Statistical inference
Statistical inference is the process of using observed data to learn about an underlying population, process, or model. It includes estimation, uncertainty quantification, hypothesis testing, and model comparison.
What is being inferred
Inference can target several different things:
- parameter values such as means, trends, or regression coefficients
- uncertainty ranges around those quantities
- the plausibility of hypotheses under a reference model
- the relative adequacy of competing probabilistic models
- the predictive behavior of a model on new or unseen data
Core tasks
- Estimate unknown quantities from limited data.
- Quantify uncertainty in those estimates.
- Evaluate competing explanations or models.
- Separate signal from noise under explicit assumptions.
Estimation and uncertainty
One major branch of inference is estimation: using data to approximate an unknown quantity and describe the uncertainty around that estimate. This can include confidence intervals, posterior intervals, predictive intervals, or uncertainty bounds derived from resampling and model assumptions.
Hypothesis testing
- Hypothesis testing asks whether observed data are sufficiently incompatible with a reference hypothesis.
Model comparison
- Information criteria and related model-comparison tools compare candidate models by balancing fit and complexity.
- Probabilistic model selection focuses on comparing how well models represent the data-generating process under explicit probabilistic assumptions.
Why it matters
Inference is where statistical methods become scientific reasoning. The same dataset can support different conclusions depending on model assumptions, sampling design, and the inferential question being asked.
Why assumptions matter
Inference is never assumption free. Results depend on issues such as independence, measurement error, sampling bias, distributional form, and whether the model structure matches the real process closely enough to support the question being asked.
Common misunderstandings
- Statistical significance is not the same as scientific importance.
- A point estimate is not the same as certainty.
- The best model under one criterion is not automatically the true model.
- Inference does not create information that was not present in the data and design.
In this garden
This note acts as the umbrella page for the statistics branch.
- Hypothesis test introduces null-model testing.
- test statistic explains how evidence is summarized.
- Information criteria covers model comparison beyond null-hypothesis tests.
- Probabilistic model selection extends the model-comparison perspective.
The note is intentionally broad because the connected pages answer different inferential questions rather than applying a single unified recipe.
See also: Hypothesis test, test statistic, information criteria, probabilistic model selection, MOC Statistics and Inference