Probability supplies a language for uncertainty and variation. It does not say why a system is uncertain until we specify the experiment, model, and state of knowledge.
Mass, density, and support¶
For discrete outcomes, probability mass assigns probabilities that sum to one. For continuous quantities, a probability density integrates to one but is not itself the probability of an exact point. Density carries reciprocal units of its variable. The support states which outcomes the model admits; behavior outside support is part of the contract.
Normalization makes the total probability one. In a finite stellar-mass power law, the normalization changes smoothly with the exponent and has a logarithmic removable limit. A correct formula must preserve values and parameter derivatives through that limit.
Summaries and dependence¶
An expectation is a probability-weighted average under a specified distribution. Variance measures squared spread about an expectation. Covariance records linear co-variation and has product units; correlation is its dimensionless normalized form. These summaries do not uniquely determine a general distribution.
Conditioning, written , changes the distribution of one quantity given information about another. It is not the same as numerical conditioning, though both concepts appear in scientific inference.
Transformations and sampling¶
A transformation changes density through its local volume factor, not merely by substituting values. Sampling produces draws from a specified distribution; it does not validate that distribution as a model of nature. Inverse-CDF sampling depends on normalization, monotonicity, support, and a reliable quantile map.
Kinds of uncertainty¶
Aleatoric uncertainty represents variation modeled as intrinsic to the data- generating process. Epistemic uncertainty represents limited knowledge about models, parameters, or missing structure. The boundary is model-dependent: a latent physical variable treated as noise in one analysis may become explicit state in another.
Predict¶
Name the random variable, support, units, normalization, expected symmetries, limiting cases, and which uncertainty is represented.
Compute¶
Evaluate density, CDF, quantile, moments, or samples with explicit boundary behavior. Use stable limiting kernels and keep random seeds and transformations in the evidence record.
Audit¶
Check normalization, nonnegativity, support boundaries, monotonic CDF behavior, CDF/quantile round trips, analytic moments where available, and parameter derivatives against an independent method.
State the warranted claim¶
Numerical normalization and round trips warrant a distribution-implementation claim. They do not show that the chosen distribution adequately represents a stellar population or measurement process.
Misconception check¶
A density can exceed one because it is not probability mass. A random draw is not evidence that a model is random “in reality,” and a normalized likelihood over data is not automatically a normalized posterior over parameters.
Continue to Models, inference, and information or Jaxstro’s Distribution kernels chapter.