Doesn’t Ohio State have electronic access to Nature? ]]>

And apologies for mistyping Bayesian.

(I do know how to spell Bayes’s name – I even know where he’s buried.)

]]>I happily take back my comment about Gould. Clearly the correlations between the data points are important (and could be quite easily built into modelling), so why wasn’t this done? It’s not much of an excuse for carrying on regardless.

]]>Of course I could correct these errors, but I rather like them.

]]>(And yet it doesn’t sort out “bloodily”! Grrr . . .)

]]>(Bloodly auto-correct has turned “sort” into “sort” . . .)

]]>. . . at least formally. The reason for the caveat is that, according to Davis, there are strong correlations between the data-points, so the constraints implied by ignoring them (as both Davis and Gould have done so far) are clearly too tight. I suggested that taking every second measurement would produce (nearly) uncorrelated results and Davis agreed; hence there’s an immediate increase in the M_BH uncertainties of sort(2) = 1.4. But reading between the lines I believe Davis suspects this would still be too accurate given the data at hand.

As for Bryn’s request for the Bayesian version of this, the ingredients (bar the as-yet unknown correlations) are already there in the original paper. If one assumes a broad, uniform prior on M_BH and M/L then the shrunken contours that Gould argued for are recovered. The contours are elliptical and evenly spaced for the constant increments of sort(chi^2), so the posterior is well approximated as a bivariate normal. As such, the marginal posterior in M_BH is just a normal with the mean and standard deviation argued for by Gould. (It’s one of those simple cases where the numbers that come out of a Bayesian calculation and a frequentist/classical calculation match, even if the philosophical interpretation differs.)

]]>Perhaps readers of this blog would appreciate it if Peter could find the time to post here a simple Baysian analysis of the problem as a worked example?

]]>I think Gould is probably correct (at least in his re-evaluation of Figure 2 of Davis et al., even if probably not in terms of his final black hole mass estimate) for the following reasons.

The caption of Figure 2 and the text both state that the contours are of chi^2 – chi^2_min, but the two example “bad” models shown in Figure 1 are labelled as having chi^2_red (which I take to be the same as chi^2/dof) that differ by ~28 from the best fit model. Both these models lie just outside the “chi^2 – chi^2_min = 25” contour in Figure 2, which implies that either i) it’s not chi^2_red in Figure 1 or ii) it is chi^2_red in Figure 2.

The data shown in Figure 1 might be able to resolve this ambiguity. The “overweight” black hole model has 14 central data-points which each appear to be ~100 km/s off the predicted values; the Supplementary Information implies uncertainties of ~10 km/s on each point (which roughly matches the error bars on the plot), and doesn’t say anything about correlations being taken into account. So that implies a chi^2 of ~1400, implying a reduced chi^2 of ~20 (but both values are very rough). At any rate, I think it makes it pretty clear that FIgure 1 does list chi^2_red, and hence that Figure 2 shows chi^2_red as well, implying Gould’s re-evaluation is correct.

That said, the comment from the discussion group seems reasonable, and I suspect is in part because the trace data must be significantly correlated. The correct uncertainty on the black hole mass would then be greater than that obtained by Gould, but I still suspect it should be considerably lower than that reported by Davis et al.

I’m not about to try modelling this system for myself, but the data should be made available by the authors – it’s a condition of publishing in Nature. And even without doing any modelling, just having the trace and the uncertainties would be enough to at least calculate chi^2 for the two bad models, hence resolving the chi^2 vs. chi^2_red ambiguity . . .

]]>The problem is that chi-squared is used by too many people who don’t know (or care) what it is they’re actually trying to do. Set it up as a Bayesian exercise and you won’t be tempted to use an inappropriate recipe.

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