## Abstract

An agent is asked to assess a real-valued variable Yp based on certain characteristics (formula present), and on a database consisting of (formula present) for (formula present). A possible approach to combine past observations of X and Y with the current values of X to generate an assessment of Y is similarity-weighted averaging. It suggests that the predicted value of (formula present) be the weighted average of all previously observed values Yi, where the weight of Yi, for every (formula present) is the similarity between the vector (formula present) p, associated with Yp, and the previously observed vector, (formula present). We axiomatize this rule. We assume that, given every database, a predictor has a ranking over possible values, and we show that certain reasonable conditions on these rankings imply that they are determined by the proximity to a similarity-weighted average for a certain similarity function. The axiomatization does not suggest a particular similarity function, or even a particular form of this function. We therefore proceed to suggest that the similarity function be estimated from past observations. We develop tools of statistical inference for parametric estimation of the similarity function, for the case of a continuous as well as a discrete variable. Finally, we discuss the relationship of the proposed method to other methods of estimation and prediction.

Original language | English |
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Title of host publication | Case-Based Predictions |

Subtitle of host publication | An Axiomatic Approach to Prediction, Classification and Statistical Learning |

Publisher | Taylor and Francis |

Pages | 211-244 |

Number of pages | 34 |

ISBN (Electronic) | 9789814366182 |

ISBN (Print) | 981436617X, 9789814366175 |

DOIs | |

State | Published - 1 Jan 2012 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2012 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.

## Keywords

- Estimation
- Similarity