Concept

Bayes factor — where it appears

The ratio of how probable the data are under one model to how probable they are under another, each averaged over its own parameters. It compares models of different complexity directly, but depends on the priors placed on those parameters.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The evidence for a period, one trial period at a time. The natural logarithm of the Bayes factor for a sinusoid at each trial period against no signal, for 120 radial velocities spread over 359 days with 2.0 m/s white noise per point and a planet of semi-amplitude 1.0 m/s at 11.3 days. The amplitude prior is Gaussian with scale 3.0 m/s in each component. At the planet's period the evidence is 14.4 — odds of about 1.8·10⁶ to one — and almost everywhere else it is near zero or negative, because a sinusoid that fits nothing is penalised for the amplitude range it wasted. The quantity that answers the question "is there a planet at some period" is not the peak but the average of the Bayes factor over the prior on the period, here uniform in the logarithm of the period from 1.5 to 1,000 days, and that average is ln B = 7.7: 6.7 lower than the peak, the Bayesian form of the look-elsewhere penalty, paid because the data had to find the period as well as the amplitude.

Odds for a planet that depend on a guess about its size

A false-alarm probability asks how often noise alone would produce a peak this tall. A Bayes factor asks something better — how much more probable the data are with a planet than without one — and pays for it with two numbers nobody measured. The width allowed for the period barely matters. The width allowed for the amplitude can turn strong evidence into none, from the same velocities.

starlight · Periodograms

Amplitudes measured near the noise, against the amplitudes that were there. A simulated survey of 6000 stars, each with one planet whose true radial-velocity semi-amplitude K is drawn from a population rising towards small amplitudes (dN/d ln K ∝ K^−0.6, from 0.2 to 20 m/s), each measured with an amplitude uncertainty of 0.6 m/s; 1500 are drawn. Across: the true amplitude; up: the amplitude fitted at the planet's period; both logarithmic. The dashed diagonal is a perfect measurement; the horizontal line is the detection threshold, 4 times the uncertainty. Points above it are the detections (filled). Well above the threshold they sit on the diagonal; near it they sit above it — the detections within one unit of the threshold have measured amplitudes 1.33 times their true ones on average — and below it the measured amplitudes stop falling at all, because an amplitude is a length and noise alone gives it about 0.75 m/s.

A prior the survey measures through its own noise

Whether a weak periodic signal is a planet depends on how large planets are expected to be, and the best source for that expectation is the population the surveys themselves have measured. But the population's small end is built from the marginal detections whose reality depends on the prior. Breaking the circle means measuring the population and the planets together, with the noise and the threshold written into the model — and doing it that way changes the answers by factors of two where they matter most.

starlight · Periodograms

Named alongside it

The objects these essays reach for when they reach for this one.

Periodogram Prior Radial-velocity Eddington bias False-alarm probability Gaussian process Hierarchical model Lindley's paradox Look elsewhere effect Occurrence rate Red noise Selection effect

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