Contents -- Foreword -- Preface -- 1. Entropy and Martingale K. B. Athreya and M. G. Nadkarni -- 1.1. Introduction -- 1.2. Relative Entropy and Gibbs-Boltzmann Measures -- 1.2.1. Entropy Maximization Results -- 1.2.2. Weak Convergence of Gibbs-Boltzmann Distribution -- 1.2.3. Relative Entropy and Conditioning -- 1.3. Measure Free Martingales, Weak Martingales, Martingales -- 1.3.1. Finite Range Case -- 1.3.2. The General Case -- 1.4. Equivalent Martingale Measures -- References -- 2. Marginal Quantiles: Asymptotics for Functions of Order Statistics G. J. Babu -- 2.1. Introduction -- 2.1.1. Streaming Data -- 2.2. Marginal Quantiles -- 2.2.1. Joint Distribution of Marginal Quantiles -- 2.2.2. Weak Convergence of Quantile Process -- 2.3. Regression under Lost Association -- 2.4. Mean of Functions of Order Statistics -- 2.5. Examples -- Acknowledgment -- References -- 3. Statistics on Manifolds with Applications to Shape Spaces R. Bhattacharya and A. Bhattacharya -- 3.1. Introduction -- 3.2. Geometry of Shape Manifolds -- 3.2.1. The Real Projective Space RPd -- 3.2.2. Kendall's (Direct Similarity) Shape Spaces Σk -- 3.2.3. Reflection (Similarity) Shape Spaces RSk m -- 3.2.4. Affine Shape Spaces ASk m -- 3.2.5. Projective Shape Spaces PΣk m -- 3.3. Fréchet Means on Metric Spaces -- 3.4. Extrinsic Means on Manifolds -- 3.4.1. Asymptotic Distribution of the Extrinsic Sample Mean -- 3.5. Intrinsic Means on Manifolds -- 3.6. Applications -- 3.6.1. Sd -- 3.6.1.1. Extrinsic Mean on Sd -- 3.6.1.2. Intrinsic Mean on Sd -- 3.6.2. RPd -- 3.6.2.1. Extrinsic Mean on RPd -- 3.6.2.2. Intrinsic Mean on RPd -- 3.6.3. Σk m -- 3.6.4. Σk2 -- 3.6.4.1. Extrinsic Mean on Σk2 -- 3.6.4.2. Intrinsic Mean on Σk2 -- 3.6.5. RΣk m -- 3.6.6. AΣk m -- 3.6.7. P0Σk m -- 3.7. Examples -- 3.7.1. Example 1: Gorilla Skulls -- 3.7.2. Example 2: Schizophrenic Children.

3.7.3. Example 3: Glaucoma Detection -- Acknowledgment -- References -- 4. Reinforcement Learning - A Bridge Between Numerical Methods and Monte Carlo V. S. Borkar -- 4.1. Introduction -- 4.2. Stochastic Approximation -- 4.3. Estimating Stationary Averages -- 4.4. Function Approximation -- 4.5. Estimating Stationary Distribution -- 4.6. Acceleration Techniques -- 4.7. Future Directions -- References -- 5. Factors, Roots and Embeddings of Measures on Lie Groups S. G. Dani -- 5.1. Introduction -- 5.2. Some Basic Properties of Factors and Roots -- 5.3. Factor Sets -- 5.4. Compactness -- 5.5. Roots -- 5.6. One-Parameter Semigroups -- References -- 6. Higher Criticism in the Context of Unknown Distribution, Non-independence and Classi.cation A. Delaigle and P. Hall -- 6.1. Introduction -- 6.2. Methodology -- 6.2.1. Higher-criticism signal detection -- 6.2.2. Generalising and adapting to an unknown null distribution -- 6.2.3. Classifiers based on higher criticism -- 6.3. Theoretical Properties -- 6.3.1. Effectiveness of approximation to hcW by hcW -- 6.3.2. Removing the assumption of independence -- 6.3.3. Delineating good performance -- 6.4. Further Results -- 6.4.1. Alternative constructions of hcW and hcW -- 6.4.2. Advantages of incorporating the threshold -- 6.5. Numerical Properties in the Case of Classification -- 6.6. Technical Arguments -- 6.6.1. Proof of Theorem 6.1 -- 6.6.2. Proof of Theorem 6.2 -- References -- Appendix -- A.1. Description of the Cross-Validation Procedure -- A.2. Proof of Theorem 6.3 -- 7. Bayesian Nonparametric Approach to Multiple Testing S. Ghosal and A. Roy -- 7.1. Bayesian Nonparametric Inference -- 7.2. Multiple Hypothesis Testing -- 7.3. Bayesian Mixture Models for p-Values -- 7.3.1. Independent case: Beta mixture model for p-values -- 7.3.2. Dependent case: Skew-normal mixture model for probit p-values.

7.4. Areas of Application -- References -- 8. Bayesian Inference on Finite Mixtures of Distributions K. Lee, J.-M. Marin, K. Mengersen and C. Robert -- 8.1. Introduction -- 8.2. Finite Mixtures -- 8.2.1. Definition -- 8.2.2. Missing data -- 8.2.3. The necessary but costly expansion of the likelihood -- 8.2.4. Exact posterior computation -- 8.3. Mixture Inference -- 8.3.1. Nonidentifiability, hence label switching -- 8.3.2. Restrictions on priors -- 8.4. Inference for Mixtures with a Known Number of Components -- 8.4.1. Reordering -- 8.4.2. Data augmentation and Gibbs sampling approximations -- 8.4.3. Metropolis-Hastings approximations -- 8.5. Inference for Mixture Models with an Unknown Number of Components -- Acknowledgments -- References -- 9. Markov Processes Generated by Random Iterates of Monotone Maps: Theory and Applications M. Majumdar -- 9.1. Introduction -- 9.2. Random Dynamical Systems -- 9.3. Evolution -- 9.4. Splitting -- 9.4.1. Splitting and Monotone Maps -- 9.4.2. An Extension and Some Applications -- 9.4.3. Extinction and Growth -- 9.5. Invariant Distributions: Computation and Estimation -- 9.5.1. An Estimation Problem -- 9.6. Growth Under Uncertainty -- 9.6.1. A Stochastic Stability Theorem in a Descriptive Model -- 9.6.2. One Sector Log-Cobb-Douglas Optimal Growth -- 9.6.2.1. The Support of the Invariant Distribution -- Acknowledgments -- References -- 10. An Invitation to Quantum Information Theory K. R. Parthasarathy -- 10.1. Introduction -- 10.2. Elements of Finite Dimensional Quantum Probability -- 10.3. Quantum Error-Correcting Codes -- 10.4. Testing Quantum Hypotheses -- References -- 11. Scaling Limits S. R. S. Varadhan -- 11.1. Introduction -- 11.2. Non-Interacting Case -- 11.3. Simple Exclusion Processes -- 11.4. Large Deviations -- References -- Author Index -- Subject Index -- Contents of Part II.