# Latent Variable Model

A Latent Variable Model is a statistical model that contains one or more latent variables.

**AKA:**Hidden Variable Statistical Model.**Context:**- It can range from being a Conditional Latent Variable Model/Discriminative Latent Variable Model to being a Joint Latent Variable Model/Generative Latent Variable Model.
- It can be trained by a Latent Variable Model Training System (that solves a latent variable model training task by implementing a latent variable model training algorithm.
- It can range from being a Syntactic Latent Variable Model to being a Semantic Latent Variable Model.

**Example(s):****See:**Variable Representation, Manifest Variable, Mixture Model, Latent Dirichlet Allocation Model.

## References

### 2016

- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/latent_variable_model Retrieved:2016-6-13.
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**latent variable model**is a statistical model that relates a set of variables (so-called*manifest variables*) to a set of latent variables. It is assumed that the responses on the indicators or manifest variables are the result of an individual's position on the latent variable(s), and that the manifest variables have nothing in common after controlling for the latent variable (local independence).

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### 2013

- http://en.wikipedia.org/wiki/Latent_variable_model
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**latent variable model**is a statistical model that relates a set of variables (so-called*manifest variables*) to a set of latent variables.It is assumed that

- the responses on the indicators or manifest variables are the result of an individual's position on the latent variable(s), and
- that the manifest variables have nothing in common after controlling for the latent variable (local independence).

- Different types of the latent variable model can be grouped according to whether the manifest and latent variables are categorical or continuous:
^{[1]}

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Manifest variables | ||
---|---|---|

Latent variables | Continuous | Categorical |

Continuous | Factor analysis | Latent trait analysis |

Categorical | Latent profile analysis | Latent class analysis |

- http://en.wikipedia.org/wiki/Latent_variable_model (continued).
- Another name for latent trait analysis is item response theory (IRT). The most simple IRT model is the Rasch model. An important part of the latent profile analysis is the mixture model.
In factor analysis and latent trait analysis the latent variables are treated as continuous normally distributed variables, and in latent profile analysis and latent class analysis as from a multinomial distribution. The manifest variables in factor analysis and latent profile analysis are continuous and in most cases, their conditional distribution given the latent variables is assumed to be normal. In latent trait analysis and latent class analysis, the manifest variables are discrete. These variables could be dichotomous, ordinal or nominal variables. Their conditional distributions are assumed to be binomial or multinomial.

Because the distribution of a continuous latent variable can be approximated by a discrete distribution, the distinction between continuous and discrete variables turns out not to be fundamental at all. Therefore there may be a psychometrical latent variable, but not a psychological psychometric variable.

- Another name for latent trait analysis is item response theory (IRT). The most simple IRT model is the Rasch model. An important part of the latent profile analysis is the mixture model.

- ↑ David J. Bartholomew, Fiona Steel, Irini Moustaki, Jane I. Galbraith (2002),
*The Analysis and Interpretation of Multivariate Data for Social Scientists*, Chapman & Hall/CRC, p. 145

### 2007

- (Jara et al., 2007) ⇒ Alejandro Jara, María José García-Zattera, and Emmanuel Lesaffre. (2007). “A Dirichlet Process Mixture Model for the Analysis of Correlated Binary Responses.” In: Computational Statistics & Data Analysis, 51(11). doi:10.1016/j.csda.2006.09.010

### 2003

- (Blei & Jordan, 2003) ⇒ David M. Blei, and Michael I. Jordan. (2003). “Modeling Annotated Data.” In: Proceedings of the 26th annual international ACM SIGIR conference on Research and development in informaion retrieval. ISBN:1-58113-646-3 doi:10.1145/860435.860460
- QUOTE: We describe three hierarchical probabilistic mixture models which aim to describe such data, culminating in
*correspondence latent Dirichlet allocation*, a latent variable model that is effective at modeling the joint distribution of both types and the conditional distribution of the annotation given the primary type.

- QUOTE: We describe three hierarchical probabilistic mixture models which aim to describe such data, culminating in