10  Individual scores

The first output family is the one with a person attached to every number. The question is concrete: whose reported score moves when the prior changes, in which direction, and by enough to matter. The prior contrast here is the focused DP against the Gaussian with the summary held at PM; summary swaps were the previous chapter’s subject, and elicitation swaps are deferred to the noise discussion in Chapter 12 because that is where they surface.

10.1 Where movement lives

The following class-conditioned aggregates use the standardized shape classes of Section 5.4. Median absolute movement is 0.136 SD on the skewed cells, 0.045 on the bimodal ones, 0.039 on the normal controls and 0.024 on the undetermined cells, against the 0.10 SD materiality bar. Skew is the only class above the bar, and Figure 10.1 shows why in the only way a median cannot: movement is not a uniform perturbation but a structured function of where a person sits on the scale.

Figure 10.1: The prior does not move everyone a little; it moves particular regions of the score scale, and which regions those are follows the shape class. Each panel pools the case-cells of one shape class; grey points are individual respondents, placed by their standardized default score (Gaussian + PM, horizontal) and by how far the prior swap moves their score (vertical, in reference-SD units; the dashed lines mark the 0.10 SD materiality convention for the median person). Colored lines trace the per-cell median movement within baseline deciles, one line per case-cell. One panel holds the single case-cell that the standardized screen reports as non-normal without a class. Under skew the lines bend at the ends: the sparse tail is pulled in and the dense shoulder pushed out, with median movement 0.153 SD in the lowest quintile and -0.076 in the highest (pooled). The bimodal panels stay within a band of about 0.03 SD, and the normal- control rows are structured but small; that latter pattern is registered exploratory observation E-02, which remains exploratory because it is a post hoc reading of a sub-material profile. Movement is displacement and not accuracy. Movement computed from the frozen per-person estimates; 8,308 respondents in total.

Under skew the per-cell traces bend at both ends: the sparse tail is pulled inward (the flexible prior, having learned that the population thins there, shrinks its stragglers toward the mass) and the dense shoulder is pushed outward. The quintile profiles of Figure 10.2 make the geometry exact: the lowest quintile’s net movement is +0.153 SD and the highest quintile’s -0.076, ends in, middle out. On the bimodal cells every quintile’s net movement stays within 0.029 SD, the flat profile that makes the next chapter’s finding surprising. The normal-control profile is the registered exploratory observation E-02, not a settled claim. It is small but distinguishable from measured seed variation: its largest quintile net is 0.064 SD, an order of magnitude above the measured Monte Carlo error (every reported movement in this family runs ten to forty times the sampler noise), which is the materiality-versus-distinguishability distinction of Chapter 3 in action. Quiet means sub-material; it does not mean zero, and a reader who needs the flexible prior to change nothing at all on a normal population is asking for more than any sampler can promise.

Figure 10.2: Under skew the prior pulls both ends inward and pushes the middle out; under bimodality the profile is nearly flat; the normal controls show exploratory E-02, a small structured down-up-down pattern. Grey lines give the net (median signed) movement of each baseline-score quintile for one case-cell under the prior swap; the orange line is the frozen class-level profile, drawn only for the three classes that have one. The skewed profile runs from +0.153 SD in quintile 1 to -0.076 in quintile 5; every bimodal quintile stays within 0.029 SD; the normal profile’s largest value is 0.064 SD, an order of magnitude above its seed-based Monte Carlo error estimate, so ‘quiet’ means sub-material, not zero. E-02 stays a registered exploratory observation: it is a post hoc reading of a sub-material profile, and a validated normal class does not make it a settled claim. Quintiles are formed on the default (Gaussian + PM) scores; profiles from the frozen movement tables, per-cell lines re-derived from the per-person estimates, classes from the standardized shape manifest.

10.2 Size of departure does not predict size of consequence

The natural expectation, written into the first edition’s blueprint, was that the farther a population departs from normal the more the prior matters. The portfolio contradicts it. The bimodal cases depart farther, 1.43 times the simulation’s own bimodal condition on the departure scale, against 0.80 for the skewed cases, and move a third as much. Within class, departure size predicts nothing (correlations of -0.05 and +0.03). Figure 10.3 displays the double dissociation.

Figure 10.3: The size of a departure from normality does not track how much the prior moves scores; the bimodal cells depart farther on median and move less. Each point is a case-cell carrying a non-normal class, positioned by its departure from normality expressed as a fraction of the simulation’s own generating condition (horizontal; 1 means as skewed, or as deeply bimodal, as the condition the simulation used) and by its median score movement under the prior swap (vertical). The skewed cells have a median f of 0.8 and move around 0.14 SD; the bimodal cells have a median f of 1.3, past the simulation’s own condition, and move about a third as much. Within class the correlation between departure and movement is -0.05 (skewed) and +0.08 (bimodal): the kind of departure moderates the observed consequence while its magnitude does not, and the correlations rest on nine and seven points. Departure fractions and classes both come from the standardized shape manifest, so the horizontal coordinate is on the same scale as the screen that assigned the panels; movement from the frozen comparison rowset.

One interpretation, consistent with the simulation, is as follows. A skewed density disagrees with the normal about density nearly everywhere, so learning it moves posterior means throughout the scale. A bimodal density of moderate separation agrees with the normal about the center of mass and disagrees mainly about a valley; at these reliabilities each person’s likelihood is too wide to assign them firmly to a mode, so the posterior mean, which averages over that uncertainty, barely moves even while the estimated population density changes a great deal. What bimodality changes is not where people are put but what curve their positions are read against, and that is the next chapter.

10.3 What this family establishes

Conditional on the standardized shape classes, individual scores move materially only in the skewed group and show an ends-in, middle-out pattern. Movement is unrelated to the stored departure magnitude within class, so a shape statistic’s magnitude is not a license to expect large score changes. The normal-control structure is E-02 and remains exploratory. Nothing in this family yet touches the question most analysts would ask first, whether those movements accumulate into different decisions; they mostly do not at the individual level, and they do elsewhere, which is the point of reporting by family.