Bayesian Networks

Another popular approach for activity recognition is the use of Bayesian networks. Bayesian networks (BNs) are well adapted to cope with the problem of uncertain environments. The main advantage of Bayesian networks is that they are capable of modeling the uncertainty of the recognition by using probabilities based on the Bayes’ theorem of probability theory

P(A|B) = P(B|A).P(A)

P(B) (2.5)

Where,

¥ P(B|A)is the conditional probability of an event B, given the event A. It is also called the likelihood.

¥ P(A)is the prior probability (or ‘unconditional’ or ‘marginal’ probability) of A. It is ”prior”

in the sense that it does not take into account any information about B.

¥ P(B)is the prior or marginal probability of B, and acts as a normalizing constant.

¥ P(A|B)is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.

A Bayesian network consists of a directed acyclic graph in which nodes represent random variables which may be discrete or continuous (i.e. parametric distribution), and arcs represent the causal relationships (i.e. conditional dependencies or links) between variables. For a link

between two variables,A→ B, the overall joint distribution is specified by the product of the prior probability P(A)and the conditional probability P(B|A). The dependencies are specified a priori and used to create the network structure. The distributionsP(A) andP(B|A)must be specified beforehand to form the network from domain knowledge.

Bayesian networks are mainly used for calculating conditional probabilities of variables related to each other by relations of cause and effect. This computation is called inference.

A particular type of inference is the updating of probabilities that occurs following a set of observations. However, depending on the network structure, these calculations are more or less complex. While the general inference problem in BNs is NP-hard, efficient algorithms for inference exist under certain BN structure assumptions.

A naive Bayes classifier is a simple probabilistic classifier based on applying Bayes’ theorem with strong (naive) independence assumptions. In naive Bayes classifier each data instance X is assigned to one ofmclasses C1, C2, ..., Cmwhich it has the highest posterior probability conditioned on X (i.e. the class which is most probable given the prior probabilities of the classes and the dataX[Squire, 2004]. That is to say, X is assigned to classC_iif and only if:

P(C_i|X)> P(C_j|X) for all j such that 1≤j≤m. (2.6)

P(C_i|X) = P(X|C_i)P(C_i)

P(X) (2.7)

Naive Bayes classifier have been often used to recognize elementary actions at the numerical level [Nevatia et al., 2004] with only one physical object. Buxton and Gong [Buxton and Gong, 1995]

proposed an interpretation of the scene based on Bayesian networks characterizing the positions and interactions between objects observed in a traffic surveillance zone.

Hongeng et al. [Hongeng and Nevatia, 2001], [Nevatia et al., 2004] have used Bayesian networks and multi-agent architecture in the recognition of complex events (fig. 2.5). These events are represented in four hierarchical layers:

- The first layer corresponds to the detection and monitoring of observed objects (or agents).

- The second corresponds to the updating of properties associated with agents (distance be-tween objects, changing speeds and directions of observed objects ...)

- The third defines the simple events (approaching another object, to slowdown,...)

- The latter defines the scenarios, using the simple events to model these scenarios, using logical rules of sequences and durations.

However only the simple events were recognized by BNs, the temporal aspect was considered by HMMs.

BNs have been successfully applied to person interaction [Park and Aggarwal, 2004] such as

‘shake hands’, parking lot surveillance [Nevatia et al., 2004], traffic monitoring [Kumar et al., 2005]

Figure2.5: Representation of the complex event ‘converse’ as described in [Nevatia et al., 2004].

and detection of left luggage [Lv et al., 2006]. However, BNs are not adapted to model tem-poral compositions of events like sequential events. In fact, a major limitation of Bayesian networks is their poor representation of temporal dynamics. Time is not straightforward mod-eled in Bayesian Network formalism, temporal representation in Bayesian networks is often done using a static representation, where time points or time slices are represented as static processes. Dynamic Bayesian Networks (DBNs, BNs for dynamic processes) have been used to represent temporal sequencing of BNs [Town, 2006]. Dynamic Bayesian Networks have been used successfully to recognize short temporal actions [Reddy et al., 2003], but the recognition process depends on time segmentation: when the frame-rate or the activity duration changes, the DBN has to be re-trained. DBNs pose specific computational challenges. In fact, DBNs are able to model temporal relationships between variables, they do this by breaking up time into relevant discrete time-steps (or time slice), and placing a structurally similar copy of the net-work within each time-step. A causal relationship between nodes in time-stepkand nodes in time-stepk+1 are then inserted (fig. 2.6). For complex networks (i.e. network with a large number of nodes, 20 or more nodes) this would not be feasible as it makes the network gets very large, very quickly and increases in complexity very quickly, in turn it makes greatly increasing the amount of computational power required to run it ([Nicholson and Korb, 2006]).

Other DBNs have been studied to recognize more complicated composite events with sampling-based inference algorithms [Laxton et al., 2007]. However, DBNs are still insufficient to de-scribe various events because their scenarios are bounded by their conditional dependency

Figure2.6: Example of a DBN as described in [Nicholson and Korb, 2006]. Each variable in a DBN is associated with a time slice tand denotedXt. Intra-slice arcs: X^t_i− > X^t_j. Inter-slice (temporal) arcs:

X^t_i−> X^t+1_i andX^t_i−> X^t+1_j .

structures: to represent all the conditional dependencies in time-series data, the network can become excessively complex. Also, the inference algorithms for DBNs involve heuris-tics such as beam search [Shi et al., 2004], [Shi et al., 2006] or greedy pruning of candidates [Laxton et al., 2007] to explore the large search space: inferring the most likely observed se-quence of events by computing the probability of future events at time tgiven the past obser-vation at timet−i. The span timet− (t−i)is typically large ([Darwiche, 2000]).

The main advantage of Bayesian Networks is that they are well adapted to model the un-certainty of the recognition by using probabilities. However, the main drawback is that they are not adapted to model temporal relations because time needs to be explicitly indicated.

One among the most popular approach able to model the uncertainty and recognize activ-ities with temporal relations and in particular sequential events is the Hidden Markov Models described in the section below.