Differential geometrical approach to Bayesian prediction theory to classical and quantum correlated systems

• Project/Area Number: 20700250
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Statistical science
• Research Institution: The University of Tokyo
• Principal Investigator: TANAKA Fuyuhiko 東京大学, 大学院・情報理工学系研究科, 助教 (90456161)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2008: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 統計的予測 / 情報幾何 / 時系列解析 / 量子情報 / 事前分布 / ベイズ予測 / 量子推定

Research Abstract

We obtain a useful parametrization of superharmonic priors in AR models, which are often used in time series analysis. It is shown to be useful not only in theoretical development but also in numerical analysis. We show that there is no alpha parallel prior except for the Jeffreys prior in the ARMA model by using differential geometry. We give a general definition of noninformative prior in a statistical model of wave functions and prove a minimax theorem with respect to this prior. We also derive the optimal estimation of wave functions and give an explicit example where our estimator has a uniformly better performance than that based on MLE.

Research Products

• [Journal Article] Noninformative prior in the quantum statistical model of pure states (2012)
  • Author(s): TANAKA Fuyuhiko
  • Journal Title: Physical Review A, American Institute of Physics Volume: Vol.85 Issue: 6 Pages: 62305-62305
  • DOI: 10.1103/physreva.85.062305
  • Peer Reviewed
• [Journal Article] Curvature form on statistical model manifolds and its application to Bayesian analysis (2012)
  • Author(s): TANAKA Fuyuhiko
  • Journal Title: Journal of Statistics Applications and Probability, Natural Sciences Publishing Volume: Vol.1 Pages: 35-43
  • URL: http://naturalspublishing.com/ContIss.asp?IssID=33
  • Peer Reviewed
• [Journal Article] Curvature form on statistical model manifolds and its application to Bayesian analysis (2012)
  • Author(s): Fuyuhiko Tanaka
  • Journal Title: Journal of Statistics Applications & Probability Volume: 1 Pages: 35-43
  • Peer Reviewed
• [Journal Article] Asymptotic expansion of the risk difference of the Bayesian spectral density in the autoregressive moving average model (2011)
  • Author(s): TANAKA Fuyuhiko and KOMAKI Fumiyasu
  • Journal Title: Sankhya Series A, Indian Statistical Institute Volume: Vol.73-A Pages: 162-184
  • Peer Reviewed
• [Journal Article] Superharmonic priors for autoregressive models (2009)
  • Author(s): TANAKA Fuyuhiko
  • Journal Title: Mathematical Engineering Technical Reports Volume: 2009-18 Pages: 1-20
  • URL: http://www.keisu.t.u-tokyo.ac.jp/research/techrep/2009.html
• [Presentation] Curvature form on statistical model manifolds and its application to Bayesian analysis (2012)
  • Author(s): Fuyuhiko Tanaka
  • Organizer: The Fourth International Conference on Mathematical Sciences (ICM 2012)
  • Place of Presentation: Al-Ain, UAE
  • Year and Date: 2012-03-11
• [Presentation] Noninformative prior in quantum pure states model (2012)
  • Author(s): Fuyuhiko Tanaka
  • Organizer: International Workshop on Anomalous Statistics, Generalized Entropies, and Information Geometry
  • Place of Presentation: 奈良
  • Year and Date: 2012-03-08
• [Presentation] Curvature form on statistical model manifolds and its application to Bayesian analysis (2012)
  • Author(s): TANAKA Fuyuhiko
  • Organizer: ICM 2012, Al Ain
• [Presentation] Noninformative prior in quantum pure states model (2012)
  • Author(s): TANAKA Fuyuhiko
  • Organizer: NEXT2012Nara, Nara
  • Place of Presentation: JAPAN
• [Presentation] Hypothesis testing of a maximally entangled state under the unknown unitary process (2011)
  • Author(s): Fuyuhiko Tanaka
  • Organizer: 15th workshop on quantum information processing (QIP2012)
  • Place of Presentation: Montreal, CANADA
  • Year and Date: 2011-12-14
• [Presentation] 波動関数のベイズ推定 (2011)
  • Author(s): 田中冬彦
  • Organizer: 統計的推測の理論と方法論、及び、最近の動向
  • Place of Presentation: 筑波
  • Year and Date: 2011-11-07
• [Presentation] 量子ベイズ予測の手法に基いた波動関数の推定 (2011)
  • Author(s): 田中冬彦
  • Organizer: 2011年度統計関連学会連合大会
  • Place of Presentation: 福岡
  • Year and Date: 2011-09-07
• [Presentation] Application of differential geometry to Bayesian estimation of spectral density (2011)
  • Author(s): Fuyuhiko Tanaka
  • Organizer: 7th International Congress on Industrial and Applied Mathematics (ICIAM 2011)
  • Place of Presentation: Vancouver, CANADA
  • Year and Date: 2011-07-21
• [Presentation] Application of differential geometry to Bayesian estimation of spectral density (2011)
  • Author(s): TANAKA Fuyuhiko
  • Organizer: ICIAM 2011, Vancouver, BC
  • Place of Presentation: CANADA
• [Presentation] スペクトル密度の推定における事前分布の選択 (2010)
  • Author(s): 田中冬彦
  • Organizer: RIMS研究集会"諸分野との協働による数理科学のフロンティア"
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-11-19
• [Presentation] ARモデル多様体上の曲率形式とその応用 (2010)
  • Author(s): 田中冬彦
  • Organizer: 2010年度統計関連学会連合大会
  • Place of Presentation: 早稲田大学 早稲田キャンパス
  • Year and Date: 2010-09-08
• [Presentation] 量子ベイズ統計における事前分布の選択について (2010)
  • Author(s): 田中冬彦
  • Organizer: 学術研究ネット定例セミナー
  • Place of Presentation: 東京芸術劇場
  • Year and Date: 2010-07-24
• [Presentation] 自己回帰過程の優調和事前分布の偏自己相関係数による表示 (2009)
  • Author(s): 田中冬彦
  • Organizer: 2009年度統計関連学会連合大会
  • Place of Presentation: 同志社大学 京田辺キャンパス
  • Year and Date: 2009-09-07
• [Presentation] 量子系における最適ベイズ予測を与える測定の必要十分条件 (2008)
  • Author(s): 田中冬彦
  • Organizer: 2008年度統計関連学会連合大会
  • Place of Presentation: 慶應義塾大学理工学部矢上キャンパス
  • Year and Date: 2008-09-10