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| Foreword | |
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| Preface | |
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| Survey of probability theory | |
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| Introduction | |
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| Experiments, Sample Spaces, and Probability | |
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| Experiments and Sample Spaces | |
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| Set Theory | |
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| Events and Probability | |
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| Conditional Probability | |
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| Binomial Coefficients | |
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| Exercises | |
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| Random Variables, Random Vectors, and Distribution Functions | |
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| Random Variables and Their Distributions | |
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| Multivariate Distributions | |
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| Sums and Integrals | |
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| Marginal Distributions and Independence | |
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| Vectors and Matrices | |
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| Expectations, Moments, and Characteristic Functions | |
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| Transformations of Random Variables | |
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| Conditional Distributions | |
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| Exercises | |
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| Some Special Univariate Distributions | |
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| Introduction | |
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| The Bernoulli Distribution | |
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| The Binomial Distribution | |
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| The Poisson Distribution | |
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| The Negative Binomial Distribution | |
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| The Hypergeometric Distribution | |
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| The Normal Distribution | |
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| The Gamma Distribution | |
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| The Beta Distribution | |
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| The Uniform Distribution | |
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| The Pareto Distribution | |
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| The t Distribution | |
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| The F Distribution | |
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| Exercises | |
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| Some Special Multivariate Distributions | |
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| Introduction | |
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| The Multinomial Distribution | |
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| The Dirichlet Distribution | |
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| The Multivariate Normal Distribution | |
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| The Wishart Distribution | |
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| The Multivariate t Distribution | |
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| The Bilateral Bivariate Pareto Distribution | |
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| Exercises | |
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| Subjective probability and utility | |
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| Subjective Probability | |
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| Introduction | |
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| Relative Likelihood | |
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| The Auxiliary Experiment | |
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| Construction of the Probability Distribution | |
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| Verification of the Properties of a Probability Distribution | |
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| Conditional Likelihoods | |
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| Exercises | |
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| Utility | |
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| Preferences among Rewards | |
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| Preferences among Probability Distributions | |
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| The Definition of a Utility Function | |
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| Some Properties of Utility Functions | |
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| The Utility of Monetary Rewards | |
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| Convex and Concave Utility Functions | |
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| The Axiomatic Development of Utility | |
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| Construction of the Utility Function | |
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| Verification of the Properties of a Utility Function | |
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| Extension of the Properties of a Utility Function to the Class P[subscript E] | |
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| Exercises | |
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| Statistical decision problems | |
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| Decision Problems | |
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| Elements of a Decision Problem | |
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| Bayes Risk and Bayes Decisions | |
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| Nonnegative Loss Functions | |
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| Concavity of the Bayes Risk | |
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| Randomization and Mixed Decisions | |
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| Convex Sets | |
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| Decision Problems in Which [similar]2 and D Are Finite | |
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| Decision Problems with Observations | |
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| Construction of Bayes Decision Functions | |
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| The Cost of Observation | |
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| Statistical Decision Problems in Which Both [Omega] and D Contain Two Points | |
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| Computation of the Posterior Distribution When the Observations Are Made in More Than One Stage | |
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| Exercises | |
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| Conjugate Prior Distributions | |
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| Sufficient Statistics | |
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| Conjugate Families of Distributions | |
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| Construction of the Conjugate Family | |
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| Conjugate Families for Samples from Various Standard Distributions | |
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| Conjugate Families for Samples from a Normal Distribution | |
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| Sampling from a Normal Distribution with Unknown Mean and Unknown Precision | |
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| Sampling from a Uniform Distribution | |
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| A Conjugate Family for Multinomial Observations | |
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| Conjugate Families for Samples from a Multivariate Normal Distribution | |
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| Multivariate Normal Distributions with Unknown Mean Vector and Unknown Precision Matrix | |
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| The Marginal Distribution of the Mean Vector | |
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| The Distribution of a Correlation | |
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| Precision Matrices Having an Unknown Factor | |
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| Exercises | |
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| Limiting Posterior Distributions | |
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| Improper Prior Distributions | |
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| Improper Prior Distributions for Samples from a Normal Distribution | |
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| Improper Prior Distributions for Samples from a Multivariate Normal Distribution | |
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| Precise Measurement | |
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| Convergence of Posterior Distributions | |
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| Supercontinuity | |
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| Solutions of the Likelihood Equation | |
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| Convergence of Supercontinuous Functions | |
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| Limiting Properties of the Likelihood Function | |
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| Normal Approximation to the Posterior Distribution | |
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| Approximations for Vector Parameters | |
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| Posterior Ratios | |
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| Exercises | |
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| Estimation, Testing Hypotheses, and Linear Statistical Models | |
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| Estimation | |
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| Quadratic Loss | |
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| Loss Proportional to the Absolute Value of the Error | |
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| Estimation of a Vector | |
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| Problems of Testing Hypotheses | |
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| Testing a Simple Hypothesis about the Mean of a Normal Distribution | |
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| Testing Hypotheses about the Mean of a Normal Distribution When the Precision Is Unknown | |
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| Deciding Whether a Parameter Is Smaller or Larger Than a Specified Value | |
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| Deciding Whether the Mean of a Normal Distribution Is Smaller or Larger Than a Specified Value | |
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| Linear Models | |
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| Testing Hypotheses in Linear Models | |
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| Investigating the Hypothesis That Certain Regression Coefficients Vanish | |
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| One-way Analysis of Variance | |
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| Exercises | |
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| Sequential decisions | |
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| Sequential Sampling | |
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| Gains from Sequential Sampling | |
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| Sequential Decision Procedures | |
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| The Risk of a Sequential Decision Procedure | |
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| Backward Induction | |
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| Optimal Bounded Sequential Decision Procedures | |
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| Illustrative Examples | |
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| Unbounded Sequential Decision Procedures | |
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| Regular Sequential Decision Procedures | |
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| Existence of an Optimal Procedure | |
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| Approximating an Optimal Procedure by Bounded Procedures | |
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| Regions for Continuing or Terminating Sampling | |
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| The Functional Equation | |
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| Approximations and Bounds for the Bayes Risk | |
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| The Sequential Probability-ratio Test | |
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| Characteristics of Sequential Probability-ratio Tests | |
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| Approximating the Expected Number of Observations | |
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| Exercises | |
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| Optimal Stopping | |
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| Introduction | |
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| The Statistician's Reward | |
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| Choice of the Utility Function | |
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| Sampling without Recall | |
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| Further Problems of Sampling with Recall and Sampling without Recall | |
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| Sampling without Recall from a Normal Distribution with Unknown Mean | |
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| Sampling with Recall from a Normal Distribution with Unknown Mean | |
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| Existence of Optimal Stopping Rules | |
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| Existence of Optimal Stopping Rules for Problems of Sampling with Recall and Sampling without Recall | |
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| Martingales | |
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| Stopping Rules for Martingales | |
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| Uniformly Integrable Sequences of Random Variables | |
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| Martingales Formed from Sums and Products of Random Variables | |
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| Regular Supermartingales | |
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| Supermartingales and General Problems of Optimal Stopping | |
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| Markov Processes | |
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| Stationary Stopping Rules for Markov Processes | |
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| Entrance-fee Problems | |
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| The Functional Equation for a Markov Process | |
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| Exercises | |
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| Sequential Choice of Experiments | |
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| Introduction | |
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| Markovian Decision Processes with a Finite Number of Stages | |
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| Markovian Decision Processes with an Infinite Number of Stages | |
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| Some Betting Problems | |
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| Two-armed-bandit Problems | |
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| Two-armed-bandit Problems When the Value of One Parameter Is Known | |
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| Two-armed-bandit Problems When the Parameters Are Dependent | |
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| Inventory Problems | |
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| Inventory Problems with an Infinite Number of Stages | |
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| Control Problems | |
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| Optimal Control When the Process Cannot Be Observed without Error | |
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| Multidimensional Control Problems | |
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| Control Problems with Actuation Errors | |
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| Search Problems | |
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| Search Problems with Equal Costs | |
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| Uncertainty Functions and Statistical Decision Problems | |
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| Sufficient Experiments | |
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| Examples of Sufficient Experiments | |
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| Exercises | |
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| References | |
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| Supplementary Bibliography | |
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| Name Index | |
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| Subject Index | |