# Is HMM discriminative or generative?

Wikipedia

1. "An HMM can be considered as the simplest dynamic Bayesian network." Here.

2. "In probability and statistics, a generative model is a model for randomly generating observable data, typically given some hidden parameters. [A generative model] specifies a joint probability distribution over observation and label sequences. Generative models are used in machine learning for either modeling data directly (i.e., modeling observed draws from a probability density function), or as an intermediate step to forming a conditional probability density function. A conditional distribution can be formed from a generative model through Bayes' rule."

3. "Discriminative models are a class of models used in machine learning for modeling the dependence of an unobserved variable y on an observed variable x. Within a statistical framework, this is done by modeling the conditional probability distribution P(y|x), which can be used for predicting y from x.

Discriminative models differ from generative models in that they do not allow one to generate samples from the joint distribution of x and y."

Wikipedia lists HMM under generative models. I cannot understand this, specifially:

$P(y|x)=\frac{P(y\cap x)}{P(x)}$ where the conditional probability uses the joint probability of x and y i.e. $P(y\cap x)$ -- and the definitions above "Discriminatory models differ from generative models in that they do not allow one to generate samples from the joint distribution of x and y". I cannot yet understand this, both models use the joint probability in some way -- and still they are somehow different. Could somehow clarify this difference?

P.s. Trying to make some sense to this thread here about HMM in machine-learning.

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Statistical classification, Generative model, Discrinative model -- suppose I have a state-machine model or petri-nets, can it be both discrinative and generative? I think so. What about HMM? Can it be both discrinative and generative? –  hhh Feb 13 at 10:33