Probability and statistics · BEGINNER LESSON
Independent and identically distributed
Independent and identically distributed, often abbreviated i.i.d., describes a set of samples in which each sample is statistically independent of the others and all samples are drawn from the same distribution. In practice, this means the observations are treated as coming from one common source with no sample-to-sample dependence or distribution change.
What it is
The i.i.d. assumption is a basic idea in statistics and machine learning. “Independent” means one data point does not help determine another. “Identically distributed” means every point follows the same probability distribution, so the data are treated as if they were generated by one shared process. The source explains that many classical theories assume samples {(X_i, Y_i)} are i.i.d., but that this is often too restrictive for real-world data.
[1] Robust learning for ridge-penalized quasi-GLMs under non-identical distributions
How it is used
When data are assumed i.i.d., methods can combine information from all samples as though they were exchangeable draws from one source. This assumption supports standard learning theory and regression analysis. The source notes that log-truncated robust regression and empirical risk minimization can be studied under this setting, even when the data contain heavy-tailed contamination or non-identical distributions.
[1] Robust learning for ridge-penalized quasi-GLMs under non-identical distributions
Why the assumption can fail
The source gives three main reasons the i.i.d. assumption is often violated: dataset bias or domain shift, where data are influenced by outliers or come from a different context; change points, where the data-generating distribution changes over time; and mixtures of multiple distinct distributions, such as in multi-armed bandit settings. It also notes that independent non-identically distributed, or i.n.i.d., data can be more realistic in industrial, complex social, and time-dependent problems.
[1] Robust learning for ridge-penalized quasi-GLMs under non-identical distributions
Limits and practical consequences
The source emphasizes that real data may be contaminated by heterogeneous, heavy-tailed errors, including cases where variance is finite or infinite. Because the exact corruption is often impossible to determine, assumptions about lower-order moments may be more practical than idealized i.i.d. modeling. The lesson here is not that i.i.d. is wrong, but that it is a simplifying model whose validity must be checked against the data context.
[1] Robust learning for ridge-penalized quasi-GLMs under non-identical distributions
SEE THE IDEA
How the i.i.d. assumption works and where it breaks
All samples are treated as coming from the same probability pattern.
Each sample does not depend on the others.
Statistics and machine learning combine samples under this assumption.
Outliers, shifts, or mixed sources can make the assumption false.
A method often starts by assuming i.i.d. data, but real data can violate one or both parts of that assumption.
The i.i.d. model simplifies analysis, but changing or mixed data sources can break it.
TRY IT YOURSELF
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Sources and further reading
This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.