Bayesian Analysis

Hellinger Distance and Non-informative Priors

Arkady Shemyakin

Full-text: Open access

Abstract

This paper introduces an extension of the Jeffreys’ rule to the construction of objective priors for non-regular parametric families. A new class of priors based on Hellinger information is introduced as Hellinger priors. The main results establish the relationship of Hellinger priors to the Jeffreys’ rule priors in the regular case, and to the reference and probability matching priors for the non-regular class introduced by Ghosal and Samanta. These priors are also studied for some non-regular examples outside of this class. Their behavior proves to be similar to that of the reference priors considered by Berger, Bernardo, and Sun, however some differences are observed. For the multi-parameter case, a combination of Hellinger priors and reference priors is suggested and some examples are considered.

Article information

Source
Bayesian Anal., Volume 9, Number 4 (2014), 923-938.

Dates
First available in Project Euclid: 21 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.ba/1416579185

Digital Object Identifier
doi:10.1214/14-BA881

Mathematical Reviews number (MathSciNet)
MR3293962

Zentralblatt MATH identifier
1327.62164

Keywords
non-informative prior Jeffreys’ rule reference prior matching probability prior Hellinger distance Hellinger information

Citation

Shemyakin, Arkady. Hellinger Distance and Non-informative Priors. Bayesian Anal. 9 (2014), no. 4, 923--938. doi:10.1214/14-BA881. https://projecteuclid.org/euclid.ba/1416579185


Export citation

References

  • Aguilar, O. and West, M. (2000). “Bayesian Dynamic Factor Models and Portfolio Allocation.” Journal of Business and Economic Statistics, 18(3): 338–357.
  • Andersen, T. G., Bollerslev, T., Diebold, F. X., and Ebens, H. (2001). “The Distribution of Realized Stock Return Volatility.” Journal of Financial Econometrics, 61: 43–76.
  • Asai, M. and McAleer, M. (2009). “The Structure of Dynamic Correlations in Multivariate Stochastic Volatility Models.” Journal of Econometrics, 150: 182–192.
  • Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., and Shephard, N. (2009). “Realized Kernels in Practice: Trades and Quotes.” Econometrics Journal, 12(3): C1–C32.
  • — (2011). “Multivariate Realized Kernels: Consistent Positive Semi-Definite Estimators of the Covariation of Equity Prices with Noise and Non-Synchronous Trading.” Journal of Econometrics, 162: 149–169.
  • Barndorff-Nielsen, O. E. and Shephard, N. (2002). “Econometric Analysis of Realized Volatility and Its Use in Estimating Stochastic Volatility.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(2): 253–280.
  • Bauer, G. H. and Vorkink, K. (2011). “Forecasting Multivariate Realized Stock Market Volatility.” Journal of Econometrics, 160: 93–101.
  • Bauwens, L., Laurent, S., and Rombouts, J. V. K. (2006). “Multivariate GARCH Models: a Survey.” Journal of Applied Econometrics, 21: 7109.
  • Bollerslev, T. (1986). “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics, 31: 307–327.
  • Brown, R. G. (1959). Statistical Forecasting for Inventory Control. McGraw-Hill.
  • Carter, C. K. and Kohn, R. (1994). “On Gibbs Sampling for State Space Models.” Biometrika, 81: 541–533.
  • Carvalho, C. M., Lopes, H. F., and Aguilar, O. (2011). “Dynamic Stock Selection Strategies: A Structured Factor Model Framework.” In Bayesian Statistics 9. Oxford University Press.
  • Chib, S., Nardari, F., and Shephard, N. (2002). “Markov Chain Monte Carlo Methods for Stochastic Volatility Models.” Journal of Econometrics, 108: 281–316.
  • Chiriac, R. and Voev, V. (2010). “Modelling and Forecasting Multivariate Realized Volatility.” Journal of Applied Econometrics, 26: 922–947.
  • Chiu, T. Y. M., Leonard, T., and Tsui, K.-W. (1996). “The Matrix-Logarithmic Covariance Model.” Journal of the American Statistical Association, 91: 198–210.
  • Choi, C. and Christensen, H. I. (2011). “Robust 3D Visual Tracking Using Particle Filters on the SE(3) Group.” In IEEE International Conference on Robotics and Automation, 4384–4390.
  • Cochrane, J. H. (2005). Asset Pricing. Princeton University Press.
  • Díaz-García, J. A. and Jáimez, R. G. (1997). “Proof of the Conjectures of H. Uhlig on the Singular Multivariate Beta and the Jacobian of a Certain Matrix Transformation.” The Annals of Statistics, 25: 2018–2023.
  • Edelman, A. (2005). “The Mathematics and Applications of (Finite) Random Matrices.” See handouts 1-4. URL http://web.mit.edu/18.325/www/handouts.html
  • Fama, E. F. and French, K. R. (1993). “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics, 33: 3–56.
  • Fox, E. B. and West, M. (2011). “Autoregressive Models for Variance Matrices: Stationary Inverse Wishart Processes.” Technical report, Duke University.
  • Früwirth-Schnatter, S. (1994). “Data Augmentation and Dynamic Linear Models.” Journal of Time Series Analysis, 15: 183–202.
  • Gourieroux, C., Jasiak, J., and Sufana, R. (2009). “The Wishart Autoregressive Process of Multivariate Stochastic Volatility.” Journal of Econometrics, 150: 167–181.
  • Harvey, A., Ruiz, E., and Shephard, N. (1994). “Multivariate Stochastic Volatility Models.” The Review of Economic Studies, 61: 247–264.
  • Hauberg, S., Lauze, F., and Pedersen, K. S. (2013). “Unscented Kalman Filtering on Riemannian Manifolds.” Journal of Mathematical Imaging and Vision, 46: 103–120.
  • Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems For Stochastic Processes. Springer.
  • Jacquier, E., Polson, N. G., and Rossi, P. E. (2004). “Bayesian Analysis of Stochastic Volatility Models with Fat-Tails and Correlated Errors.” Journal of Econometrics, 122: 185–212.
  • Jin, X. and Maheu, J. M. (2012). “Modelling Realized Covariances and Returns.” URL http://homes.chass.utoronto.ca/ jmaheu/jin-maheu.pdf
  • Julier, S. J. and Uhlmann, J. K. (1997). “New Extensions of the Kalman Filter to Nonlinear Systems.” In Signal Processing, Sensor Fusion, and Target Recognition VI, volume 3068.
  • Kalman, R. E. (1960). “A New Approach to Linear Filtering and Prediction Problems.” Journal of Basic Engineering, 82 (Series D): 35–45.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer.
  • Koopman, S. J., Jungbacker, B., and Hol, E. (2005). “Forecasting Daily Variability of the S&P 100 Stock Index using Historical, Realised and Implied Volatility Measurements.” Journal of Empirical Finance, 12: 445–475.
  • Lence, S. H. and Hayes, D. J. (1994a). “The Empirical Minimum-Variance Hedge.” American Journal of Agricultural Economics, 76: 94–104.
  • — (1994b). “Parameter-Based Decision Making Under Estimation Risk: An Application to Futures Trading.” The Journal of Finance, 49: 345–357.
  • Liu, Q. (2009). “On Portfolio Optimization: How and When Do We Benefit from High-Frequency Data?” Journal of Applied Econometrics, 24: 560–582.
  • Loddo, A., Ni, S., and Sun, D. (2011). “Selection of Multivariate Stochastic Volatility Models via Bayesian Stochastic Search.” Journal of Business and Economic Statistics, 29: 342–355.
  • Mikusiński, P. and Taylor, M. D. (2002). An Introduction to Multivariate Analysis. Birkhäuser.
  • Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley.
  • Philipov, A. and Glickman, M. E. (2006). “Multivariate Stochastic Volatility via Wishart Processes.” Journal of Business and Economic Statistics, 24: 313–328.
  • Pitt, M. K. and Walker, S. G. (2005). “Constructing Stationary Time Series Models Using Auxiliary Variables with Applications.” Journal of the American Statistical Association, 100(470): 554–564.
  • Prado, R. and West, M. (2010). Time Series: Modeling, Computation, and Inference, chapter Multivariate DLMs and Covariance Models, 263–319. Chapman & Hall/CRC.
  • Quintana, J. M. and West, M. (1987). “An Analysis of International Exchange Rates Using Multivariate DLMs.” The Statistician, 36: 275–281.
  • Shephard, N. (1994). “Local Scale Models: State Space Alternative to Integrated GARCH Processes.” Journal of Econometrics, 60: 181–202.
  • — (2005). Stochastic Volatility: Selected Readings. Oxford University Press.
  • Srivastava, A. and Klassen, E. (2004). “Bayesian and Geometric Subspace Tracking.” Advances in Applied Probability, 36(1): 43–56.
  • Taylor, S. J. (1982). Financial Returns Modelled by the Product of Two Stochastic Processes–a Study of Daily Sugar Prices 1961-1979, 203–226. Amersterdam: North-Holland.
  • Tompkins, F. and Wolfe, P. J. (2007). “Bayesian Filtering on the Stiefel Manifold.” In Computational Advances in Multi-Sensor Adaptive Processing, 261 – 264.
  • Triantafyllopoulos, K. (2008). “Multivariate stochastic volatility with Bayesian dynamic linear models.” Journal of Statistical Planning and Inference, 138: 1021–1037.
  • Tyagi, A. and Davis, J. W. (2008). “A Recursive Filter For Linear Systems on Riemannian Manifolds.” In IEEE Conference on Computer Vision and Pattern Recognition.
  • Uhlig, H. (1994). “On Singular Wishart and Singular Multivariate Beta Distributions.” The Annals of Statistics, 22(1): 395–495.
  • — (1997). “Bayesian Vector Autoregressions with Stochastic Volatility.” Econometrica, 65(1): 59–73.
  • West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models. Springer Verlag.
  • Windle, J. (2013). “Forecasting High-Dimensional Variance-Covariance Matrices with High-Frequency Data and Sampling Pólya-Gamma Random Variates for Posterior Distributions Derived from Logistic Likelihoods.” Ph.D. thesis, The University of Texas at Austin.
  • Windle, J., Carvalho, C. M., Scott, J. G., and Sun, L. (2013). “Efficient Data Augmentation in Dynamic Models for Binary and Count Data.” URL http://arxiv.org/abs/1308.0774