از ویکی‌پدیا، دانشنامهٔ آزاد
در احتمالات بیزی توزیع پیشین جفری (به انگلیسی : Jeffrey's prior احتمال پیشین است که متناسب با ریشه دوم دترمینان اطلاع فیشر است:
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{\displaystyle p\left({\vec {\theta }}\right)\propto {\sqrt {\det {\mathcal {I}}\left({\vec {\theta }}\right)}}.\,}
تغییر پارامتر [ ویرایش ] برای تغییر یک پارامتر اگر بخواهیم
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{\displaystyle p(\varphi )\propto {\sqrt {I(\varphi )}}\,}
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{\displaystyle p(\theta )\propto {\sqrt {I(\theta )}}\,}
بدست آوریم، با استفاده از قانون تغییر متغیرها داریم:
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{\displaystyle {\begin{aligned}p(\varphi )&=p(\theta )\left|{\frac {d\theta }{d\varphi }}\right|\propto {\sqrt {I(\theta )\left({\frac {d\theta }{d\varphi }}\right)^{2}}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d\ln L}{d\theta }}\right)^{2}\right]\left({\frac {d\theta }{d\varphi }}\right)^{2}}}\\&={\sqrt {\operatorname {E} \!\left[\left({\frac {d\ln L}{d\theta }}{\frac {d\theta }{d\varphi }}\right)^{2}\right]}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d\ln L}{d\varphi }}\right)^{2}\right]}}={\sqrt {I(\varphi )}}.\end{aligned}}}
برای تغییر متغیر چندین متغیر نیز مشابه همین عمل می کنیم:
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{\displaystyle {\begin{aligned}p({\vec {\varphi }})&=p({\vec {\theta }})\left|\det {\frac {\partial \theta _{i}}{\partial \varphi _{j}}}\right|\\&\propto {\sqrt {\det I({\vec {\theta }})\,{\det }^{2}{\frac {\partial \theta _{i}}{\partial \varphi _{j}}}}}\\&={\sqrt {\det {\frac {\partial \theta _{k}}{\partial \varphi _{i}}}\,\det \operatorname {E} \!\left[{\frac {\partial \ln L}{\partial \theta _{k}}}{\frac {\partial \ln L}{\partial \theta _{l}}}\right]\,\det {\frac {\partial \theta _{l}}{\partial \varphi _{j}}}}}\\&={\sqrt {\det \operatorname {E} \!\left[\sum _{k,l}{\frac {\partial \theta _{k}}{\partial \varphi _{i}}}{\frac {\partial \ln L}{\partial \theta _{k}}}{\frac {\partial \ln L}{\partial \theta _{l}}}{\frac {\partial \theta _{l}}{\partial \varphi _{j}}}\right]}}\\&={\sqrt {\det \operatorname {E} \!\left[{\frac {\partial \ln L}{\partial \varphi _{i}}}{\frac {\partial \ln L}{\partial \varphi _{j}}}\right]}}={\sqrt {\det I({\vec {\varphi }})}}.\end{aligned}}}
![{\displaystyle {\begin{aligned}p(\varphi )&=p(\theta )\left|{\frac {d\theta }{d\varphi }}\right|\propto {\sqrt {I(\theta )\left({\frac {d\theta }{d\varphi }}\right)^{2}}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d\ln L}{d\theta }}\right)^{2}\right]\left({\frac {d\theta }{d\varphi }}\right)^{2}}}\\&={\sqrt {\operatorname {E} \!\left[\left({\frac {d\ln L}{d\theta }}{\frac {d\theta }{d\varphi }}\right)^{2}\right]}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d\ln L}{d\varphi }}\right)^{2}\right]}}={\sqrt {I(\varphi )}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7ef177ac482ff371ab7e4849afdb0980c7f6f2)
![{\displaystyle {\begin{aligned}p({\vec {\varphi }})&=p({\vec {\theta }})\left|\det {\frac {\partial \theta _{i}}{\partial \varphi _{j}}}\right|\\&\propto {\sqrt {\det I({\vec {\theta }})\,{\det }^{2}{\frac {\partial \theta _{i}}{\partial \varphi _{j}}}}}\\&={\sqrt {\det {\frac {\partial \theta _{k}}{\partial \varphi _{i}}}\,\det \operatorname {E} \!\left[{\frac {\partial \ln L}{\partial \theta _{k}}}{\frac {\partial \ln L}{\partial \theta _{l}}}\right]\,\det {\frac {\partial \theta _{l}}{\partial \varphi _{j}}}}}\\&={\sqrt {\det \operatorname {E} \!\left[\sum _{k,l}{\frac {\partial \theta _{k}}{\partial \varphi _{i}}}{\frac {\partial \ln L}{\partial \theta _{k}}}{\frac {\partial \ln L}{\partial \theta _{l}}}{\frac {\partial \theta _{l}}{\partial \varphi _{j}}}\right]}}\\&={\sqrt {\det \operatorname {E} \!\left[{\frac {\partial \ln L}{\partial \varphi _{i}}}{\frac {\partial \ln L}{\partial \varphi _{j}}}\right]}}={\sqrt {\det I({\vec {\varphi }})}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f6685b4eb747214e06e1576c660a55ee0d2e15)