# Mixture Density Network

A Mixture Density Network is a feedforward neural network combined with a mixture density model.

**Context:**- It can represent arbitrary conditional probability distributions.

**Counter-Example(s):****See:**Probability Distribution Space.

## References

### 2015

- (Vilnis & McCallum, 2015) ⇒ Luke Vilnis, and Andrew McCallum. (2015). “Word Representations via Gaussian Embedding.” In: arXiv preprint arXiv:1412.6623 submitted to ICRL 2015.
- QUOTE: However, these Bayesian methods apply Bayes’ rule to observed data to infer the latent distributions, whereas our model works directly in the space of probability distributions and discriminatively trains them. This allows us to go beyond the Bayesian approach and use arbitrary (and even asymmetric) training criteria, and is more similar to methods that learn kernels (Lanckriet et al., 2004) or function-valued neural networks such as mixture density networks (Bishop, 1994).

### 1994

- (Bishop, 1994b) ⇒ Christopher M. Bishop. (1994). “Mixture Density Networks." Technical Report. Aston University, Birmingham.
- ABSTRACT: Minimization of a sum-of-squares or cross-entropy error function leads to network outputs which approximate the conditional averages of the target data, conditioned on the input vector. For classifications problems, with a suitably chosen target coding scheme, these averages represent the posterior probabilities of class membership, and so can be regarded as optimal. For problems involving the prediction of continuous variables, however, the conditional averages provide only a very limited description of the properties of the target variables. This is particularly true for problems in which the mapping to be learned is multi-valued, as often arises in the solution of inverse problems, since the average of several correct target values is not necessarily itself a correct value. In order to obtain a complete description of the data, for the purposes of predicting the outputs corresponding to new input vectors, we must model the conditional probability distribution of the target data, again conditioned on the input vector. In this paper we introduce a new class of network models obtained by combining a conventional neural network with a mixture density model. The complete system is called a Mixture Density Network, and can in principle represent arbitrary conditional probability distributions in the same way that a conventional neural network can represent arbitrary functions. We demonstrate the effectiveness of Mixture Density Networks using both a toy problem and a problem involving robot inverse kinematics.