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

1. Estimate unknown quantities from limited data.
2. Quantify uncertainty in those estimates.
3. Evaluate competing explanations or models.
4. 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.

Where the reasoning happens

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.

The notes around this one answer different questions

There is no single inferential recipe, and treating the surrounding notes as steps in one would be misleading. Hypothesis testing asks whether data is compatible with a null model, and test statistics are how that compatibility gets summarised into a number. Information criteria ask a different question entirely — which of several models predicts best — and answer it on a continuous scale rather than by accepting or rejecting. Probabilistic model selection takes that further by treating the choice as an inference problem in its own right.

The distinction matters because the questions are not interchangeable, and the most common misuse in climate work is answering one with the machinery of another. Amrhein and colleagues make the case that the threshold habit is what does most of the damage.

See also: Hypothesis test, test statistic, information criteria, probabilistic model selection, MOC Statistics and Inference