Astronomy · From raw signals to supported claims

Selection, completeness and censored measurements

Calculate how an inclusion rule changes a sample’s distribution and why a retained nondetection contributes an event probability rather than a value at its threshold.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

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After reading, you can

Define a selection function for a stated source population and observing process.
Normalize a distribution conditional on inclusion in a fixed-size sample.
Distinguish censored measurements, omitted targets and measured low-significance values.

Start with the population and the inclusion rule

An astronomical catalogue is produced by an observing and processing system. A source must be in the covered region and time, generate a usable signal, pass extraction and selection, and survive any publication or analysis filters. Its presence is evidence about that process as well as about the sky.

A selection function specifies the probability of inclusion conditional on stated quantities: for example source brightness, morphology, colour, sky position, observing conditions and the measurement procedure. Completeness describes inclusion or recovery for a defined source domain and rule. Purity or contamination concerns how many retained candidates belong to the desired class; it answers a different question.

Real selection commonly uses noisy measured quantities. A source whose true flux is above a nominal threshold can fall below it in a realization, while a fainter source can scatter upward. A sharp cut in measured flux does not automatically become a sharp cut in true flux.

The Gaia EDR3/DR3 gaps documentation gives a dated concrete example: different data products have different observing and processing gaps, and quality filtering also affects inclusion. Its release-specific tables are not a completeness model for every later catalogue. Always match the selection information to the data product actually analysed.

A fixed-size selected sample needs its own normalization

Consider an original toy with an exactly measured dimensionless source property x between zero and one. Suppose its parent density is fθ(x) = θx^(θ−1), with θ > 0. Retain only sources with x ≥ 0.5. This example deliberately has no measurement noise, contamination or uncertainty in the threshold.

The inclusion probability under this parent model is qθ = integral from 0.5 to 1 of fθ(x) dx = 1 − (0.5)^θ. Conditional on inclusion, the density is gθ(x) = fθ(x)/qθ on the selected range, and zero outside it. The denominator depends on θ, so it matters when comparing candidate parent populations.

Suppose the same two independently selected values are x = 0.8 and 0.9, and condition on this fixed selected sample size. For θ = 1, q1 = 0.5 and g1 = 2 throughout the selected interval, giving a density product of 4. For θ = 2, q2 = 0.75 and g2(x) = 2x/0.75, giving (1.6/0.75)(1.8/0.75) = 5.12.

The conditional likelihood ratio for θ = 2 over θ = 1 is therefore 5.12/4 = 1.28. Using unconditioned parent densities would give 2.88/1 = 2.88, which compares the wrong observed law. A density value of two is permissible: its integral over a range supplies probability, while the value at one exact continuous point is not a probability.

Original exactly measured, fixed-size selected toy. Both candidates use the same retained x values, 0.8 and 0.9, and inclusion rule x ≥ 0.5.
Parent parameter	Inclusion probability qθ	Selected density product	Parent density product
θ = 1	0.5	4	1
θ = 2	0.75	5.12	2.88

Do not extend the toy beyond its recorded information

The toy conditions on the number selected. It supplies no total parent count and no number of nondetections. If those counts are part of the observations, their sampling model can carry additional population information and belongs in the likelihood. The conditional two-value calculation is not a complete survey-count analysis.

Kelly’s measurement-error treatment explicitly models inclusion based on measured values and integrates over unobserved quantities. Loredo and Hendry likewise derive population inference from the detection and measurement process. Their detailed formulas have stated hypotheses; an efficiency factor cannot be inserted everywhere by analogy.

In particular, when the same data are used for both detection and characterization, an extra selection factor can count information twice if the likelihood has already accounted for that event. Derive the expression from the information actually retained and the conditioning used.

Artificial-source injection and recovery can help characterize completeness, but the result inherits the injected morphologies, brightness distribution, noise, location, timing and extraction settings. A domain with zero sensitivity supplies no direct recovered population there. Dividing by a completeness value of zero cannot discover its missing sources.

Censoring differs from omitting a target

A censored measurement retains a target and partial information about its measurement, such as a recorded threshold event. Truncation removes objects or measurements from the retained sample under an inclusion rule. A dataset can involve both, and the corresponding information must be distinguished.

For a separate original example, assume a target was selected independently of the present flux measurement. Let its measured flux density Y, in millijanskys (mJy), follow a Gaussian model with mean μ and known standard deviation 2 mJy. Retain only the information Y < 3 mJy for this target; its actual measured value is unreported.

The likelihood contribution is the probability of that event: L(μ) = Φ((3 mJy − μ)/(2 mJy)), where Φ is the standard normal cumulative distribution function. It integrates the measurement density over all values compatible with the retained information. It is dimensionless.

For candidate mean flux densities 1, 3 and 5 mJy, the event probabilities are approximately 0.841344746, 0.5 and 0.158655254. A true mean of 5 mJy is therefore compatible with this recorded event, although less favoured than 1 mJy by this likelihood comparison. The record does not assert that the unknown true mean is certainly below 3 mJy.

Probability of the retained event Y < 3 mJy under a Gaussian measurement with standard deviation 2 mJy. These synthetic probabilities are neither a posterior nor a universal upper-limit convention.
Candidate μ (mJy)	Standardized threshold	P(Y < 3 mJy \| μ)
1	1	0.841344746
3	0	0.500000000
5	−1	0.158655254

A threshold is not a substitute observation

Replacing a censored value by zero, by the threshold, or by half the threshold manufactures a point observation. Evaluating a Gaussian density at the threshold also represents different information: that density has units mJy⁻¹, while the probability of the retained event has no unit.

If an actual low-significance forced measurement and an appropriate likelihood are available, they can retain more information than a threshold flag. Conversely, a published upper limit may come from a confidence or posterior construction rather than the particular censoring rule used here. Read its definition before treating it as a measured flux or a probability event.

For a population analysis, preserve the sample definition, masks, observing coverage, detection and quality rules, retained measurements and limit conventions. Explain whether selection depends on the same noisy data used for inference and whether nondetection counts are included.

Selection is not a cosmetic correction applied after fitting. It helps define the observed data law. The next lesson follows that principle into the assumptions behind a fitted physical model and shows how even a missing normalization can reverse a likelihood comparison.

Worked examples

Compare parent models using the selected law

For the retained values 0.8 and 0.9 above 0.5, what is the conditional likelihood ratio of θ = 2 over θ = 1?

The selected probabilities are q1 = 0.5 and q2 = 0.75.
Divide each parent density by its candidate’s qθ before multiplying: L1 = 4 and L2 = 5.12.
Form L2/L1 = 5.12/4.

1.28. The unconditioned ratio 2.88 omits the parameter-dependent selection normalization.

Use the information in a retained nondetection

Under Y ∼ Normal(μ, (2 mJy)²), what likelihood contribution does Y < 3 mJy give for μ = 5 mJy?

Standardize the threshold: (3 − 5)/2 = −1.
Integrate the standard normal density below −1.
Use Φ(−1) ≈ 0.158655254 rather than a density evaluated at 3 mJy.

Approximately 0.158655254, a dimensionless event probability. It is not the posterior probability that μ is below the threshold.

Try the reasoning

Does high purity imply that almost every source in the desired population was recovered?
Show the answer and reasoning
No.
Purity concerns the retained candidates; completeness concerns inclusion of the stated source population.
Is a conditional density of 2 impossible because probabilities cannot exceed one?
Show the answer and reasoning
No.
A density is integrated over an interval to obtain probability; its point value is not itself an event probability.
An independently selected target is retained with only Y below a threshold. Is this the same record as omitting the target entirely?
Show the answer and reasoning
No.
The retained target and threshold event supply censored information that an omitted record does not contain.
Can division by a completeness of zero reconstruct a population without additional assumptions?
Show the answer and reasoning
No.
That observing process provides no recovered data for the missing domain.

What to carry forward

A selection function belongs to a stated population, observing process and data product.
Conditioned samples require the corresponding normalized observed law.
Use the likelihood of the information retained: a censoring event, a measured value and an omitted target are distinct records.

References

Teaching resources and research supporting this lesson’s explanations and examples.