Канал: Carlos Fernandez-Granda

Dimensionality Reduction via Principal Component Analysis

Dimensionality Reduction via Principal Component Analysis

The Mathematics Behind Principal Component Analysis

The Mathematics Behind Principal Component Analysis

Principal Component Analysis

Principal Component Analysis

The Covariance Matrix

The Covariance Matrix

Overview of Hypothesis Testing

Overview of Hypothesis Testing

P-Value Abuse! Practical Significance and P Hacking

P-Value Abuse! Practical Significance and P Hacking

HYPOTHESIS TESTING AND CAUSAL INFERENCE

HYPOTHESIS TESTING AND CAUSAL INFERENCE

MULTIPLE TESTING: EVALUATING NBA PLAYERS

MULTIPLE TESTING: EVALUATING NBA PLAYERS

No Model? No Problem! The Permutation Test

No Model? No Problem! The Permutation Test

THE POWER OF HYPOTHESIS TESTS

THE POWER OF HYPOTHESIS TESTS

Statistical Significance: Why the P Value Controls False Positives

Statistical Significance: Why the P Value Controls False Positives

A Two-Sample Test for Antetokounmpo's Free Throws

A Two-Sample Test for Antetokounmpo's Free Throws

THE NULL HYPOTHESIS AND THE P VALUE

THE NULL HYPOTHESIS AND THE P VALUE

ESTIMATION OF POPULATION PARAMETERS FROM RANDOM SAMPLES

ESTIMATION OF POPULATION PARAMETERS FROM RANDOM SAMPLES

Bootstrap Confidence Intervals

Bootstrap Confidence Intervals

THE BOOTSTRAP

THE BOOTSTRAP

CONFIDENCE INTERVALS FOR PROBABILITIES AND PROPORTIONS

CONFIDENCE INTERVALS FOR PROBABILITIES AND PROPORTIONS

CONFIDENCE INTERVALS

CONFIDENCE INTERVALS

How Not To Estimate Risk

How Not To Estimate Risk

THE CENTRAL LIMIT THEOREM

THE CENTRAL LIMIT THEOREM

Why Sums And Averages Tend To Look Gaussian

Why Sums And Averages Tend To Look Gaussian

The Law of Large Numbers Can Fail!

The Law of Large Numbers Can Fail!

The Law of Large Numbers and Deviation Inequalities

The Law of Large Numbers and Deviation Inequalities

THE STANDARD ERROR

THE STANDARD ERROR

RANDOM SAMPLING AND ESTIMATION BIAS

RANDOM SAMPLING AND ESTIMATION BIAS

Overview of Correlation

Overview of Correlation

CORRELATION DOES NOT (USUALLY) IMPLY CAUSATION

CORRELATION DOES NOT (USUALLY) IMPLY CAUSATION

A Geometric Analysis of Correlation

A Geometric Analysis of Correlation

Uncorrelation and Independence

Uncorrelation and Independence

PROPERTIES OF THE CORRELATION COEFFICIENT

PROPERTIES OF THE CORRELATION COEFFICIENT

SIMPLE LINEAR REGRESSION

SIMPLE LINEAR REGRESSION

CORRELATION AND COVARIANCE

CORRELATION AND COVARIANCE

The Correlation Coefficient

The Correlation Coefficient

OVERVIEW OF AVERAGING

OVERVIEW OF AVERAGING

Do ALL CAPS titles increase YouTube views? The Average Treatment Effect

Do ALL CAPS titles increase YouTube views? The Average Treatment Effect

OPTIMAL REGRESSION VIA THE CONDITIONAL MEAN

OPTIMAL REGRESSION VIA THE CONDITIONAL MEAN

ITERATED EXPECTATION AND THE CONDITIONAL MEAN

ITERATED EXPECTATION AND THE CONDITIONAL MEAN

THE CONDITIONAL MEAN FUNCTION

THE CONDITIONAL MEAN FUNCTION

THE VARIANCE, THE STANDARD DEVIATION AND THE MEAN SQUARE

THE VARIANCE, THE STANDARD DEVIATION AND THE MEAN SQUARE

Properties of the Mean

Properties of the Mean

MEAN OF PARAMETRIC DISTRIBUTIONS AND THE METHOD OF MOMENTS

MEAN OF PARAMETRIC DISTRIBUTIONS AND THE METHOD OF MOMENTS

EXTREME VALUES DISTORT THE MEAN

EXTREME VALUES DISTORT THE MEAN

THE MEAN: AVERAGING RANDOM VARIABLES

THE MEAN: AVERAGING RANDOM VARIABLES

HOW NOT TO PREDICT AN ELECTION

HOW NOT TO PREDICT AN ELECTION

Bayesian Models

Bayesian Models

Dependence (and Independence) Between Discrete and Continuous Variables

Dependence (and Independence) Between Discrete and Continuous Variables

Joint Distribution of Discrete and Continuous Random Variables

Joint Distribution of Discrete and Continuous Random Variables

Modeling Multiple Continuous Variables (Overview)

Modeling Multiple Continuous Variables (Overview)

Gaussian Random Vectors: Marginal and Conditional Distributions

Gaussian Random Vectors: Marginal and Conditional Distributions

GAUSSIAN RANDOM VECTORS: DEFINITION AND GEOMETRIC INTUITION

GAUSSIAN RANDOM VECTORS: DEFINITION AND GEOMETRIC INTUITION

Joint Simulation of Random Variables

Joint Simulation of Random Variables

MODELING DEPENDENCE (AND INDEPENDENCE) BETWEEN CONTINUOUS RANDOM VARIABLES

MODELING DEPENDENCE (AND INDEPENDENCE) BETWEEN CONTINUOUS RANDOM VARIABLES

Marginal Distribution Of Continuous Random Variables

Marginal Distribution Of Continuous Random Variables

Probability Density Of Random Vectors

Probability Density Of Random Vectors

Multivariate Continuous Random Variables

Multivariate Continuous Random Variables

CONVERGENCE OF MARKOV CHAINS

CONVERGENCE OF MARKOV CHAINS

Markov Chains

Markov Chains

Why Probabilistic Modeling is Hard: The Curse of Dimensionality and Naive Bayes

Why Probabilistic Modeling is Hard: The Curse of Dimensionality and Naive Bayes

Modeling Multiple Discrete Variables (Overview)

Modeling Multiple Discrete Variables (Overview)

WAS COURTNEY LEE BETTER THAN STEPH CURRY? SIMPSON'S PARADOX

WAS COURTNEY LEE BETTER THAN STEPH CURRY? SIMPSON'S PARADOX

CAUSAL INFERENCE: CONFOUNDERS AND HOW TO ADJUST FOR THEM

CAUSAL INFERENCE: CONFOUNDERS AND HOW TO ADJUST FOR THEM

CAUSAL INFERENCE: POTENTIAL OUTCOMES AND RANDOMIZATION

CAUSAL INFERENCE: POTENTIAL OUTCOMES AND RANDOMIZATION

Conditional Distributions and Independence of Discrete Random Variables

Conditional Distributions and Independence of Discrete Random Variables

MARGINAL DISTRIBUTIONS OF DISCRETE RANDOM VARIABLES

MARGINAL DISTRIBUTIONS OF DISCRETE RANDOM VARIABLES

Multivariate Discrete Random Variables

Multivariate Discrete Random Variables

What is the Mystery Function? (Inverse Transform Sampling)

What is the Mystery Function? (Inverse Transform Sampling)

Fitting Continuous Models to Data via Maximum Likelihood

Fitting Continuous Models to Data via Maximum Likelihood

The Gaussian Distribution

The Gaussian Distribution

The Exponential Distribution

The Exponential Distribution

THE HISTOGRAM AND KERNEL DENSITY ESTIMATION: NONPARAMETRIC DENSITY ESTIMATORS

THE HISTOGRAM AND KERNEL DENSITY ESTIMATION: NONPARAMETRIC DENSITY ESTIMATORS

FUNCTIONS OF CONTINUOUS RANDOM VARIABLES

FUNCTIONS OF CONTINUOUS RANDOM VARIABLES

PROBABILITY DENSITY

PROBABILITY DENSITY

THE CUMULATIVE DISTRIBUTION FUNCTION AND QUANTILES

THE CUMULATIVE DISTRIBUTION FUNCTION AND QUANTILES

BOREL SETS ARE NOT THAT SCARY: MATHEMATICAL DEFINITION OF CONTINUOUS RANDOM VARIABLES

BOREL SETS ARE NOT THAT SCARY: MATHEMATICAL DEFINITION OF CONTINUOUS RANDOM VARIABLES

Overview of Continuous Random Variables

Overview of Continuous Random Variables

PARAMETRIC VS. NONPARAMETRIC MODELS

PARAMETRIC VS. NONPARAMETRIC MODELS

MAXIMUM LIKELIHOOD: FITTING DISCRETE MODELS TO DATA

MAXIMUM LIKELIHOOD: FITTING DISCRETE MODELS TO DATA

The Poisson Distribution: Modeling Earthquakes

The Poisson Distribution: Modeling Earthquakes

THE GEOMETRIC DISTRIBUTION: MODELING DURANT'S FREE-THROW STREAKS

THE GEOMETRIC DISTRIBUTION: MODELING DURANT'S FREE-THROW STREAKS

The Binomial Distribution

The Binomial Distribution

Probability Mass Function: The User Interface of Discrete Random Variables

Probability Mass Function: The User Interface of Discrete Random Variables

Under the Hood: Mathematical Definition of Discrete Random Variables

Under the Hood: Mathematical Definition of Discrete Random Variables

RANDOM VARIABLES ARE WHAT? OVERVIEW OF DISCRETE RANDOM VARIABLES

RANDOM VARIABLES ARE WHAT? OVERVIEW OF DISCRETE RANDOM VARIABLES

Life Is Not A Homework Problem: The Monte Carlo Method

Life Is Not A Homework Problem: The Monte Carlo Method

CONDITIONAL INDEPENDENCE

CONDITIONAL INDEPENDENCE

Independence Between Events: Tom Brady Has (Probably) Nothing To Do With Category 5 Hurricanes

Independence Between Events: Tom Brady Has (Probably) Nothing To Do With Category 5 Hurricanes

Estimating Probabilities From Data

Estimating Probabilities From Data

CONDITIONAL PROBABILITY

CONDITIONAL PROBABILITY

PROBABILITY SPACES

PROBABILITY SPACES

Overview of Probability

Overview of Probability

Trainspotting is the opposite of Star Wars: Matrix Completion For Collaborative Filtering

Trainspotting is the opposite of Star Wars: Matrix Completion For Collaborative Filtering

Low-Rank Models For Matrix Data

Low-Rank Models For Matrix Data

Dimensionality Reduction Via Principal Component Analysis

Dimensionality Reduction Via Principal Component Analysis

Why Eigenvectors? The Mathematics Behind Principal Component Analysis

Why Eigenvectors? The Mathematics Behind Principal Component Analysis

Clustering Via Gaussian Mixture Models

Clustering Via Gaussian Mixture Models

Gaussian Discriminant Analysis And Diagnosis of Alzheimer's

Gaussian Discriminant Analysis And Diagnosis of Alzheimer's

Uncorrelation and Independence Between Random Variables

Uncorrelation and Independence Between Random Variables

Properties of the Correlation Coefficient

Properties of the Correlation Coefficient

Linear Minimum-Mean-Squared Error Estimation

Linear Minimum-Mean-Squared Error Estimation

Correlation

Correlation

Building and Interpreting Confidence Intervals

Building and Interpreting Confidence Intervals

The Central Limit Theorem

The Central Limit Theorem

The Law of Large Numbers Can Fail: When Not to Trust the Sample Mean

The Law of Large Numbers Can Fail: When Not to Trust the Sample Mean

Markov and Chebyshev's Inequalities: How to Control Probabilities Using the Mean and Variance

Markov and Chebyshev's Inequalities: How to Control Probabilities Using the Mean and Variance

The Law of Large Numbers

The Law of Large Numbers

Conditional Expectation

Conditional Expectation

The Mean Square, Variance and Standard Deviation of a Random Variable

The Mean Square, Variance and Standard Deviation of a Random Variable

Properties of the Mean of a Random Variable

Properties of the Mean of a Random Variable

Mean of the Bernoulli, Geometric, Poisson, Exponential and Gaussian Distributions

Mean of the Bernoulli, Geometric, Poisson, Exponential and Gaussian Distributions

The Mean or Expected Value of a Random Variable

The Mean or Expected Value of a Random Variable

Bayesian Parametric Modeling

Bayesian Parametric Modeling

How to Predict an Election

How to Predict an Election

Joint Distribution of Discrete and Continuous Random Variables

Joint Distribution of Discrete and Continuous Random Variables

Gaussian Random Vectors

Gaussian Random Vectors

Marginal and Conditional Distributions of Continuous Random Variables

Marginal and Conditional Distributions of Continuous Random Variables

Multivariate Continuous Random Variables, Joint CDF, Joint PDF, and Multidimensional KDE

Multivariate Continuous Random Variables, Joint CDF, Joint PDF, and Multidimensional KDE

The Curse of Dimensionality and the Naive Bayes Classifier

The Curse of Dimensionality and the Naive Bayes Classifier

Markov Chains

Markov Chains

Marginal and Conditional Distributions of Discrete Random Variables

Marginal and Conditional Distributions of Discrete Random Variables

Multivariate Discrete Random Variables

Multivariate Discrete Random Variables

Simulating Continuous Distributions via Inverse Transform Sampling

Simulating Continuous Distributions via Inverse Transform Sampling

Maximum Likelihood Estimation for Continuous Parametric Models

Maximum Likelihood Estimation for Continuous Parametric Models

The Exponential and Gaussian Distributions

The Exponential and Gaussian Distributions

The Histogram and Kernel Density Estimation

The Histogram and Kernel Density Estimation

The Probability Density Function

The Probability Density Function

Continuous Random Variables, the Cumulative Distribution Function and Quantiles

Continuous Random Variables, the Cumulative Distribution Function and Quantiles

Comparing Parametric and Nonparametric Models

Comparing Parametric and Nonparametric Models

Maximum Likelihood Estimation for Discrete Parametric Models

Maximum Likelihood Estimation for Discrete Parametric Models

The Bernoulli, Binomial, Geometric and Poisson Distributions

The Bernoulli, Binomial, Geometric and Poisson Distributions

Discrete Random Variables Without the Agonizing Pain

Discrete Random Variables Without the Agonizing Pain

The Monte Carlo Method

The Monte Carlo Method

Independence and Conditional Independence

Independence and Conditional Independence

Estimating Probabilities from Data

Estimating Probabilities from Data

Conditional Probability

Conditional Probability

Definition of Probability

Definition of Probability

Properties of expectation

Properties of expectation

The mean and the median

The mean and the median

Principal component analysis

Principal component analysis

Gaussian random vectors

Gaussian random vectors

The covariance matrix

The covariance matrix

Deep learning for denoising electron-microscope images

Deep learning for denoising electron-microscope images

Convolutional neural networks for image denoising

Convolutional neural networks for image denoising

The Wiener filter

The Wiener filter

Stationary signals and principal component analysis

Stationary signals and principal component analysis

Linear translation-invariant models and convolution

Linear translation-invariant models and convolution

Denoising via thresholding

Denoising via thresholding

Multiresolution analysis based on wavelets

Multiresolution analysis based on wavelets

The short time Fourier transform (or spectrogram)

The short time Fourier transform (or spectrogram)

Time-frequency analysis via windowing

Time-frequency analysis via windowing

Fourier transformations in multiple dimensions

Fourier transformations in multiple dimensions

The discrete Fourier transform

The discrete Fourier transform

The sampling theorem for bandlimited signals

The sampling theorem for bandlimited signals

The Fourier series

The Fourier series

Analysis of the lasso for a simple example

Analysis of the lasso for a simple example

Subgradients

Subgradients

Convexity

Convexity

Sparse regression via the lasso

Sparse regression via the lasso

Regularization via early stopping in linear models

Regularization via early stopping in linear models

Ridge regression

Ridge regression

Training and test error of linear regression models

Training and test error of linear regression models

Analysis of linear-regression coefficients

Analysis of linear-regression coefficients

Mean squared error estimation

Mean squared error estimation

Ordinary least squares

Ordinary least squares