Generative Model Training Algorithm

A Generative Model Training Algorithm is a probabilistic learning algorithm that can be implemented by a generative model training system to produce a generative model (by directly estimating the prior probability of the target class and predictor variables).

• Context:
  • It can be Produced by inducing the Conditional Probability of the Training Examples given the Target Values and the Probability of the Target Values.
  • It can be Trained to optimize p(t,x)=p(x|t)p(t).
  • It can (usually) be slow/complicated to get the sum over all possible states especially when both [math]\displaystyle{ x }[/math] and/or [math]\displaystyle{ y }[/math] are Complex High-Dimensional Random Objects.
  • It can (typically) apply Bayes Rule.
  • It can find the value of the weights that are most likely to account the data that we have seen (the Maximum Likelihood).
  • It can range from (typically) being a Generative Classification Algorithm to being a Generative Ranking Algorithm to being a Generative Estimation Algorithm.
  • …
• Example(s):
  • a Linear Generative Classification, such as linear discriminant analysis and naive-Bayes.
  • an HMM Training Algorithm.
  • …
• Counter-Example(s):
  • any Discriminative Learning Algorithm, such as logistic regression.
• See: Generative Model Inferencing Algorithm, Generative Grammar.

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References

2014

• http://en.wikipedia.org/wiki/Linear_classifier#Generative_models_vs._discriminative_models
  • There are two broad classes of methods for determining the parameters of a linear classifier [math]\displaystyle{ \vec w }[/math]. Methods of the first class model conditional density functions [math]\displaystyle{ P(\vec x|{\rm class}) }[/math]. Examples of such algorithms include:
    • Linear Discriminant Analysis (or Fisher's linear discriminant) (LDA) — assumes Gaussian conditional density models
    • Naive Bayes classifier with multinomial or multivariate Bernoulli event models.

2011

• (Sammut & Webb, 2011) ⇒ Claude Sammut (editor), and Geoffrey I. Webb (editor). (2011). “Generative Learning .” In: (Sammut & Webb, 2011) p.455

2009

• (Wick et al., 2009) ⇒ Michael Wick, Aron Culotta, Khashayar Rohanimanesh, and Andrew McCallum. (2009). “An Entity Based Model for Coreference Resolution.” In: Proceedings of the SIAM International Conference on Data Mining (SDM 2009).
  • Statistical approaches to coreference resolution can be broadly placed into two categories: generative models, which model the joint probability, and discriminative models that model that conditional probability. These models can be either supervised (uses labeled coreference data for learning) or unsupervised (no labeled data is used). Our model falls into the category of discriminative and supervised.

2004

• (Bouchard & Triggs, 2004) ⇒ Guillaume Bouchard, and Bill Triggs. (2004). “The Trade-off Between Generative and Discriminative Classifiers.” In: Proceedings of COMPSTAT 2004.
  • QUOTE: … In supervised classification, inputs [math]\displaystyle{ x }[/math] and their labels [math]\displaystyle{ y }[/math] arise from an unknown joint probability [math]\displaystyle{ p(x,y) }[/math]. If we can approximate [math]\displaystyle{ p(x,y) }[/math] using a parametric family of models [math]\displaystyle{ G = \{p_θ(x,y),\theta \in \Theta\} }[/math], then a natural classifier is obtained by first estimating the class-conditional densities, then classifying each new data point to the class with highest posterior probability. This approach is called generative classification.

    However, if the overall goal is to find the classification rule with the smallest error rate, this depends only on the conditional density [math]\displaystyle{ p(y \vert x) }[/math]. Discriminative methods directly model the conditional distribution, without assuming anything about the input distribution p(x). Well known generative-discriminative pairs include Linear Discriminant Analysis (LDA) vs. Linear logistic regression and naive Bayes vs. Generalized Additive Models (GAM). Many authors have already studied these models e.g. [5,6]. Under the assumption that the underlying distributions are Gaussian with equal covariances, it is known that LDA requires less data than its discriminative counterpart, linear logistic regression [3]. More generally, it is known that generative classifiers have a smaller variance than.

    Conversely, the generative approach converges to the best model for the joint distribution p(x,y) but the resulting conditional density is usually a biased classifier unless its p_θ(x) part is an accurate model for p(x). In real world problems the assumed generative model is rarely exact, and asymptotically, a discriminative classifier should typically be preferred [9, 5]. The key argument is that the discriminative estimator converges to the conditional density that minimizes the negative log-likelihood classification loss against the true density p(x, y) [2]. For finite sample sizes, there is a bias-variance tradeoff and it is less obvious how to choose between generative and discriminative classifiers.

1999

• (Jaakkola & Haussler, 1999) ⇒ Tommi S. Jaakkola, and David Haussler. (1999). “Exploiting Generative Models in Discriminative Classifiers.” In: Proceedings of the 1998 conference on Advances in Neural Information Processing Systems II. ISBN:0-262-11245-0
  • QUOTE: Generative probability models such as hidden Markov models provide a principled way of treating missing information and dealing with variable length sequences. On the other hand, discriminative methods such as support vector machines enable us to construct flexible decision boundaries and often result in classification performance superior to that of the model based approaches.

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