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Publications about Invariant
Result of the query in the list of publications :
Article |
1 - Invariant Bayesian estimation on manifolds.
I. H. Jermyn. Annals of Statistics, 33(2): pages 583--605, April 2005.
Keywords : Bayesian estimation, MAP, MMSE, Invariant, Metric, Jeffrey's.
@ARTICLE{jermyn_annstat05,
|
| author |
= |
{Jermyn, I. H.}, |
| title |
= |
{Invariant Bayesian estimation on manifolds}, |
| year |
= |
{2005}, |
| month |
= |
{April}, |
| journal |
= |
{Annals of Statistics}, |
| volume |
= |
{33}, |
| number |
= |
{2}, |
| pages |
= |
{583--605}, |
| url |
= |
{http://dx.doi.org/10.1214/009053604000001273}, |
| pdf |
= |
{ftp://ftp-sop.inria.fr/ariana/Articles/jermyn_annstat05.pdf}, |
| keyword |
= |
{Bayesian estimation, MAP, MMSE, Invariant, Metric, Jeffrey's} |
| } |
Abstract :
A frequent and well-founded criticism of the maximum em a posteriori (MAP) and minimum mean squared error (MMSE) estimates of a continuous parameter param taking values in a differentiable manifold paramspace is that they are not invariant to arbitrary `reparametrizations' of paramspace. This paper clarifies the issues surrounding this problem, by pointing out the difference between coordinate invariance, which is a em sine qua non for a mathematically well-defined problem, and diffeomorphism invariance, which is a substantial issue, and then provides a solution. We first show that the presence of a metric structure on paramspace can be used to define coordinate-invariant MAP and MMSE estimates, and we argue that this is the natural way to proceed. We then discuss the choice of a metric structure on paramspace. By imposing an invariance criterion natural within a Bayesian framework, we show that this choice is essentially unique. It does not necessarily correspond to a choice of coordinates. In cases of complete prior ignorance, when Jeffreys' prior is used, the invariant MAP estimate reduces to the maximum likelihood estimate. The invariant MAP estimate coincides with the minimum message length (MML) estimate, but no discretization or approximation is used in its derivation. |
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Technical and Research Report |
1 - On Bayesian Estimation in Manifolds.
I. H. Jermyn. Research Report 4607, Inria, France, November 2002.
Keywords : Rare event, Bayesian estimation, Invariant.
@TECHREPORT{4607,
|
| author |
= |
{Jermyn, I. H.}, |
| title |
= |
{On Bayesian Estimation in Manifolds}, |
| year |
= |
{2002}, |
| month |
= |
{November}, |
| institution |
= |
{Inria}, |
| type |
= |
{Research Report}, |
| number |
= |
{4607}, |
| address |
= |
{France}, |
| url |
= |
{http://www.inria.fr/rrrt/rr-4607.html}, |
| pdf |
= |
{ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4607.pdf}, |
| ps |
= |
{ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4607.ps.gz}, |
| keyword |
= |
{Rare event, Bayesian estimation, Invariant} |
| } |
Résumé :
Il est fréquemment dit que les estimées au sens du maximum a posteriori (MAP) et du minimum de l'erreur quadratique moyenne (MMSE) d'un paramètre continu ne sont pas invariantes relativement aux «reparamètrisations» de l'espace des paramètres . Ce rapport clarifie les questions autour de ce problème, en soulignant la différence entre l'invariance aux changements de coordonnées, qui est une condition sine qua non pour un problème mathématiq- uement bien défini, et l'invariance aux difféomorphismes, qui est une question significative, et fournit une solution. On montre d'abord que la présence d'une structure métrique sur peut être utilisée pour définir les estimées aux sens du MAP et du MMSE qui sont invariantes aux changements de coordonnées, et on explique pourquoi cela est la fa on naturelle et nécessaire pour le faire. Le problème de l'estimation et les quantités géométriques qui y sont associées sont tous définis d'une fa on clairement invariante aux changements de coordonnées. On montre que la même estimée au sens du MAP est obtenue en utilisant soit la `maximisation d'une densité' soit une fonction de perte delta, définie de fa on invariante. Puis, on discute le choix d'une métrique pour . En imposant un critère d'invariance qui est naturel dans le cadre bayesien, on montre que ce choix est unique. Il ne correspond pas nécessairement à un choix de coordonnées. L'estimée au sens du MAP qui en résulte coincide avec l'estimée fondée sur la longueur minimum de message (MML), mais la demonstration n'utilise pas de discrétisation ou d'approximation. |
Abstract :
It is frequently stated that the maximum a posteriori (MAP) and minimum mean squared error (MMSE) estimates of a continuous parameter are not invariant to arbitrary «reparametrizations» of the parameter space . This report clarifies the issues surrounding this problem, by pointing out the difference between coordinate invariance, which is a sine qua non for a mathematically well-defined problem, and diffeomorphism invariance, which is a substantial issue, and provides a solution. We first show that the presence of a metric structure on can be used to define coordinate-invari- ant MAP and MMSE estimates, and we argue that this is the natural and necessary way to proceed. The estimation problem and related geometrical quantities are all defined in a manifestly coordinate-invariant way. We show that the same MAP estimate results from `density maximization' or from using an invariantly-defined delta function loss. We then discuss the choice of a metric structure on . By imposing an invariance criterion natural within a Bayesian framework, we show that this choice is essentially unique. It does not necessarily correspond to a choice of coordinates. The resulting MAP estimate coincides with the minimum message length (MML) estimate, but no discretization or approximation is used in its derivation. |
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