Bayes for proton structure; priors
The rule aimed at function inference; prior design and prior sensitivity, honestly.
Every number this lab publishes — the bands on a valence curve, the $\log Z = -381.9$ on a result card, the verdict that one parametrization beats another — is a single theorem applied at industrial scale. This chapter derives that theorem from the product rule of probability, builds the likelihood our fits actually evaluate from the Gaussian data model of §3.4, establishes the posterior as the answer to the inverse problem of §3.1, and then spends serious time on the part practitioners argue about: priors — what a uniform box really asserts, why "flat" is not a property of nature but of coordinates, when the prior is harmless and when it is the whole answer, and the function-space subtlety that makes priors for curves genuinely tricky.
From the product rule to the theorem
Bayes' theorem is not an axiom on top of probability theory; it is two lines of algebra inside it. Write $\theta$ for our parameter vector (13 for the LH toy, 52 for the MSHT20 port of §3.1) and $D$ for the data vector.
The product rule factorizes a joint density in two equivalent orders:
$$ p(\theta, D) \;=\; p(D \mid \theta)\, p(\theta) \;=\; p(\theta \mid D)\, p(D) $$
Equate the two factorizations and divide by $p(D)$ (nonzero for data actually observed):
$$ p(\theta \mid D) \;=\; \frac{p(D \mid \theta)\, \pi(\theta)}{Z}, \qquad Z \equiv p(D) = \int p(D \mid \theta)\, \pi(\theta)\, d\theta $$
Names: $\pi(\theta)$ the prior, $p(D\mid\theta) \equiv L(\theta)$ the likelihood (a function of $\theta$ at fixed, observed $D$), $p(\theta\mid D)$ the posterior, and $Z$ the evidence or marginal likelihood — the normalization now, the star of §4.3 later. $\blacksquare$
Section 3.4 built the experimental covariance $C$ (statistical, correlated systematics, $t_0$ treatment). The data model is multivariate normal: $D \sim \mathcal N\big(T(\theta),\, C\big)$ with $T(\theta)$ the theory prediction through the FK pipeline of §3.3. The density, written in full with its normalization:
$$ L(\theta) \;=\; (2\pi)^{-N/2}\, |C|^{-1/2}\, \exp\!\Big[-\tfrac12\, r^{\mathsf T} C^{-1} r\Big], \qquad r \equiv D - T(\theta) $$
The quadratic form is exactly the $\chi^2$ of §3.4, so
$$ L(\theta) \;=\; G \cdot e^{-\chi^2(\theta)/2}, \qquad G \equiv (2\pi)^{-N/2} |C|^{-1/2} \;=\; \text{independent of } \theta $$
The determinant and the $(2\pi)^{-N/2}$ live entirely in $G$: with $N = 648$ points, $\ln G$ is a large constant, but a constant. $\blacksquare$
Insert $L = G\,e^{-\chi^2/2}$ into Bayes' theorem:
$$ p(\theta \mid D) = \frac{G\, e^{-\chi^2(\theta)/2}\, \pi(\theta)} {G \int e^{-\chi^2(\theta')/2}\, \pi(\theta')\, d\theta'} = \frac{e^{-\chi^2(\theta)/2}\, \pi(\theta)}{\int e^{-\chi^2/2}\, \pi\, d\theta'} $$
$G$ cancels. For parameter inference only differences $\Delta\chi^2$ between points in parameter space matter — which is why frequentist fitting (§5.1) can ignore normalizations entirely. But the evidence keeps it: $\ln Z = \ln G + \ln \int e^{-\chi^2/2}\pi\, d\theta$. The absolute $\log Z$ values on our cards therefore carry a convention-dependent constant. It is harmless for exactly one reason: when two models are compared on the same data with the same covariance, $\ln G$ is identical on both sides and $\Delta \ln Z$ is normalization-free. That is why every evidence verdict in this lab — PPDF-9's $-109$, PPDF-12's $+1453$ — is a difference at fixed data, and why evidences must never be compared across different datasets. $\blacksquare$
The posterior is the answer
The posterior is a probability density over the full 13- or 52-dimensional parameter space, and everything we report is a functional of it. Three functionals do all the work:
Marginals. The distribution of one parameter, everything else integrated out: $p(\theta_i \mid D) = \int p(\theta \mid D)\, \prod_{j \neq i} d\theta_j$. This integral is what makes Bayesian errors honest: a parameter that is tightly pinned at fixed values of the others but degenerate with them gets a wide marginal, automatically.
Credible intervals. A 68% credible interval $[\theta_-, \theta_+]$ satisfies $\int_{\theta_-}^{\theta_+} p(\theta_i \mid D)\, d\theta_i = 0.68$ — a direct probability statement about the parameter, not a coverage statement about hypothetical repeated experiments (the contrast with §5.1's Hessian intervals is the subject of §5.3).
Pushforwards. Most of what we care about is not a parameter but a derived quantity: the curve $x f(x, Q_0^2; \theta)$ at each $x$, a convolved cross-section, a sum-rule integrand. For any $g(\theta)$, the induced distribution and its expectations follow from the sample-based propagation identity: given weighted posterior samples $\{\theta^{(s)}, w_s\}$ (nested sampling hands us exactly these, §4.2),
Apply it with $g = xf(x;\theta)$ at 100 values of $x$ and take percentiles instead of means: that is, literally, how every band on our result plots is drawn. The identity is exact in the limit of many samples and requires no linearization — the crucial advantage over Hessian propagation when the posterior is non-Gaussian (§5.3).
Priors: boxes, volumes, and reference bands
Our fits use uniform box priors: $\pi(\theta) = 1/V_\pi$ inside a hyper-rectangle with widths $W_i$, zero outside, so $V_\pi = \prod_i W_i$. Two design choices hide in that sentence. First, the volume: it divides $Z$ directly, so doubling a width you did not need costs $\ln 2$ of evidence per parameter — the Occam mechanism §4.3 makes exact. Second, the placement: for the MSHT20 port we use reference-band priors — boxes centred on the published MSHT20 central values with widths set to a generous multiple of the published uncertainties — so that "the prior contains the answer" is an audited statement, not a hope.
Let $\theta$ carry a flat prior on $[a, b]$, and pass to $u = \ln\theta$ (a natural move for a normalization that ranges over decades). Conservation of probability under a change of variables demands $p(u)\,|du| = p(\theta)\,|d\theta|$, i.e.
$$ p(u) \;=\; p(\theta)\,\Big|\frac{d\theta}{du}\Big| \;=\; \frac{1}{b-a}\, e^{u} $$
— exponential in $u$, not flat. Concretely: flat on $\theta \in [0.01, 1]$ puts mass $0.9/0.99 \approx 91\%$ in the top decade $[0.1, 1]$ and only $9\%$ in the bottom one. The person who declared "no preference in $\theta$" declared a 10:1 preference in $\ln\theta$. There is no coordinate-free notion of ignorance; a flat prior is a choice of coordinates, and honesty consists of declaring it (and testing sensitivity to it), not of pretending it away. $\blacksquare$
Where the prior matters — and where it cannot
The theory of prior sensitivity is a tale of two directions.
Constrained directions: the likelihood wins. Suppose along some direction the likelihood is Gaussian with width $\sigma_L$, centred well inside a box of width $W$. The posterior is the likelihood truncated to the box; the truncation changes its moments only by tail terms of order $e^{-(W/2\sigma_L)^2/2}$. For $W = 10\,\sigma_L$ that is $e^{-12.5} \sim 10^{-6}$ — the posterior is exponentially indifferent to the box. Widen $W$ by any factor: the numerator $e^{-\chi^2/2}\pi$ rescales uniformly, the normalization rescales identically, the posterior does not move. (The evidence does move — by exactly $-\ln(\text{factor})$ — which is the whole content of Worked Example 1.)
Flat directions: the prior is the posterior. If the likelihood does not vary along a direction ($\chi^2$ constant), the posterior density along it is proportional to $\pi$ — the data have added nothing, and the "uncertainty" you quote there is a verbatim copy of your prior width. This is not a pathology; it is the formalism telling the truth. Our fits live in this regime for real: PPDF-12's Bayesian complexity of 13.7 (derived in §4.3) says the data constrain about fourteen directions of the 52 — the remaining directions report the prior back to us, and the marginals in those directions must be read as assumptions, not measurements.
Because $Z \propto 1/V_\pi$ per constrained direction, a prior chosen after seeing the evidence table is a thumb on the scale: shrink your favourite model's boxes and its $\ln Z$ climbs by $\sum_i \ln(\text{shrink factor})$ with no physics added. The defence is procedural, not mathematical: priors (placement, widths, parametrization of "flat") are fixed and recorded before the evidence is computed, and prior-sensitivity scans are reported alongside verdicts (§4.3 quantifies when a verdict is prior-proof). This is the same blind-analysis discipline as the pre-registered cuts of a particle search — the analysis choices that could steer the answer are frozen while you are still ignorant of it. Our closure certificates (PPDF-11, §4.4) are run under exactly the priors later used on real data, for exactly this reason.
A prior on parameters is a prior on curves
PDF fitting adds a subtlety absent from textbook examples: the object of inference is a function, and the prior we write down lives on coefficients. The map between them is nonlinear and $x$-dependent, so an innocent-looking box on parameters induces a structured, non-uniform prior on curves. The simplest toy already shows it. Take $f(x) = a + bx$ on $x \in [0,1]$ with independent flat priors $a, b \sim U[-1,1]$. At $x = 0$ the curve value is $f = a$: uniform on $[-1,1]$. At $x = 1$ it is $f = a + b$: the sum of two uniforms — a triangular density on $[-2,2]$, peaked at 0. Same "uninformative" box, different induced law at every $x$: the envelope widens linearly while the density develops a central peak.
For realistic parametrizations the distortion compounds: a flat box on a small-$x$ exponent $\alpha$ induces an envelope on $xf(x)$ whose width at $x = 10^{-3}$ is a factor $10^{\Delta\alpha \cdot 3}$-ish in ratio — decades of curve-space prior volume concentrated where there is no data. This is why prior design in this lab is done by plotting the prior pushforward — draw parameters from $\pi$, plot the resulting curve family next to the reference band — before any likelihood is ever evaluated, and why the wiki entry on priors shows those envelope plots for every parametrization we run.
Worked example 1 — one parameter, everything by hand
Setup: one datum $d = 2.0$ with Gaussian error $\sigma = 0.5$; the model predicts the value $\theta$ itself, so $L(\theta) = \exp[-(2.0-\theta)^2/(2\times0.25)]$ (normalization absorbed). Prior: flat on $[0, 4]$, so $\pi = 1/4$.
Posterior: $p(\theta\mid D) \propto e^{-(\theta-2)^2/0.5}$ on $[0,4]$ — a Gaussian with mean 2.0 and width 0.5, truncated at $\pm 4\sigma$ (truncated mass $\sim 6\times10^{-5}$: negligible). The 68% credible interval is $[1.5, 2.5]$; 95% is $[1.02, 2.98]$.
Evidence: $Z = \frac14 \int_0^4 e^{-(\theta-2)^2/0.5}\, d\theta \approx \frac14 \sqrt{2\pi}\,(0.5) = \frac{1.2533}{4} = 0.3133$, so $\ln Z = -1.161$.
Widen the prior to $[-18, 22]$ (width 40, ten times larger, still containing the peak): the posterior is unchanged — the likelihood was already negligible at the old edges. But $Z = 1.2533/40 = 0.03133$: $\;\ln Z = -3.463 = -1.161 - \ln 10$. Evidence drops by exactly $\ln 10 = 2.303$ while the inference does not move a hair. That asymmetry — posteriors robust, evidences volume-sensitive — is the single most important fact in this chapter, and §4.3 turns it into the Occam factor.
Worked example 2 — a degenerate likelihood: our valence story in miniature
Setup: $L \propto \exp\big[-(\theta_1+\theta_2-2)^2 / (2 \times 0.1^2)\big]$, priors $\theta_1, \theta_2 \sim U[0,2]$. The data measure the sum to $\pm 0.1$ and say nothing else.
Rotate: $u = (\theta_1+\theta_2)/\sqrt2$, $v = (\theta_1-\theta_2)/\sqrt2$. The likelihood is Gaussian in $u$ (width $0.1/\sqrt2 = 0.071$, centred at $\sqrt2$) and constant in $v$: the posterior is a ridge — razor-thin across, prior-flat along.
The sum is measured: prior on $\theta_1+\theta_2$ (sum of two uniforms) is triangular on $[0,4]$ with $\sigma_\pi = 2/\sqrt6 = 0.816$; posterior width 0.1 — a compression factor of 8.2. Constrained direction: likelihood wins.
The difference is not: along the ridge the posterior reproduces the prior conditioned on the ridge: $\theta_1 - \theta_2 = 2\theta_1 - 2$ with $\theta_1$ essentially uniform — no information added.
The marginals lie in wait: $p(\theta_1 \mid D) \propto \int_0^2 e^{-(\theta_1+\theta_2-2)^2/0.02} d\theta_2$. For $\theta_1 \in [0.3, 1.7]$ the inner Gaussian sits wholly inside the range, and the integral is the same constant $\approx \sqrt{2\pi}\,(0.1)$: the marginal is flat, with $\sigma(\theta_1) \approx 2/\sqrt{12} = 0.577$ — indistinguishable from the prior. A reader of the $\theta_1$ marginal alone would conclude "the data taught us nothing," while the $u$ marginal shows a crisp $\pm 0.071$ measurement. This is precisely our valence situation: DIS data constrain particular flavour combinations hard while individual parameters stay prior-wide — the mechanism behind the $-808$ evidence collapse when constrained flavour directions are removed (PPDF-8), and the reason our result cards show combination marginals, not just raw-parameter ones.