From a few detections to a population

Say which planets the rate counts

Choose a region $B$ in period $P$ , measured in days, and radius $R$ , measured in nominal equatorial Earth radii. Let $f_B$ count planets in that region per searched star. It can exceed one because a star can host several planets.

The number alone does not specify how planets are distributed inside the region. Let $h(P,R)$ be a normalized shape: integrating it over the chosen region gives one. If the shape is uniform in logarithmic period and radius, a fixed multiplicative interval receives the same probability wherever it lies inside the bounds.

A logarithmic coordinate changes by $d\ln P=dP/P$ , so a density that is constant in logarithms has factors $1/P$ and $1/R$ when written in the linear coordinates. These factors are the change-of-variable weights, sometimes called Jacobian factors. They prevent wide linear cells from receiving an unintended population model.

h(P,R)=\frac{1}{PR\ln(P_{\max}/P_{\min})\ln(R_{\max}/R_{\min})},\qquad \int_B h(P,R)\,dP\,dR=1

These finite bounds and logarithmic shape specify the population assumed in this calculation.

Reading: A Probabilistic Approach to Kepler Completeness and Reliability for Exoplanet Occurrence Rates · IAU 2015 Resolution B3 nominal conversion constants

Turn the search into an expected count

For searched star $j$ , let $q_j(P,R)$ be the probability that a planet enters the final catalog. Average this probability using the population shape, then add over every searched star. The result $E_B$ is an effective searched-star exposure.

Suppose the independent number of planets around each star follows a Poisson distribution. Selecting each planet with its modeled probability thins that Poisson process. Summing the selected counts over stars gives a Poisson discovery count with mean $f_BE_B$ .

This assumption is convenient for the synthetic experiment, where planet counts, orbital directions and measurement noise are generated independently. Real system architectures can produce correlated transit opportunities. A model of planets per star and a model of planetary systems should therefore be identified separately.

E_B=\sum_{j=1}^{N_\star}\int_B q_j(P,R)h(P,R)\,dP\,dR,\qquad p(n\mid f_B,E_B)=\frac{(f_BE_B)^n e^{-f_BE_B}}{n!}

The observed count

n

is an integer. The effective exposure counts weighted searched stars. Observing duration has already entered each star’s selection probability.

Reading: A Probabilistic Approach to Kepler Completeness and Reliability for Exoplanet Occurrence Rates

Normalize the uncertainty in the rate

A likelihood describes how the integer count changes as the possible rate changes. To turn it into a posterior density for that rate, supply a prior and normalize. Use a Gamma prior with positive shape $\alpha$ and rate $\beta$ . Its density is proportional to $f_B^{\alpha-1}e^{-\beta f_B}$ on positive rates. Its total probability after normalization is one, so this is a proper prior.

Multiplying the prior by the Poisson likelihood adds $n$ to the exponent that determines the shape and adds $E_B$ to the exponential rate. The resulting Gamma distribution is normalized by its Gamma-function factor. The word rate here names the Gamma distribution’s second parameter, which is different from the planet occurrence being inferred.

In the default synthetic record, five thousand searched stars produce 3,038 generated planets and 79 detections. The effective exposure is about $121$ stars. The prior has shape and rate one half. The posterior median is about $0.651$ planet per star, with a central 95-percent interval about $[0.518,0.805]$ . The generating mean $0.600$ lies inside that interval. One successful example is not a general coverage test.

f_B\mid n,E_B\sim\operatorname{Gamma}(\alpha+n,\beta+E_B)

For

E_B>0

, or for

E_B=0,n=0

, the posterior follows directly from multiplying and normalizing. A positive count with exactly zero selection has zero likelihood and signals a broken model or input.

Frozen synthetic records, the same 5000 searched stars

Case	Detected planets	Supported exposure	Posterior median
Shallow transits hide	$79$	$121$	$0.651$
Quieter detector	$97$	$143$	$0.677$
Longer season	$98$	$142$	$0.69$
Repeated gaps	$61$	$95.4$	$0.638$
True radius valley	$96$	$142$	$0.676$

Noise changes retain the generated population; the valley preset changes its generating shape. The likelihood uses integer detections.

Reading: A Probabilistic Approach to Kepler Completeness and Reliability for Exoplanet Occurrence Rates

A doubtful candidate still has an integer true state

Suppose three candidate classifications have externally supplied probabilities $0.9$ , $0.6$ and $0.2$ of being planets. Each object’s true state is still zero or one. Under independent classifications, multiplying their two-state probability polynomials gives the probabilities of zero, one, two and three true planets: $0.032$ , $0.344$ , $0.516$ and $0.108$ .

The expected count is $1.7$ , but that expectation is not the observed integer count in a Poisson likelihood. One conditional approach computes the normalized posterior for each integer catalog and averages those posteriors with the external classification probabilities. The result is a mixture of Gamma densities, not a Gamma density obtained by putting a fractional count into the shape.

This is a modular calculation: the classifications are treated as supplied information and are not updated by the inferred occurrence. A joint generative model must instead describe how planets and false events enter the candidate sample. For example, a Poisson count with mean $f_BE_B+\eta$ cannot separately determine occurrence and a false-event mean $\eta$ from one total alone. An independently modeled control sample can add the missing information.

p(f_B\mid\text{external classifications},E_B)=\sum_{n=0}^{N_{\rm cand}}\pi_n\,\operatorname{Gamma}(f_B;\alpha+n,\beta+E_B)

\pi_n

are normalized probabilities over integer catalogs. Independence and the origin of the external probabilities must be stated.

Reading: A Probabilistic Approach to Kepler Completeness and Reliability for Exoplanet Occurrence Rates · Reliability Correction is Key for Robust Kepler Occurrence Rates

Zero information should stay uninformative

When no planet is detected but the exposure is positive, the data favour smaller rates and give a finite upper credible limit under the stated prior. When both the count and exposure are zero, the likelihood is constant in occurrence: the posterior stays exactly the prior.

An uncalibrated region is different from a region known to have zero recovery. The real tutorial calculates exposure where its empirical response has support. Elsewhere it allows any recovery probability from zero to one. The resulting minimum and maximum exposures are carried through the normalized integer-catalog mixture.

The endpoints give a family of conditional posteriors. Their outer 95-percent quantiles form an envelope across those conditions. There is no probability distribution over the unknown response, so this envelope should not be read as one uniquely defined 95-percent posterior interval. Its breadth shows what the measured calibration has left unresolved.

Reading: A Probabilistic Approach to Kepler Completeness and Reliability for Exoplanet Occurrence Rates

Radius uncertainty moves planets between cells

A radius estimate can scatter across a bin edge. The probability that a planet from true cell $i$ appears in observed cell $j$ forms a migration matrix $M_{ji}$ . Include outside cells as well, so the probabilities for each true cell sum to one. Predict the observed cells by applying this measurement model after survey selection.

Planets around the same star share its radius uncertainty. Moving that stellar radius changes every associated planet radius together. It also changes geometric transit probability and the depth predicted for a fixed physical planet. Changing only catalog membership while holding selection fixed misses part of this effect.

The synthetic radius display gives half the log-radius variance to a shared host term and half to independent planet terms. Its total-count calculation uses every detection. Radius migration redistributes the plotted bins and preserves that total. The real tutorial separately shifts stellar radii and recomputes both membership and exposure; unavailable stellar/transit covariance remains an explicit limitation.

Reading: The California-Kepler Survey III. A Gap in the Radius Distribution of Small Planets · Shallower radius valley around low-mass hosts

From a few detections to a population

Say which planets the rate counts

Turn the search into an expected count

Normalize the uncertainty in the rate

A doubtful candidate still has an integer true state

Zero information should stay uninformative

Radius uncertainty moves planets between cells

Exercises and explanations

Try the prediction

Change the detector and watch the missing planets

Sources and calculation scope