Independence
Two events or random quantities are independent when knowing the outcome or value of one does not change the probability or distribution of the other.
Grokking Machine Learning
A probabilistic relationship in which the occurrence of one event does not change the probability of another event.
- How it is used
- Uses independence to estimate probabilities for multiple email words in a spam-detection model.
- How deep it goes
- Defines independence, states the product rule, and applies the assumption in estimates and exercises.
- Examples
- Uses gender and eye color, email words, coin and dice outcomes, and weather-event pairs.
- What you can do after
- Evaluate whether event pairs are independent and estimate joint probabilities under an independence assumption.
Effective Data Analysis
Independence is a condition in which the occurrence or value of one observation does not affect another observation.
- How it is used
- Frames independence as an assumption to check when choosing a parametric test.
- How deep it goes
- Defines the assumption and illustrates violations, without providing a formal diagnostic procedure.
- Examples
- Uses influenced survey answers, coworkers comparing A/B interfaces, and control participants observing intervention outcomes.
- What you can do after
- Evaluate/understand whether participant interactions threaten independence before applying a standard parametric test.
Statistics Every Programmer Needs
Independence is the condition that one random quantity or error does not depend on another.
- How it is used
- Uses independence to select probability rules and validate linear-regression assumptions.
- How deep it goes
- Explains repeated-trial intuition, rule selection, conditional-probability boundaries, and regression assumptions without formal tests or proofs.
- Examples
- Uses coin flips, die rolls, repeated card selection with replacement, and regression residuals as concrete examples.
- What you can do after
- Evaluate whether probability multiplication applies and recognize independence as a regression assumption requiring validation.
Read how Statistics Every Programmer Needs explains independence →
Math and Architectures of Deep Learning
A variable that is statistically unrelated to another variable, with their joint probability equal to the product of their marginal probabilities.
- How it is used
- Uses independence to construct joint distributions and simplify conditional probabilities in machine-learning probability reasoning.
- How deep it goes
- Defines independence, derives the product criterion, checks tables numerically, and contrasts it with dependence; it does not develop continuous-variable cases.
- Examples
- Uses adult residents’ weight, height, and home distance, with probability tables and products of marginal probabilities.
- What you can do after
- Evaluate/understand: Test independence from joint and marginal probabilities and compute joint probabilities for independent events.
Read how Math and Architectures of Deep Learning explains independence →
Causal AI
Independence is a probabilistic relationship in which knowing the value of one variable does not change the distribution of another variable.
- How it is used
- Simplifies probability models and supplies testable constraints for causal modeling, statistical modeling, regularization, and predictive analysis.
- How deep it goes
- Defines marginal and conditional independence, derives factor reductions and expectation identities, and stops short of detailed test algorithms.
- Examples
- Uses blood types, joint-distribution factors, conditional-expectation equations, statistical tests, and instrumental-variable assumptions.
- What you can do after
- Evaluate probability and causal models by simplifying factors, interpreting predictive absence, and testing consequential independence assumptions.
Causal Inference for Data Science
Independence is a probabilistic property in which knowledge of one random variable does not change the probability distribution of another.
- How it is used
- Uses independence to justify conditioning, adjustment, and causal comparisons between treated and untreated groups.
- How deep it goes
- Defines independence formally, demonstrates it with coin tosses, and extends it to conditional independence and potential outcomes.
- Examples
- Uses fair and forced coin tosses, randomized treatment groups, confounded age groups, and predictive outcome variables.
- What you can do after
- Evaluate causal assumptions by checking whether treatment, potential outcomes, and covariates are independent or conditionally independent.
Read how Causal Inference for Data Science explains independence →
Experimentation for Engineers
Independence is the probabilistic property that the outcome of one random event does not affect the outcome of another.
- How it is used
- Protects A/B-test standard-error and z-score calculations from dependent individual measurements.
- How deep it goes
- Defines independence through iid assumptions and traces repeated samples to false positives, but stops short of a general correction method.
- Examples
- Uses repeated NumPy measurements, overly frequent server sampling, and a die-roll analogy for lucky experimental results.
- What you can do after
- Diagnose/debug with repeated-value checks and interpret apparently significant A/B results as potentially dependent or lucky.
Read how Experimentation for Engineers explains independence →
Related concepts
- multiplication ruleused by · The multiplication rule determines a combined count or probability by multiplying the numbers of choices or the relevant probabilities at successive stages.
- conditional probabilitycontrasts with · Conditional probability is the probability of an event or variable value given that specified information, an event, or another variable’s value is known.
- conditional independencecontrasts with · Two variables are conditionally independent given specified information when their conditional joint distribution factors into the product of their conditional distributions.
- conditioninguses · Conditioning is restricting or guiding a probabilistic or generative process using specified information, variables, values, or events.
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