the possible orderings, which is convenient as it is smaller than the space of network structures. Multiple orderings are then sampled and evaluated. This method has been proven to be the best available in literature when the number of variables is huge.
Another method consists of focusing on the sub-class of decomposable models, for which the MLE has a closed-form. It is then possible to discover a consistent structure for hundreds of variables.
Learning Bayesian networks with bounded treewidth is necessary to allow exact, tractable inference since the worst-case inference complexity is exponential in the treewidth k (under the exponential time hypothesis). Yet, as a global property of the graph, it considerably increases the difficulty of the learning process. In this context, it is possible to use K-tree for effective learning.
Conclusion
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