# model

*Entry of the [Dictionary of Applied Machine Learning](https://dictionaryofml.org) (course edition), by Alexander Jung, Aalto University. CC BY 4.0. Updated 2026-10-29. [HTML](https://dictionaryofml.org/terms/model.html) · [typeset PDF](https://dictionaryofml.org/terms/model.pdf) · [LaTeX](https://dictionaryofml.org/terms/model.tex) · [Python demo](https://dictionaryofml.org/terms/model.py) · [notebook](https://dictionaryofml.org/terms/model.ipynb)*

Math is LaTeX with the book's macros expanded, so no notation package is needed to read it.

## Abstract

In machine learning, the word *model* is used in different ways. Formally, a model is a hypothesis space $\mathcal{H}$: the set of candidate hypothesis maps from which a machine learning method picks (or learns) one. In applied machine learning literature and in software libraries, the word *model* also refers to the trained predictor, i.e., to the learned hypothesis returned by a machine learning algorithm; the EU AI Act uses this sense in the compound term general-purpose AI model. A different type of mathematical model is a probabilistic model, which specifies a family of probability distributions that characterize a data generation process.

## Definition

Consider the problem of predicting tomorrow's maximum daytime temperature from today's morning temperature. Machine learning methods learn a hypothesis map $\widehat{h}$ that maps today's morning temperature, the feature, to an accurate prediction of tomorrow's maximum temperature, the label. This learned hypothesis is chosen from a (typically large) set of candidate hypothesis maps, i.e., a hypothesis space $\mathcal{H}$.

The hypothesis space underlying a machine learning method is its model: the set of candidate hypothesis maps, chosen in advance, from which the method picks one (Shalev-Shwartz and Ben-David, 2014, Sect. 2.3). For example, linear regression and logistic regression use the linear model as their hypothesis space. A decision tree method uses a hypothesis space that consists of hypothesis maps generated by a flow chart. Large language model systems use a hypothesis space consisting of non-linear maps represented by an artificial neural network.

Model training refers to the process of finding a hypothesis in the hypothesis space that yields accurate predictions. A trained (or fitted) model is a learned hypothesis $\widehat{h} = \mathcal{A}(\mathcal{D}^{(\rm train)})$, the output of the map $\mathcal{A}$ by which the machine learning method turns a training set $\mathcal{D}^{(\rm train)}$ into a hypothesis. In applied machine learning literature and in software libraries, the word *model* typically refers to a trained model, i.e., to a specific hypothesis map $\widehat{h}$ (Goodfellow et al., 2016, Sect. 5.1; Murphy, 2012, Sect. 3.5.2; Prince, 2023). The EU AI Act uses the word *model* in the same sense (i.e., a learned hypothesis) in the compound term general-purpose AI model (European Parliament and Council of the European Union, 2024, Art. 3(63)).

A different type of mathematical model is a probabilistic model, which specifies a family of probability distributions. Each such probability distribution defines a data generator: data points are realizations of independent and identically distributed RVs with the given probability distribution.

Fig. 1 illustrates these three uses of the word *model*: the hypothesis space, the trained model, and the probabilistic model.

![Figure 1 of the entry model](https://dictionaryofml.org/terms/fig/model-fig1.svg)

*Figure 1: Three uses of the word *model* in machine learning practice. (a) Hypothesis space consisting of three linear maps (solid, dashed, dotted). (b) Trained model: the learned hypothesis $\widehat{h} = \mathcal{A}(\mathcal{D}^{(\rm train)})$ selected from the hypothesis space in (a) by a machine learning map $\mathcal{A}$ acting on a training set $\mathcal{D}^{(\rm train)}$ of data points (blue circles). (c) Probabilistic model consisting of two probability distributions (solid and dashed outlines) for the generation of data points with feature $x$ and label $y$*

**Synonyms:** hypothesis space, hypothesis class, model class.

**See also:** [hypothesis space](https://dictionaryofml.org/terms/hypospace.html), [hypothesis](https://dictionaryofml.org/terms/hypothesis.html), [training](https://dictionaryofml.org/terms/training.html), probabilistic model, parametric model, model parameter, general-purpose AI model.

## References

- Shalev-Shwartz and Ben-David (2014). Understanding Machine Learning: From Theory to Algorithms. *Cambridge Univ. Press*. <https://doi.org/10.1017/cbo9781107298019>
- Goodfellow et al. (2016). Deep Learning. *MIT Press*.
- Murphy (2012). Machine Learning: A Probabilistic Perspective. *MIT Press*.
- Prince (2023). Understanding Deep Learning. *MIT Press*.
- European Parliament and Council of the European Union (2024). Regulation (EU) 2024/1689 of the European Parliament and of the Council of 13 June 2024 laying down harmonised rules on artificial intelligence and amending Regulations (EC) No 300/2008, (EU) No 167/2013, (EU) No 168/2013, (EU) 2018/858, (EU) 2018/1139 and (EU) 2019/2144 and Directives 2014/90/EU, (EU) 2016/797 and (EU) 2020/1828 (Artificial Intelligence Act) (Text with EEA relevance). <https://eur-lex.europa.eu/eli/reg/2024/1689/oj/eng>

## Cite this entry

```bibtex
@incollection{dictml_model,
  author = {Jung, Alexander},
  editor = {Olioumtsevits, Konstantina and Schnoor, Ekkehard},
  title = {model},
  booktitle = {Dictionary of Applied Machine Learning},
  publisher = {Springer Nature Singapore},
  year = {2026},
  doi = {10.1007/978-981-92-2012-0_2-1},
  isbn = {978-981-92-2012-0},
  url = {https://doi.org/10.1007/978-981-92-2012-0_2-1}
}
```