Graphical models¶

Bayesian network¶

A directed acyclic graph where the nodes represent random variables.

Not to be confused with Bayesian neural networks.

The chain rule for Bayesian networks¶

The joint distribution for all the variables in a network is equal to the product of the distributions for all the individual variables, conditional on their parents.

$P(X_1,...,X_n) = \prod_i P(X_i|Par(X_i))$

where $Par(X_i)$ denotes the parents of the node $X_i$ in the graph.

Boltzmann Machines¶

Restricted Boltzmann Machines (RBMs)¶

Trained with contrastive divergence.

Deep Belief Networks (DBNs)¶

Deep Belief Machines (DBMs)¶

Clique¶

A subset of a graph where the nodes are fully-connected, ie each node has an edge with every other node in the set.

Conditional Random Field (CRF)¶

Discriminative model that can be seen as a generalization of logistic regression.

Common applications of CRFs include image segmentation and named entity recognition.

Used in

Seed, Expand and Constrain: Three Principles for Weakly-Supervised Image Segmentation, Kolesnikov and Lampert (2016)

Linear Chain CRFs¶

A simple sequential CRF.

Hidden Markov Model (HMM)¶

A simple generative sequence model in which there is an observable state and a latent state, which must be inferred.

At each time step the model is in a latent state $x_t$ and outputs an observation $y_t$ . The observation is solely a function of the latent state, as is the probability distribution over the next state, $x_{t+1}$ . Hence the model obeys the Markov property.

The model is defined by:

A matrix $T$ of transition probabilities where $T_{ij}$ is the probability of going from state i to state j.
A matrix $E$ of emission probabilities where $E_{ij}$ is the probability of emitting observation j in state i.

The parameters can be learnt with the Baum-Welch algorithm.

Markov chain¶

A simple state transition model where the next state depends only on the current state. At any given time, if the current state is node i, there is a probability $T_{ij}$ of transitioning to node j, where $T$ is the transition matrix.

Source: Wikipedia

Markov property¶

A process is said to have the Markov property if the next state depends only on the current state, not any of the previous ones.

Markov Random Field (MRF)¶

A type of undirected graphical model which defines the joint probability distribution over a set of variables. Each variable is represented by one node in the graph.

One use for an MRF could be to model the distribution over the pixel values for a set of images. In order to keep the model tractable edges are only drawn between neighbouring pixels.

Naive Bayes Model¶

A simple classifier that models all of the features as independent, given the label.

$P(Y|X_1,...,X_n) = P(Y)\prod_{i=1}^n P(Y|X_i)$