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tokens: smaller than the symbols. Right: Similar to the left panel, but showing the distribution of . All spectra were visually inspected for the presence of Ca ii triplet emission. The spread in EW values is extremely large at low S/N values due to noise (compare the left and right panels of Fig. 3 ), but rapidly tightens as the S/N increases, making it trivial to identify genuine emission features such as SDSS J1228+1040. Closer to the bulk EW distribution, it becomes difficult to detect Ca ii triplet emission such as that of SDSS J0738+1835 and SDSS J1043+0855 (see Fig. A1 for the EW values). We therefore chose to limit the sample used for the following statistical analysis to spectra with S/N ≥ 5, wherein if there were any additional emission features with an EW similar to the gaseous debris discs in our sample we should have detected them. Assuming that we can only detect gaseous emission in the spectra of systems above S/N = 5, the number of white dwarfs in our sample reduces to 7705, which we use for the calculations in the next section. 2.1 The occurrence rate of gaseous discs at white dwarfs We find no new gaseous discs in our magnitude and S/N limited Gaia /SDSS sample, resulting in a total of five detected gaseous components to debris discs out of 7705 systems. Using Bayes’ theorem we calculate the probability distribution, p ( f | n, t ), of an occurrence rate, f , of a gaseous debris disc at a white dwarf, given n detections out of t systems as where p ( n | f, t ) = f n (1 − f ) t − n is the binomial distribution, p ( f ) is the prior distribution of f , and p ( n, t ) is the probability distribution of detecting n gaseous disc hosts out of t systems which is a constant. We assume an uninformative prior distribution known as the Jeffreys prior (Jeffreys 1946 ), which for the binomial distribution is where B is the beta function. Substituting this prior into equation ( 1 ) results in Taking the median and 1σ intervals of this probability distribution (which is independent of the proportionality constant) with n = 5, and t = 7705, we find that 0.067 ± per cent of white dwarfs host a debris disc with a detectable gaseous component (Fig. 4A ). This is significantly lower than the occurrence rate of detectable dust discs around white dwarfs, which has been estimated by various studies to be 1–3 per cent (Farihi et al. 2009 ; Girven et al. 2011 ; Rocchetto et al. 2015 ; Rebassa-Mansergas et al. 2019 ; Wilson et al. 2019 ). For further discussion, we adopt the estimate of Wilson et al. ( 2019 ), 1.5 ± per cent, calculated using an unbiased sample of 195 white dwarfs observed with Spitzer containing three dusty debris discs. We note, however, that using equation ( 3 ) with n = 3 and t = 195, we obtain a slightly different occurrence rate of 1.6 ± per cent, and the probability distribution of this occurrence rate is shown in Fig. 4(B) . These two occurrence rates of dusty debris discs at white dwarfs are in agreement, and the difference results