Optimal modeling of 1D azimuth correlations in the context of Bayesian inference

Analysis and interpretation of spectrum and correlation data from high-energy nuclear collisions is currently controversial because two opposing physics narratives derive contradictory implications from the same data, one narrative claiming collision dynamics is dominated by dijet production and projectile-nucleon fragmentation, the other claiming collision dynamics is dominated by a dense, flowing QCD medium. Opposing interpretations seem to be supported by alternative data models, and current model-comparison schemes are unable to distinguish between them. There is clearly need for a convincing new methodology to break the deadlock. In this study we introduce Bayesian inference (BI) methods applied to angular correlation data as a basis to evaluate competing data models. For simplicity the data considered are projections of two-dimensional (2D) angular correlations onto a 1D azimuth from three centrality classes of 200-GeV Au-Au collisions. We consider several data models typical of current model choices, including Fourier series (FS) and a Gaussian plus various combinations of individual cosine components. We evaluate model performance with BI methods and with power-spectrum analysis. We find that FS-only models are rejected in all cases by Bayesian analysis, which always prefers a Gaussian. A cylindrical quadrupole $cos\left(2\varphi \right)$ is required in some cases but rejected for 0%–5%-central Au-Au collisions. Given a Gaussian centered at the azimuth origin, “higher harmonics” $cos\left(m\varphi \right)$ for $m>2$ are rejected. A model consisting of $\mathrm{Gaussian}+\mathrm{dipole}cos\left(\varphi \right)+\mathrm{quadrupole}cos\left(2\varphi \right)$ provides good 1D data descriptions in all cases.

Figure 1

(Left) 2D angular autocorrelations from 200-GeV Au-Au collisions for (a) 83%–94% ($\sim N-N$ collisions) and (c) 0%–5% centralities. (Right) Two-dimensional model fits to the histograms in the left panels obtained with Eq. (14).

Figure 2

One-dimensional projection onto azimuth (points) from the 2D data histogram for 0%–5% central 200-GeV Au-Au collisions in Fig. 1. Statistical errors at 0 and $\pi$ are $\sqrt{2}$ larger than the others owing to symmetrization of data on the periodic variable. The binwise statistical errors 0.0037 have been multiplied by 2 to make them visible outside the data points. The (red) dashed curve is obtained from a fit to the data with the basic Model of Eq. (20). A fit with an FS-only model including four or more terms would appear identical on the scale of this plot. A similar remark applies to corresponding data plots for two other centrality bins.

Figure 3

Power spectrum values $P_m$ (points) derived from the data in Fig. 2 via Eq. (12). The (red) dashed curve is the Gaussian PS described by Eq. (21) with width and amplitude corresponding to the fitted Gaussian in Fig. 2. Interval $m\ge 5$ is consistent with a “white-noise” PS (dotted line) representing the statistical noise in Fig. 2.

Figure 4

The PS for residuals in the form ($\mathrm{data}-\mathrm{Gaussian}$) from Fig. 2 consistent with the white-noise part of the PS in Fig. 3, with mean approximately 0.001. The negative PS value for $m=1$ (not shown) corresponds to the AS dipole amplitude $-A_D^{\prime }$ from the basic-Model fit in Fig. 2.

Figure 5

${\chi }^2$ (upper solid curve and points) and information $2I$ (dashed curve and points with uncertainty band) vs number of parameters $K$ for FS (FS-only) models. The sum (log evidence, $-2LE$, dotted curve) is also included. The lower solid curve is ${\chi }^2$ values for fits to residuals ($\mathrm{data}-\mathrm{basic}\mathrm{Model}$) from Fig. 2 consistent with the trend $11-K$ expected for no signal (noise only) in the data.

Figure 6

${\chi }^2$ values vs number of parameters $K$ for several data models. The general trend is monotonic decrease with increasing number of model parameters, responding only to statistical noise with ${\chi }^2\approx 11-K$ for $K>4$.

Figure 7

Negative log evidence $-2LE$ vs number of parameters $K$ for several models. The basic Model (solid square) is strongly favored over all others (lowest $-LE$). The hatched band indicates the common uncertainty of priors assigned to cosine terms in all models. FS-only models for all $K$ (solid dots and line) are strongly rejected by the evidence.

Figure 8

Normalized $p\left(H_l\right|D^{*})$ from Eq. (22) for several models indexed by $l$. As in Fig. 7 the basic Model (solid square) is strongly favored over all other models while FS-only models for all $K$ are strongly rejected.

Figure 9

One-dimensional projection onto azimuth (points) from the 2D data histogram for 9%–18% central 200-GeV Au-Au collisions. The (red) dashed curve is a fit to the data with the basic Model of Eq. (20) plus independent quadrupole component $A_{Q}2cos\left(2{\varphi }_{\mathrm{\Delta }}\right)$. The binwise statistical errors are 0.0026, not visible outside the points.

Figure 10

Power spectrum values $P_m$ (points) derived from the data in Fig. 9 via Eq. (12). The (red) dashed curve is the Gaussian PS described by Eq. (21) with amplitude and width from the basic $\mathrm{Model}+\mathrm{quadrupole}$ fitted to data in Fig. 9. The interval $m\ge 5$ is consistent with a “white-noise” PS (dotted line) representing the statistical noise in Fig. 9.

Figure 11

${\chi }^2$ values vs number of parameters $K$ for several data models. The general trend is again monotonic decrease with increasing parameter number.

Figure 12

Negative log evidence $-2LE$ vs number of parameters $K$ for several models. The basic $\mathrm{Model}+\mathrm{quadrupole}$ (solid diamond) is strongly favored over others (lowest $-LE$, largest evidence). The basic Model alone (solid square) is strongly rejected by the evidence, as are FS-only models for all $K$ (blue points and line).

Figure 13

1D projection onto azimuth (points) from the 2D data histogram for 83%–94% central 200-GeV Au-Au collisions in Fig. 1. The (red) dashed curve is a fit to the data with the basic Model of Eq. (20). The statistical errors are 0.0026.

Figure 14

Power spectrum values $P_m$ (points) derived from the data in Fig. 13 via Eq. (12). The (red) dashed curve is the Gaussian PS described by Eq. (21) with Gaussian width and amplitude corresponding to the basic Model fitted to data in Fig. 13. The interval $m>2$ is consistent with a “white-noise” PS (dotted line) representing the statistical noise in Fig. 13.

Figure 15

${\chi }^2$ (upper solid curve and points) and information $2I$ (dashed curve and points) vs number of parameters $K$ for FS-only models. The sum $-2LE$ (dotted curve and points) is also included. The ${\chi }^2$ trend for basic-Model fit residuals (lower solid curve) is approximately consistent with the expected noise trend $11-K$.

Figure 16

${\chi }^2$ values vs number of parameters $K$ for several data models. The basic Model (solid square) is equivalent to the FS-only model with $K=3$ (solid dot).

Figure 17

Negative log evidence $-2LE$ vs number of parameters $K$ for several models. The $K=3$ basic Model (solid square) is favored over all others (lowest $-LE$, largest evidence), especially over the $K=2$ FS-only model expected to prevail for this centrality (lowest solid dot).

Figure 18

Joint parameter-data space for two data models. These panels are zoomed out from the scale of Fig. 19 to reveal the PDFs distributed over the entire parameter and data spaces. Given prior intervals ${\mathrm{\Delta }}_k$ the model angles ${\theta }_{kn}$ determine the magnitudes of the predicted data intervals ${\mathrm{\Delta }}_n$, the predicted data volume $V_D\left(H\right)$, and therefore the evidence $E\left(D^{*}\right|H)$. The ${\theta }_{kn}$, and hence the Jacobian of a model function, largely determine the predictivity of a model. For these two models the typical $tan\left({\theta }_{kn}\right)$'s differ by factor 12.

Figure 19

Schematic representation of the local relation in space $w\times D$ between data $D$ and model parameters $w$ with specific elements $D_n$ and $w_k$, especially the errors. The solid diagonal represents model function $D\left(w\right)$. The hatched band arises from data errors, specifically ${\sigma }_n$, corresponding then to parameter errors, specifically ${\sigma }_k$. Angle ${\theta }_{kn}$ relating data and model errors is approximately a Jacobian element characterizing the algebraic structure of the data model. The dash-dotted lines represent parameter-prior and data PDFs.

Figure 20

(Left) Periodic arrays of SS (dash-dotted) and AS (dashed) peaks. The SS peaks are Gaussians. The AS peaks are described by a dipole. The dotted sinusoid corresponds to the $m=2$ Fourier component of the SS peaks. (Right) Evaluation of Eq. (D2) for four values of $m$, with ${\sigma }_{{\varphi }_{\mathrm{\Delta }}}=0.65$.