Cinderella and her sisters

The results of the full observing run of the Advanced LIGO detectors (September 12, 2015, to January 19, 2016) have been presented on June 8[4], slightly updating some of the February's digits. Figure 3 summarizes detector performances and results, with some important numbers (within this context) reminded in the caption.

Figure: The Monster (GW150914), Cinderella (LVT151012) and the third sister (GW151226), visiting us in 2015 (Fig. 1 of [4] - see text for the reason of the names). The published `significance' of the three events (Table 1 of [4]) is, in the order, ``$> 5.3\,\sigma$'', ``$1.7\,\sigma$'' and ``$> 5.3\,\sigma$'', corresponding to the following p-values: $7.5\times 10^{-8}$, $0.045$, $7.5\times 10^{-8}$. The log of the Bayes factors are instead (Table 4 of [4]) approximately 289, 23 and 60, corresponding to Bayes factors about $3\times 10^{125}$, $10^{10}$ and $10^{26}$.

The busy plot on the left side shows the sensitivity curves of the two interferometers (red and blue curves, with plenty of resonant peaks) and how the three signals fall inside them (bands with colors matching the wave forms of the right plot). In short, the two curves tell us that a signal of a given frequency can be distinguished from the noise if its amplitude is above them. Therefore all initial parts of the waves, when the black holes begin to spiral around each other at low frequency, are unobservable, and the bands below $\approx 20\,$Hz are extrapolations from the physical models. Later, when the frequency increases, the wave enters the sensitivity range,²¹ which extends up to a given frequency, after which we `loose' it. The lower and upper boundary frequencies depend on the amplitude of the signal, as it also happens in acoustics.

The plot on the right shows finally the `waves'²² from the instant they enter the optimal 30Hz sensitivity region (the acoustic analogy depicted in footnote20 might help):

• The wave indicated by GW150914 (the `Monster', with GW standing for gravitational wave and 150914 for the detection date, September 14, 2015) is characterized by high amplitude, but short duration in the sensitivity region, because it fades out at a few hundred hertz.
• GW151226 instead, although of smaller intensity, has a longer `life' (about 1.7 seconds) in the `audible' spectrum, and therefore the signature of a BBH merger is also very recognizable.
• Then there is the October 12 event, LVT151012, which has an amplitude comparable to that of GW151226, but smaller duration. It has, nevertheless, about 20 oscillations in the sensitivity region, an information that, combined with the peculiar shape of the signal (remarkably the crests get closer as time passes, while the amplitude increases, until something `catastrophic' seems to happen) and the fact that two practically `identical' and `simultaneous' signals have been observed by the two interferometers 3000 km apart, makes the experts highly confident that this is also a gravitational wave.

However, even if at a first sight it does not look dissimilar from GW151226 (but remember that the waves in Fig.3 do not show raw data!), the October 12 event, hereafter referred as Cinderella, is not ranked as GW, but, more modestly, as LVT, for LIGO-Virgo Trigger. The reason of the downgrading is that `she' cannot wear a ``$> 5\sigma$'s dress'' to go together with the `sisters' to the `sumptuous ball of the Establishment.' In fact Chance has assigned `her' only a poor, unpresentable $1.7\,\sigma$ ranking, usually considered in the Particle Physics community not even worth a mention in a parallel session of a minor conference by an undergraduate student.²³ But, despite the modest `statistical significance', experts are highly confident, because of physics reasons²⁴(and of their understanding of background), that this is also a gravitational wave radiated by a BBH merger, much more than the 87% quoted in [4].²⁵Indeed the most useful number experimentalists can provide to the scientific community to quantify how the experimental data alone favor the 'Signal' hypothesis is the Bayes factor, as expounded in the preamble. And this factor is very large also for Cinderella: $\approx 10^{10}$. This means that, even if your initial odds Signal Vs Noise were one to one million, the observation of the LIGO interferometers turns them into 10,000 to 1, i.e. a probability of BBH merger of 99.99%.²⁶

Now the question is, how can a modest $1.7\sigma$ effect be compatible with a Bayes factor as large as $10^{10}$? The solution to this apparent paradox will be given in the next section, but I anticipate the answer: p-values and BF's are two different things, and there is no simple, general rule, inside probability theory, that relates them.