Cosmology · Cosmic maps, uncertainty and fair comparisons
Sampling and selection can reshape a catalogue
Pass equal hypothetical populations through different inclusion probabilities to see why a large observed sample need not represent the population behind it.
Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.
Back to the chapterAfter reading, you can
- Define a selection function as a conditional inclusion probability.
- Calculate expected selected counts for a simple population.
- State the conditions and limits of correcting counts with inverse probabilities.
An object must pass several gates to enter a sample
A catalogue lists objects that entered a defined measurement and selection process. Some were too faint to detect, some lay outside the survey footprint, and some detections were later removed by quality or classification cuts. The resulting list is not automatically a miniature copy of the underlying population.
A selection function describes the probability that an object enters the specified sample, conditional on relevant properties. Those properties can include brightness, position, colour or measurement quality. The sample definition matters: the selection for all detections can differ from the selection for a later scientific subsample.
Detection incompleteness is also different from contamination. Missing a real faint object concerns inclusion; accepting an object from the wrong class concerns what the listed objects actually are. Both can matter, but one correction does not automatically fix the other.
To infer a population, a model must connect its objects to the chance of observing and retaining them. The methodological analysis by Rix and colleagues uses this logic explicitly. The numbers below are a chosen illustration, rather than a completeness calibration for an actual survey.
Equal populations can yield unequal selected counts
Choose a population with 50 objects in a hypothetical bright class and 50 in a faint class. Suppose every bright object enters the sample, with inclusion probability p = 1, while each faint object has inclusion probability p = 0.2.
For a class with N objects and inclusion probability p, the expected selected count is Np. The bright class therefore contributes 50 × 1 = 50 expected selections and the faint class 50 × 0.2 = 10. The expectation describes an average over repetitions of the stated selection process.
The ratio of these expected counts is 50 bright objects to 60 total, or about 83.3 percent bright. The underlying population was 50 percent bright. This arithmetic compares expected counts; it is not a claim that every realized sample has exactly 60 objects or that the expected realized fraction must equal that ratio.
A finite sample can fluctuate around the expected counts. Even so, repeatedly applying this unequal selection does not make its catalogue composition representative merely by accumulating more objects. Precision about the selected sample and representativeness of the population are different goals.
| Class | Underlying count N | Inclusion probability p | Expected selected Np |
|---|---|---|---|
| Bright | 50 | 1 | 50 |
| Faint | 50 | 0.2 | 10 |
A correction requires the right probabilities
If the positive inclusion probabilities are known correctly, an included object can be assigned weight 1/p for an estimate of the population count. A bright selected object has weight 1, while a faint selected object has weight 5 in this model.
For the expected counts in the table, 50/1 = 50 and 10/0.2 = 50. The expectation of this weighted count recovers the supplied population counts. A particular realization need not do so exactly: if only eight faint objects were selected, their weighted count would be 40.
Large weights also amplify uncertainty. One uncertain or misclassified low-probability object can have a large effect. A guessed selection probability does not become correct because dividing by it returns a plausible-looking answer.
The method requires matching the probability to the actual target sample and relevant object properties. A single average probability may hide strong dependence on brightness or sky location. Any use of these weights must carry the uncertainty in the selection model and in the observed classifications.
A completely unobserved region cannot be recovered by division
Now suppose one class has inclusion probability p = 0. Whatever its population count, its expected selected count is N × 0 = 0. The selected catalogue alone cannot distinguish an absent population from a large population that had no chance to enter.
The inverse weight 1/p is undefined at zero. No numerical weight applied to nonexistent detections can supply the missing observations. An inference about that class needs additional data or explicit modelling assumptions, whose consequences should be stated.
The same issue can appear in a part of the sky outside a footprint or behind an excluded region. More observations within the already selected region do not automatically identify what lies in an area that remains inaccessible. A whole-population statement must explain how such gaps are treated.
A practical boundary may involve probabilities so small or so poorly known that the correction is unreliable, even without being exactly zero. Good population studies therefore define where their selection model can support an inference and where it cannot.
Compare catalogues through their selection histories
Two surveys can report different class fractions because they target different sky areas, depths or subsamples. That difference may be informative about selection rather than a contradiction about the Universe. First define the target population and the entry conditions for each catalogue.
For a first reading, retain the equal 50-and-50 population and the unequal 50-and-10 expected detections. It shows the entire mechanism: the sky population is filtered through inclusion probabilities before appearing in a list. A population model and a selection model must meet in the comparison with data.
Worked reasoning
Correct one possible realization
Eight faint objects are selected in a realization of the stated p = 0.2 model. What inverse-probability weighted count do they give? Must it equal the true count 50?
- Each selected faint object receives weight 1/0.2 = 5.
- Eight such objects give weighted count 8 × 5 = 40.
- The estimator recovers the population count in expectation under the supplied model, not in every finite realization.
The weighted count is 40. It need not equal 50 in this particular realization.
Try the reasoning
What is the expected selected count from 50 faint objects with p = 0.2?
Show the answer and reasoning
10.
Expected selected count is Np = 50 × 0.2. It is not a guarantee for one catalogue.
Can inverse weighting determine the population of a class with inclusion probability zero from its selected count alone?
Show the answer and reasoning
No.
Every population count produces zero selected objects in this case, and 1/p is undefined.
Does a much larger catalogue automatically remove a brightness-dependent selection bias?
Show the answer and reasoning
No.
More selected objects can reduce some sampling noise while preserving the same unequal entry probabilities.
What to carry forward
- A catalogue reflects detection, coverage and later selection cuts.
- Expected selected counts depend jointly on the population and its inclusion probabilities.
- Inverse weighting requires a valid positive-probability selection model and does not fill completely unobserved populations by itself.
References
Sources supporting the explanations, physical distinctions and examples in this lesson.