Right Haar prior

In Bayesian statistics, the right Haar prior is an objective prior that has various useful and interesting mathematical properties. In particular, Bayesian predictions generated using right Haar priors are perfectly calibrated, i.e., the predicted probabilities correspond exactly to frequencies, as proven by Severini et al in 2002. This makes the right Haar prior a natural choice when making predictions.

Right Haar priors only exist for a limited set of statistical models that have appropriate group structure. Models with the appropriate group structure include one parameter distributions with a location parameter or a scale parameter, and two parameter distributions with location and scale parameters. They also include models related to these models by log or exponential transformations, and versions of these models in which the location parameter is replaced by a linear predictor and/or the scale parameter is replaced by a log-linear predictor.

Right Haar prior definition

Certain statistical models form locally compact Hausdorf topological groups. Examples are the exponential distribution, the continuous uniform distribution, the normal distribution, the log-normal distribution, the Gumbel distribution and the Weibull distribution. The group structure exists for these distributions because there is transformation of the random variable and parameters of the model which forms a sharply transitive group action. The existence of this group action means that the set of all models form a homogeneous space. Because these models are locally compact topological groups, Haar's theorem shows that there must exist left and right invariant measures, known as Haar measures. These measures have various applications. In statistics, the right Haar measure is known as the right Haar prior.

Right Haar prior prediction

The key relevance of the right Haar prior is that a Bayesian prediction generated using the right Haar prior has the property that the predicted probabilities are perfectly calibrated. In other words:

• For any true parameter value, the predicted probabilities correspond exactly to frequencies.
• The predictions, although Bayesian, have perfect frequentist properties.
• A reliability diagram, in which observed frequencies are plotted against nominal probabilities, will show converge to exact reliability as the sample size increases.

This result was proven in 2002 by Severini et al. based on an earlier result from 1966 due to Hora and Buehler, which itself depended on an earlier result from 1961 due to Fraser.

Examples of right Haar priors

The following table lists a number of commonly used distributions, the transformation of the parameters that defines a sharply transitive group action, and the resulting right Haar prior:The fourth column in the table specfies whether there is a closed-form expression for the resulting prediction, or whether the prediction has be to evaluated using numerical methods.

Model	Transformation	Right Haar Prior	Prediction
Exponential	${\lambda }^{\prime }=b\lambda$	$1/\lambda$	Closed-form
Normal	$({\mu }^{\prime },{\sigma }^{\prime })=(a+b\mu ,b\sigma )$	$1/\sigma$	Closed-form
Log-normal	$({\mu }^{\prime },{\sigma }^{\prime })=(a+b\mu ,b\sigma )$	$1/\sigma$	Closed-form
Gumbel	$({\mu }^{\prime },{\beta }^{\prime })=(a+b\mu ,b\beta )$	$1/\beta$	Numerical
Weibull	$({\lambda }^{\prime },k^{\prime })=(b{\lambda }^c,k/c)$	$1/(k\lambda )$	Numerical

Other examples of models with right Haar priors are:

• Any other model with just a scale parameter, such as the half-normal distribution
• Any model with just a location parameter, such as the normal distribution with known standard deviation
• Any model with location and scale parameters, such as the GEV distribution with known shape parameter.
• Any of these models with predictors on the location parameter, such as Gaussian simple linear regression

Applications

Right Haar prior predictions for a number of commonly used distributions are given in the R package fitdistcp.[1]

References

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