Probabilistic inference and factor graphs
This documents presents a high-level overview of probabilistic inference and an introduction to factor graphs, a model used by DeepDive to perform probabilistic inference.
Probabilistic inference is the task of deriving the probability of one or more random variables taking a specific value or set of values. For example, a Bernoulli (Boolean) random variable may describe the event that John has cancer. Such a variable could take a value of 1 (John has cancer) or 0 (John does not have cancer). DeepDive uses probabilistic inference to estimate the probability that the random variable takes value 1: a probability of 0.78 would mean that John is 78% likely to have cancer.
A factor graph is a type of probabilistic graphical model. A factor graph has two types of nodes:
Factors, which define the relationships between variables in the graph. Each factor can be connected to many variables and comes with a factor function to define the relationship between these variables. For example, if a factor node is connected to two variables nodes
B, a possible factor function could be
imply(A,B), meaning that if the random variable
1, then so must the random variable
B. Each factor function has a weight associated with it, which describes how much influence the factor has on its variables in relative terms. In other words, the weight encodes the confidence we have in the relationship expressed by the factor function. If the weight is high and positive, we are very confident in the function that the factor encodes; if the weight is high and negative, we are confident that the function is incorrect. The weight can be learned from training data, or assigned manually.
A possible world is an assignment to every variable in a factor graph. The possible worlds are not usually equiprobable; rather, each possible world has a different probability. The probability of a possible world is proportional to a weighted combination of all factor functions in the graph, evaluated at the assignments specified by the possible world. The weights can be assigned statically or learned automatically. In the latter case, some training data is needed. Training data define a set of possible worlds and, intuitively, the learning process chooses the weights by maximizing the probabilities of these possible worlds.
Marginal inference is the task of inferring the probability of one variable taking a particular value. Using the law of total probability, it is straightforward to express this probability as the sum of the probabilities of possible worlds that contain the requested value for that variable.
Exact inference is an intractable problem on factor graphs, but a commonly used
method in this domain is Gibbs sampling. The process starts from a random
possible world and iterates over each variable
v, updating its value
by taking into account the factor functions of the factors that
v is connected
to and the values of the variables connected to those factors (this is known as
the Markov blanket of
v). After enough iterations
over the random variables, we can compute the number of iterations during which
each variable had a specific value and use the ratio between this quantity and
the total number of iterations as an estimate of the probability of the variable
taking that value.
Inference in DeepDive
DeepDive allows the user to write inference rules to specify how to create the factor graph. A rule expresses concepts like "If John smokes then he is likely to have cancer" and, in other words, describes the factor function of a factor and which variables are connected to this factor. Each rule has a weight (either computed by DeepDive or assigned by the user), which represents the confidence in the correctness of the rule. If a rule has a high positive weight, then the variables appearing in the rule are likely to take on values that would make the rule evaluate to true. In the above example, if the rule "If John smokes then he is likely to have cancer" has a high weight and we are sure that John smokes, then we are also reasonably confident that John has cancer. However, if we are not sure whether or not John smokes, then we can not be sure about him having cancer either. In the latter case, both "John does not have cancer," and "John has cancer" would make the rule evaluate to true.
This is a subtle but very important point. Contrary to many traditional machine learning algorithms, which often assume that prior knowledge is exact and make predictions in isolation, DeepDive performs joint inference: it determines the values of all events at the same time. This allows events to influence each other if they are (directly or indirectly) connected through inference rules. Thus, the uncertainty of one event (John smoking) may influence the uncertainty of another event (John having cancer). As the relationships among events become more complex this model becomes very powerful. For example, one could imagine the event "John smokes" being influenced by whether or not John has friends who smoke. This is particularly useful when dealing with inherently noisy signals, such as human language.
We refer the reader interested in additional details to other good resources:
- A Simple Factor Graph Tutorial
- An Introduction to Conditional Random Fields for Relational Learning
- PGM class on Coursera
- Graphical Models Lecture at CMU
- Towards High-Throughput Gibbs Sampling at Scale: A Study across Storage Managers
- Factor Graphs and the Sum-Product Algorithm
- Scalable Probabilistic Databases with Factor Graphs and MCMC