The paper ‘Bayesian time series analysis of terrestrial impact cratering’ was published in Mon. Not. R. Astron. Soc. 416, 1163–1180 (2011). There is an error in the write-up of the method in sections 3.2 and 3.4 of the article. It is essentially just an error in the notation used, but this may interfere with a complete understanding of the method. There is no mistake in the concept or in the code, so this does not affect the results, discussion or conclusions in any way.

The issue is with equation (5) (section 3.2, Bailer-Jones 2011), the equation for the measurement model. Roughly speaking the measurement model gives the probability of getting the measured data given the true value. More precisely, it is the probability of measuring the event at time forumla given both the true time of the event, forumla, and the uncertainty, forumla, in the measurement forumla. In other words, rather than being forumla, the measurement model is 

5
formula
which is normalized with respect to forumla (and not forumla as was written in the article). Comparing the above to equation (5) in Bailer-Jones (2011), the right-hand-sides are identical to what we had before (the order of forumla and forumla is irrelevant for a Gaussian). In fact, the left-hand-sides are also identical if we consider the data to be forumla rather than forumla, and implicitly conditioned on the measurement uncertainty, forumla. The correct perspective, which was not clear in Bailer-Jones (2011), is that we should think of the measurement model as a model of forumla conditioned not only on the (unknown) true time, forumla, but also on a fixed (measured/estimated) noise model parameter, forumla. Yet as we neither infer nor marginalize over forumla in this work, it has no practical consequence whether we regard this as measured data or a fixed parameter.

This changes the notation in section 3.4 of Bailer-Jones (2011), but not the actual content of the equations or the calculations. One can just replace forumla with forumla and remember that there is an implicit conditioning on forumla. None the less, for completeness, the second, third and fourth paragraphs of this section with the corrected notation are given below.

The consequence of this more precise notation is that it reminds us that the evidence, forumla, is actually conditioned on the fixed measurement uncertainties. This was not explicit in the original article, but the results, conclusions and discussion are entirely unaffected.

AMENDED PARAGRAPHS OF SECTION 3.4 (BAILER-JONES 2011)

…The probability of observing data forumla from model forumla with parameters forumla is forumla, the likelihood for one event. The time series model predicts the true age of an event, which is unknown. Applying the rules of probability we marginalize over this to get 

6
formula
The last step follows from conditional independence: in the first term – the measurement model – once forumla is specified forumla becomes conditionally independent of forumla and forumla; in the second term – the time series model – forumla is independent of forumla. As the data are fixed, we consider both terms as functions of forumla. Both must be properly normalized probability density functions.

If we have a set of forumla events for which the ages and uncertainties have been estimated independently of one another, then the probability of observing these data forumla, the likelihood, is 

7
formula
where forumla. The principle of this calculation is illustrated in Fig. 3: the likelihood of an event for a given model is the integral of the probability distribution for the event (equation 5) over the time series model, forumla. Specific cases for the latter are introduced in Section 3.6 below.

Figure 3

Principle of the likelihood calculation (equation 7).

Figure 3

Principle of the likelihood calculation (equation 7).

The evidence is obtained by marginalizing the likelihood over the parameter prior probability distribution, forumla, 

8
formula
(where forumla drops out of the second term due to conditional independence). For a given set of data (crater time series), we calculate this evidence for the different models we wish to compare, each parametrized by some parameters forumla. The parameter prior forumla encapsulates our prior knowledge (i.e. independent of the data) of the probabilities of different parameters, normally established from the context of the problem (see Section 3.7)…

REFERENCES

Bailer-Jones
C. A. L.
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2011
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MNRAS
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416
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1163