Introduction to probability for data science:
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| 245 | 1 | 0 | |a Introduction to probability for data science |c Stanley H. Chan, Purdue University |
| 264 | 1 | |a [Ann Arbor, MI] |b Michigan Publishing |c [2021] | |
| 264 | 4 | |c © 2021 | |
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Datensatz im Suchindex
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| adam_text | Contents 1 Mathematical Background 1 1.1 Infinite Series........................................................................................................... 2 1.1.1 Geometric Series......................................................................................... 3 1.1.2 Binomial Series............................................................................................ 6 1.2 Approximation........................................................................................................ 10 1.2.1 Taylor approximation................................................................................ 11 1.2.2 Exponential series...................................................................................... 12 1.2.3 Logarithmic approximation...................................................................... 13 1.3 Integration .............................................................................................................. 15 1.3.1 Odd and even functions............................................................................. 15 1.3.2 Fundamental Theorem of Calculus.......................................................... 17 1.4 Linear Algebra........................................................................................................ 20 1.4.1 Why do we need linear algebra in data science?.........................................20 1.4.2 Everythingyou need to know about linear algebra................................ 21 1.4.3 Inner products and
norms......................................................................... 24 1.4.4 Matrix calculus............................................................................................ 28 1.5 Basic Combinatorics............................................................................................... 31 1.5.1 Birthday paradox......................................................................................... 31 1.5.2 Permutation ............................................................................................... 33 1.5.3 Combination............................................................................................... 34 1.6 Summary................................................................................................................. 37 1.7 Reference................................................................................................................. 38 1.8 Problems ................................................................................................................. 38 2 Probability 43 2.1 Set Theory............................................................................................................... 44 2.1.1 Why study set theory?................................................................................ 44 2.1.2 Basic concepts of a set................................................................................... 45 2.1.3 Subsets........................................................................................................ 47 2.1.4 Empty set and universal
set.......................................................................... 48 2.1.5 Union........................................................................................................... 48 2.1.6 Intersection.................................................................................................. 50 2.1.7 Complement and difference...................................................................... 52 2.1.8 Disjoint and partition................................................................................... 54 2.1.9 Set operations ............................................................................................ 56 2.1.10 Closing remarks about set theory............................................................. 57 vii
CONTENTS 2.2 2.3 2.4 2.5 2.6 2.7 Probability Space.............................................................. 2.2.1 Sample space ........................................................ 2.2.2 Event space ........................................................... 2.2.3 Probability law ..................................................... 2.2.4 Measure zero sets................................................. 2.2.5 Summary of the probability space................... Axioms of Probability..................................................... 2.3.1 Why these three probability axioms?................ 2.3.2 Axioms through the lens of measure................ 2.3.3 Corollaries derived from the axioms ................ Conditional Probability.................................................. 2.4.1 Definition of conditional probability................ 2.4.2 Independence........................................................ 2.4.3 Bayes’ theorem and the law of total probability 2.4.4 The Three Prisoners problem............................ Summary.......................................................................... References.......................................................................... Problems .......................................................................... 58 59 61 66 71 74 74 75 76 77 80 81 85 89 92 95 96 97 3 Discrete Random Variables 103 3.1 Random Variables..................................................................................................... 105 3.1.1 A motivating
example...................................................................................105 3.1.2 Definition of a random variable................................................................... 105 3.1.3 Probability measure on randomvariables ..................................................107 3.2 Probability Mass Function.........................................................................................110 3.2.1 Definition of probability massfunction........................................................ 110 3.2.2 PMF and probability measure...................................................................... 110 3.2.3 Normalization property............................................................................... 112 3.2.4 PMF versus histogram...................................................................................113 3.2.5 Estimating histograms from real data .......................................................117 3.3 Cumulative Distribution Functions (Discrete).......................................................121 3.3.1 Definition of the cumulative distribution function.................................... 121 3.3.2 Properties of the CDF...................................................................................123 3.3.3 Converting between PMF and CDF .......................................................... 124 3.4 Expectation................................................................................................................. 125 3.4.1 Definition of
expectation................................................................................125 3.4.2 Existence of expectation . . . ................................................................130 3.4.3 Properties of expectation .............................................................................130 3.4.4 Moments and variance................................................................................... 133 3.5 Common Discrete Random Variables.......................................................................136 3.5.1 Bernoulli random variable............................................................................ 137 3.5.2 Binomial random variable............................................................................ 143 3.5.3 Geometric random variable......................................................................... 149 3.5.4 Poisson random variable................................................................................152 3.6 Summary................................................................................ 2g4 3.7 References......................................................................................... 205 3.8 Problems................................. Vlil
CONTENTS 4 Continuous Random Variables 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 5 Joint Distributions 5.1 171 Probability Density Function..................................................................................172 4.1.1 Some intuitions about probability density functions.................................172 4.1.2 More in-depth discussion about PDFs...................................................... 174 4.1.3 Connecting with the PMF............................................................................178 Expectation, Moment, and Variance..................................................................... 180 4.2.1 Definition and properties ........................................................................... 180 4.2.2 Existence of expectation...............................................................................183 4.2.3 Moment and variance..................................................................................184 Cumulative Distribution Function .........................................................................185 4.3.1 CDF for continuous random variables...................................................... 186 4.3.2 Properties of CDF........................................................................................188 4.3.3 Retrieving PDF from CDF........................................................................ 193 4.3.4 CDF: Unifying discrete and continuous random variables .....................194 Median, Mode, and
Mean........................................................................................ 196 4.4.1 Median.......................................................................................................... 196 4.4.2 Mode............................................................................................................. 198 4.4.3 Mean............................................................................................................. 199 Uniform and Exponential Random Variables......................................................... 201 4.5.1 Uniform random variables............................................................................202 4.5.2 Exponential random variables......................................................................205 4.5.3 Origin of exponential random variables...................................................... 207 4.5.4 Applications of exponential random variables.......................................... 209 Gaussian Random Variables.....................................................................................211 4.6.1 Definition of a Gaussian random variable................................................ 211 4.6.2 Standard Gaussian........................................................................................213 4.6.3 Skewness and kurtosis..................................................................................216 4.6.4 Origin of Gaussian random variables ...................................................... 220 Functions of Random
Variables...............................................................................223 4.7.1 General principle...........................................................................................223 4.7.2 Examples.......................................................................................................225 Generating Random Numbers..................................................................................229 4.8.1 General principle...........................................................................................229 4.8.2 Examples.......................................................................................................230 Summary................................................................................................................... 235 Reference...................................................................................................................236 Problems .................................................................................................... 237 Joint 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 5.1.7 5.2 Joint 241 PMF and Joint PDF........................................................................................244 Probability measure in 2D........................................................................... 244 Discrete random variables........................................................................... 245 Continuous random variables..................................................................... 247
Normalization.................................................................................................248 Marginal PMF and marginal PDF............................................................ 250 Independent random variables .................................................................. 251 Joint CDF ....................................................................................................255 Expectation.......................................................................................................257 ix
CONTENTS 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.2.1 Definition and interpretation...................................... 5.2.2 Covariance and correlation coefficient ...................... 5.2.3 Independence and correlation...................................... 5.2.4 Computing correlation from data............................... Conditional PMF and PDF..................................................... 5.3.1 Conditional PMF........................................................... 5.3.2 Conditional PDF........................................................... Conditional Expectation........................................................... 5.4.1 Definition....................................................................... 5.4.2 The law of total expectation ...................................... Sum of Two Random Variables............................................... 5.5.1 Intuition through convolution...................................... 5.5.2 Main result.................................................................... 5.5.3 Sum of common distributions...................................... Random Vectors and Covariance Matrices............................ 5.6.1 PDF of random vectors............................................... 5.6.2 Expectation of random vectors.................................. 5.6.3 Covariance matrix........................................................ 5.6.4 Multidimensional Gaussian......................................... Transformation of Multidimensional Gaussians................... 5.7.1 Linear transformation of mean and
covariance .... 5.7.2 Eigenvalues and eigenvectors...................................... 5.7.3 Covariance matrices are always positive semi-definite 5.7.4 Gaussian whitening ..................................................... Principal-Component Analysis ............................................... 5.8.1 The main idea: Eigendecomposition......................... 5.8.2 The eigenface problem.................................................. 5.8.3 What cannot be analyzed by PCA? ......................... Summary................................................................................... References................................................................................... Problems ................................................................................... 6 Sample Statistics 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 261 263 265 266 267 271 275 275 276 280 280 281 282 286 286 288 289 290 293 293 295 297 299 303 303 309 311 312 313 314 319 Moment-Generating and Characteristic Functions.................................................324 6.1.1 Moment-generating function......................................................................... 324 6.1.2 Sum of independent variables viaMGF ..................................................... 327 6.1.3 Characteristic functions................................................................................329 6.2 Probability Inequalities ............................................................................................ 333 6.2.1 Union
bound.................................................................................................. 333 6.2.2 The Cauchy-Schwarz inequality................................................................... 335 6.2.3 Jensen’s inequality.........................................................................................336 6.2.4 Markov’s inequality ......................................................................................339 6.2.5 Chebyshev’s inequality...................................................................................34I 6.2.6 Chernoff’s bound........................................................................... 343 6.2.7 Comparing Chernoff and Chebyshev.......................................................... 344 6.2.8 Hoeffding’s inequality................................................................................. ... 6.3 Law of Large Numbers.................................................................... շցլ 6.3.1 Sample average........................................................ 351 x
CONTENTS 6.3.2 Weak law of large numbers (WLLN).........................................................354 6.3.3 Convergence in probability ........................................................................ 356 6.3.4 Can we prove WLLN using Chernoff’s bound? .......................................358 6.3.5 Does the weak law of large numbers always hold?....................................359 6.3.6 Strong law of large numbers........................................................................ 360 6.3.7 Almost sure convergence.............................................................................. 362 6.3.8 Proof of the strong law of large numbers................................................... 364 6.4 Central Limit Theorem ...........................................................................................366 6.4.1 Convergence in distribution........................................................................ 367 6.4.2 Central Limit Theorem .............................................................................. 372 6.4.3 Examples.......................................................................................................377 6.4.4 Limitation of the Central Limit Theorem................................................ 378 6.5 Summary...................................................................................................................380 6.6 References...................................................................................................................381 6.7 Problems
...................................................................................................................383 7 Regression 389 7.1 Principles of Regression...........................................................................................394 7.1.1 Intuition: How to fit a straight line?......................................................... 395 7.1.2 Solving the linear regression problem......................................................... 397 7.1.3 Extension: Beyond a straight line............................................................... 401 7.1.4 Overdetermined and underdetermined systems.......................................409 7.1.5 Robust linear regression.............................................................................. 412 7.2 Overfitting ................................................................................................................418 7.2.1 Overview of overfitting................................................................................. 419 7.2.2 Analysis of the linear case........................................................................... 420 7.2.3 Interpreting the linear analysis results...................................................... 425 7.3 Bias and Variance TVade-Off.....................................................................................429 7.3.1 Decomposing the testing error .................................................................. 430 7.3.2 Analysis of the bias .....................................................................................433 7.3.3
Variance..........................................................................................................436 7.3.4 Bias and variance on the learning curve................................................... 438 7.4 Regularization ..........................................................................................................440 7.4.1 Ridge regularization.....................................................................................440 7.4.2 LASSO regularization..................................................................................449 7.5 Summary...................................................................................................................457 7.6 References...................................................................................................................458 7.7 Problems ...................................................................................................................459 8 Estimation 465 8.1 Maximum-Likelihood Estimation........................................................................... 468 8.1.1 Likelihood function........................................................................................468 8.1.2 Maximum-likelihood estimate..................................................................... 472 8.1.3 Application 1: Social network analysis...................................................... 478 8.1.4 Application 2: Reconstructing images ......................................................481 8.1.5 More examples of ML
estimation...............................................................484 8.1.6 Regression versus ML estimation...............................................................487 8.2 Properties of ML Estimates.................................................................................... 491 xi
CONTENTS 8.3 8.4 8.5 8.6 8.7 8.2.1 Estimators .............................................................. 8.2.2 Unbiased estimators............................................... 8.2.3 Consistent estimators ............................................ 8.2.4 Invariance principle ............................................... Maximum A Posteriori Estimation.................................. 8.3.1 The trio of likelihood, prior, and posterior .... 8.3.2 Understanding the priors ..................................... 8.3.3 MAP formulation and solution............................ 8.3.4 Analyzing the MAP solution............................... 8.3.5 Analysis of the posterior distribution................... 8.3.6 Conjugate prior........................................................ 8.3.7 Linking MAP with regression............................... Minimum Mean-Square Estimation.................................. 8.4.1 Positioning the minimum mean-square estimation 8.4.2 Mean squared error ............................................... 8.4.3 MMSE estimate = conditional expectation . . . 8.4.4 MMSE estimator for multidimensional Gaussian 8.4.5 Linking MMSE and neural networks................... Summary............................................................................. References............................................................................. Problems ............................................................................. 9 Confidence and Hypothesis 9.1 9.2 9.3 9.4 9.5 9.6 Xll 491 492 494 500 502 503 504 506 508 511 513 517 520 520 522 523 529 533 534
535 536 541 Confidence Interval................................................................................................... 543 9.1.1 The randomness of an estimator.............................................................. 543 9.1.2 Understanding confidence intervals........................................................... 545 9.1.3 Constructing a confidence interval........................................................... 548 9.1.4 Properties of the confidence interval........................................................ 551 9.1.5 Student’s ¿-distribution ..............................................................................554 9.1.6 Comparing Student’s ¿-distribution and Gaussian.................................. 558 Bootstrapping............................................................................................................ 559 9.2.1 A brute force approach ..............................................................................560 9.2.2 Bootstrapping ............................................................................................... 562 Hypothesis Testing..................................................................................................... 566 9.3.1 What is a hypothesis?................................................................................... 566 9.3.2 Critical-value test ......................................................................................... 567 9.3.3 p-value test..................................................................................................... 571 9.3.4
Я-test and T-test............................................................................................ 574 Neyman-Pearson Test ............................................................................................... 577 9.4.1 Null and alternative distributions.................................................................577 9.4.2 Type 1 and type 2 errors .............................................................................579 9.4.3 Neyman-Pearson decision.............................................................................582 ROC and Precision-Recall Curve.................................................. 589 9.5.1 Receiver Operating Characteristic (ROC)................................................. 589 9.5.2 Comparing ROC curves...................... 5ցշ 9.5.3 The ROC curve in practice..........................................................................5gg 9.5.4 The Precision-Recall (PR) curve.................................................................601 Summary.............................................................. հՈր
CONTENTS 9.7 Reference...................................................................................................................606 9.8 Problems ................................................................................................................... 607 10 Random Processes 611 10.1 Basic Concepts..........................................................................................................612 10.1.1 Everything you need to know abouta random process............................612 10.1.2 Statistical and temporal perspectives......................................................... 614 10.2 Mean and Correlation Functions ........................................................................... 618 10.2.1 Mean function..............................................................................................618 10.2.2 Autocorrelation function...............................................................................622 10.2.3 Independent processes..................................................................................629 10.3 Wide-Sense Stationary Processes........................................................................... 630 10.3.1 Definition of a WSS process........................................................................ 631 10.3.2 Properties of Rx (r).....................................................................................632 10.3.3 Physical interpretation of Rx{t)............................................................... 633 10.4 Power Spectral Density
...........................................................................................636 10.4.1 Basic concepts..............................................................................................636 10.4.2 Origin of the power spectral density......................................................... 640 10.5 WSS Process through LTI Systems........................................................................ 643 10.5.1 Review of linear time-invariant systems................................................... 643 10.5.2 Mean and autocorrelation through LTI Systems.......................................644 10.5.3 Power spectral density through LTIsystems.............................................. 646 10.5.4 Cross-correlation through LTI Systems...................................................... 649 10.6 Optimal Linear Filter..............................................................................................653 10.6.1 Discrete-time random processes.................................................................. 653 10.6.2 Problem formulation.....................................................................................654 10.6.3 Yule-Walker equation ..................................................................................656 10.6.4 Linear prediction...........................................................................................658 10.6.5 Wiener filter.................................................................................................662 10.7
Summary...................................................................................................................669 10.8 Appendix................................................................................................................... 670 10.8.1 The Mean-Square Ergodic Theorem......................................................... 674 10.9 References................................................................................................................... 675 lO.lOProblems ................................................................................................................... 676 A Appendix 681 xiii
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Contents 1 Mathematical Background 1 1.1 Infinite Series. 2 1.1.1 Geometric Series. 3 1.1.2 Binomial Series. 6 1.2 Approximation. 10 1.2.1 Taylor approximation. 11 1.2.2 Exponential series. 12 1.2.3 Logarithmic approximation. 13 1.3 Integration . 15 1.3.1 Odd and even functions. 15 1.3.2 Fundamental Theorem of Calculus. 17 1.4 Linear Algebra. 20 1.4.1 Why do we need linear algebra in data science?.20 1.4.2 Everythingyou need to know about linear algebra. 21 1.4.3 Inner products and
norms. 24 1.4.4 Matrix calculus. 28 1.5 Basic Combinatorics. 31 1.5.1 Birthday paradox. 31 1.5.2 Permutation . 33 1.5.3 Combination. 34 1.6 Summary. 37 1.7 Reference. 38 1.8 Problems . 38 2 Probability 43 2.1 Set Theory. 44 2.1.1 Why study set theory?. 44 2.1.2 Basic concepts of a set. 45 2.1.3 Subsets. 47 2.1.4 Empty set and universal
set. 48 2.1.5 Union. 48 2.1.6 Intersection. 50 2.1.7 Complement and difference. 52 2.1.8 Disjoint and partition. 54 2.1.9 Set operations . 56 2.1.10 Closing remarks about set theory. 57 vii
CONTENTS 2.2 2.3 2.4 2.5 2.6 2.7 Probability Space. 2.2.1 Sample space . 2.2.2 Event space . 2.2.3 Probability law . 2.2.4 Measure zero sets. 2.2.5 Summary of the probability space. Axioms of Probability. 2.3.1 Why these three probability axioms?. 2.3.2 Axioms through the lens of measure. 2.3.3 Corollaries derived from the axioms . Conditional Probability. 2.4.1 Definition of conditional probability. 2.4.2 Independence. 2.4.3 Bayes’ theorem and the law of total probability 2.4.4 The Three Prisoners problem. Summary. References. Problems . 58 59 61 66 71 74 74 75 76 77 80 81 85 89 92 95 96 97 3 Discrete Random Variables 103 3.1 Random Variables. 105 3.1.1 A motivating
example.105 3.1.2 Definition of a random variable. 105 3.1.3 Probability measure on randomvariables .107 3.2 Probability Mass Function.110 3.2.1 Definition of probability massfunction. 110 3.2.2 PMF and probability measure. 110 3.2.3 Normalization property. 112 3.2.4 PMF versus histogram.113 3.2.5 Estimating histograms from real data .117 3.3 Cumulative Distribution Functions (Discrete).121 3.3.1 Definition of the cumulative distribution function. 121 3.3.2 Properties of the CDF.123 3.3.3 Converting between PMF and CDF . 124 3.4 Expectation. 125 3.4.1 Definition of
expectation.125 3.4.2 Existence of expectation . . . .130 3.4.3 Properties of expectation .130 3.4.4 Moments and variance. 133 3.5 Common Discrete Random Variables.136 3.5.1 Bernoulli random variable. 137 3.5.2 Binomial random variable. 143 3.5.3 Geometric random variable. 149 3.5.4 Poisson random variable.152 3.6 Summary. 2g4 3.7 References. 205 3.8 Problems. Vlil
CONTENTS 4 Continuous Random Variables 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 5 Joint Distributions 5.1 171 Probability Density Function.172 4.1.1 Some intuitions about probability density functions.172 4.1.2 More in-depth discussion about PDFs. 174 4.1.3 Connecting with the PMF.178 Expectation, Moment, and Variance. 180 4.2.1 Definition and properties . 180 4.2.2 Existence of expectation.183 4.2.3 Moment and variance.184 Cumulative Distribution Function .185 4.3.1 CDF for continuous random variables. 186 4.3.2 Properties of CDF.188 4.3.3 Retrieving PDF from CDF. 193 4.3.4 CDF: Unifying discrete and continuous random variables .194 Median, Mode, and
Mean. 196 4.4.1 Median. 196 4.4.2 Mode. 198 4.4.3 Mean. 199 Uniform and Exponential Random Variables. 201 4.5.1 Uniform random variables.202 4.5.2 Exponential random variables.205 4.5.3 Origin of exponential random variables. 207 4.5.4 Applications of exponential random variables. 209 Gaussian Random Variables.211 4.6.1 Definition of a Gaussian random variable. 211 4.6.2 Standard Gaussian.213 4.6.3 Skewness and kurtosis.216 4.6.4 Origin of Gaussian random variables . 220 Functions of Random
Variables.223 4.7.1 General principle.223 4.7.2 Examples.225 Generating Random Numbers.229 4.8.1 General principle.229 4.8.2 Examples.230 Summary. 235 Reference.236 Problems . 237 Joint 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 5.1.7 5.2 Joint 241 PMF and Joint PDF.244 Probability measure in 2D. 244 Discrete random variables. 245 Continuous random variables. 247
Normalization.248 Marginal PMF and marginal PDF. 250 Independent random variables . 251 Joint CDF .255 Expectation.257 ix
CONTENTS 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.2.1 Definition and interpretation. 5.2.2 Covariance and correlation coefficient . 5.2.3 Independence and correlation. 5.2.4 Computing correlation from data. Conditional PMF and PDF. 5.3.1 Conditional PMF. 5.3.2 Conditional PDF. Conditional Expectation. 5.4.1 Definition. 5.4.2 The law of total expectation . Sum of Two Random Variables. 5.5.1 Intuition through convolution. 5.5.2 Main result. 5.5.3 Sum of common distributions. Random Vectors and Covariance Matrices. 5.6.1 PDF of random vectors. 5.6.2 Expectation of random vectors. 5.6.3 Covariance matrix. 5.6.4 Multidimensional Gaussian. Transformation of Multidimensional Gaussians. 5.7.1 Linear transformation of mean and
covariance . 5.7.2 Eigenvalues and eigenvectors. 5.7.3 Covariance matrices are always positive semi-definite 5.7.4 Gaussian whitening . Principal-Component Analysis . 5.8.1 The main idea: Eigendecomposition. 5.8.2 The eigenface problem. 5.8.3 What cannot be analyzed by PCA? . Summary. References. Problems . 6 Sample Statistics 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 261 263 265 266 267 271 275 275 276 280 280 281 282 286 286 288 289 290 293 293 295 297 299 303 303 309 311 312 313 314 319 Moment-Generating and Characteristic Functions.324 6.1.1 Moment-generating function. 324 6.1.2 Sum of independent variables viaMGF . 327 6.1.3 Characteristic functions.329 6.2 Probability Inequalities . 333 6.2.1 Union
bound. 333 6.2.2 The Cauchy-Schwarz inequality. 335 6.2.3 Jensen’s inequality.336 6.2.4 Markov’s inequality .339 6.2.5 Chebyshev’s inequality.34I 6.2.6 Chernoff’s bound. 343 6.2.7 Comparing Chernoff and Chebyshev. 344 6.2.8 Hoeffding’s inequality. . 6.3 Law of Large Numbers. շցլ 6.3.1 Sample average. 351 x
CONTENTS 6.3.2 Weak law of large numbers (WLLN).354 6.3.3 Convergence in probability . 356 6.3.4 Can we prove WLLN using Chernoff’s bound? .358 6.3.5 Does the weak law of large numbers always hold?.359 6.3.6 Strong law of large numbers. 360 6.3.7 Almost sure convergence. 362 6.3.8 Proof of the strong law of large numbers. 364 6.4 Central Limit Theorem .366 6.4.1 Convergence in distribution. 367 6.4.2 Central Limit Theorem . 372 6.4.3 Examples.377 6.4.4 Limitation of the Central Limit Theorem. 378 6.5 Summary.380 6.6 References.381 6.7 Problems
.383 7 Regression 389 7.1 Principles of Regression.394 7.1.1 Intuition: How to fit a straight line?. 395 7.1.2 Solving the linear regression problem. 397 7.1.3 Extension: Beyond a straight line. 401 7.1.4 Overdetermined and underdetermined systems.409 7.1.5 Robust linear regression. 412 7.2 Overfitting .418 7.2.1 Overview of overfitting. 419 7.2.2 Analysis of the linear case. 420 7.2.3 Interpreting the linear analysis results. 425 7.3 Bias and Variance TVade-Off.429 7.3.1 Decomposing the testing error . 430 7.3.2 Analysis of the bias .433 7.3.3
Variance.436 7.3.4 Bias and variance on the learning curve. 438 7.4 Regularization .440 7.4.1 Ridge regularization.440 7.4.2 LASSO regularization.449 7.5 Summary.457 7.6 References.458 7.7 Problems .459 8 Estimation 465 8.1 Maximum-Likelihood Estimation. 468 8.1.1 Likelihood function.468 8.1.2 Maximum-likelihood estimate. 472 8.1.3 Application 1: Social network analysis. 478 8.1.4 Application 2: Reconstructing images .481 8.1.5 More examples of ML
estimation.484 8.1.6 Regression versus ML estimation.487 8.2 Properties of ML Estimates. 491 xi
CONTENTS 8.3 8.4 8.5 8.6 8.7 8.2.1 Estimators . 8.2.2 Unbiased estimators. 8.2.3 Consistent estimators . 8.2.4 Invariance principle . Maximum A Posteriori Estimation. 8.3.1 The trio of likelihood, prior, and posterior . 8.3.2 Understanding the priors . 8.3.3 MAP formulation and solution. 8.3.4 Analyzing the MAP solution. 8.3.5 Analysis of the posterior distribution. 8.3.6 Conjugate prior. 8.3.7 Linking MAP with regression. Minimum Mean-Square Estimation. 8.4.1 Positioning the minimum mean-square estimation 8.4.2 Mean squared error . 8.4.3 MMSE estimate = conditional expectation . . . 8.4.4 MMSE estimator for multidimensional Gaussian 8.4.5 Linking MMSE and neural networks. Summary. References. Problems . 9 Confidence and Hypothesis 9.1 9.2 9.3 9.4 9.5 9.6 Xll 491 492 494 500 502 503 504 506 508 511 513 517 520 520 522 523 529 533 534
535 536 541 Confidence Interval. 543 9.1.1 The randomness of an estimator. 543 9.1.2 Understanding confidence intervals. 545 9.1.3 Constructing a confidence interval. 548 9.1.4 Properties of the confidence interval. 551 9.1.5 Student’s ¿-distribution .554 9.1.6 Comparing Student’s ¿-distribution and Gaussian. 558 Bootstrapping. 559 9.2.1 A brute force approach .560 9.2.2 Bootstrapping . 562 Hypothesis Testing. 566 9.3.1 What is a hypothesis?. 566 9.3.2 Critical-value test . 567 9.3.3 p-value test. 571 9.3.4
Я-test and T-test. 574 Neyman-Pearson Test . 577 9.4.1 Null and alternative distributions.577 9.4.2 Type 1 and type 2 errors .579 9.4.3 Neyman-Pearson decision.582 ROC and Precision-Recall Curve. 589 9.5.1 Receiver Operating Characteristic (ROC). 589 9.5.2 Comparing ROC curves. 5ցշ 9.5.3 The ROC curve in practice.5gg 9.5.4 The Precision-Recall (PR) curve.601 Summary. հՈր
CONTENTS 9.7 Reference.606 9.8 Problems . 607 10 Random Processes 611 10.1 Basic Concepts.612 10.1.1 Everything you need to know abouta random process.612 10.1.2 Statistical and temporal perspectives. 614 10.2 Mean and Correlation Functions . 618 10.2.1 Mean function.618 10.2.2 Autocorrelation function.622 10.2.3 Independent processes.629 10.3 Wide-Sense Stationary Processes. 630 10.3.1 Definition of a WSS process. 631 10.3.2 Properties of Rx (r).632 10.3.3 Physical interpretation of Rx{t). 633 10.4 Power Spectral Density
.636 10.4.1 Basic concepts.636 10.4.2 Origin of the power spectral density. 640 10.5 WSS Process through LTI Systems. 643 10.5.1 Review of linear time-invariant systems. 643 10.5.2 Mean and autocorrelation through LTI Systems.644 10.5.3 Power spectral density through LTIsystems. 646 10.5.4 Cross-correlation through LTI Systems. 649 10.6 Optimal Linear Filter.653 10.6.1 Discrete-time random processes. 653 10.6.2 Problem formulation.654 10.6.3 Yule-Walker equation .656 10.6.4 Linear prediction.658 10.6.5 Wiener filter.662 10.7
Summary.669 10.8 Appendix. 670 10.8.1 The Mean-Square Ergodic Theorem. 674 10.9 References. 675 lO.lOProblems . 676 A Appendix 681 xiii |
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| discipline | Soziologie Mathematik |
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| index_date | 2024-07-03T18:43:38Z |
| indexdate | 2024-07-10T09:17:30Z |
| institution | BVB |
| isbn | 9781607857464 |
| language | English |
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| spelling | Chan, Stanley H. 19XX- Verfasser (DE-588)1246939622 aut Introduction to probability for data science Stanley H. Chan, Purdue University [Ann Arbor, MI] Michigan Publishing [2021] © 2021 xiv, 689 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Begleitende Website enthält Video- und Übungsmaterial sowie kostenfreie Downloadmöglichkeit des E-Books für Einzelpersonen Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Data Science (DE-588)1140936166 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Data Science (DE-588)1140936166 s Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s DE-604 Erscheint auch als Online-Ausgabe, PDF 978-1-60785-747-1 https://probability4datascience.com/index.html kostenfrei Begleitmaterial Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033008329&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chan, Stanley H. 19XX- Introduction to probability for data science Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Data Science (DE-588)1140936166 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)1140936166 (DE-588)4123623-3 |
| title | Introduction to probability for data science |
| title_auth | Introduction to probability for data science |
| title_exact_search | Introduction to probability for data science |
| title_exact_search_txtP | Introduction to probability for data science |
| title_full | Introduction to probability for data science Stanley H. Chan, Purdue University |
| title_fullStr | Introduction to probability for data science Stanley H. Chan, Purdue University |
| title_full_unstemmed | Introduction to probability for data science Stanley H. Chan, Purdue University |
| title_short | Introduction to probability for data science |
| title_sort | introduction to probability for data science |
| topic | Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Data Science (DE-588)1140936166 gnd |
| topic_facet | Wahrscheinlichkeitsrechnung Data Science Lehrbuch |
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