= Solution
\b[Within-model inference.] The data $\mathbf d$ are the measurements, $H$ specifies a <statistical model> including its sampling assumptions, and $\theta$ is its unknown parameter or parameter vector. The posterior $p(\theta\mid\mathbf d,H)$ describes parameter uncertainty after the observation. The <likelihood> $\mathcal L(\theta)=p(\mathbf d\mid\theta,H)$ describes how that observed data set depends on a proposed parameter. The parameter prior $\pi(\theta\mid H)=p(\theta\mid H)$ encodes information before these data. The evidence $Z_H=p(\mathbf d\mid H)$ is the prior predictive <probability density> of the data and normalizes the posterior:
$$
\boxed{Z_H=\int\mathcal L(\theta)\pi(\theta\mid H)\,d\theta.}
$$
For continuous observations the <likelihood> and evidence are densities in data space, rather than <probabilities> of one exact observation. The evidence compares different models only when they use the same data and reference measure.
\b[Uniform parameter priors.] Normalize each model's top-hat prior, so $\pi_H=1/(\theta_2-\theta_1)$ inside its allowed interval and zero elsewhere. Then
$$
\boxed{Z_H=\frac1{\theta_2-\theta_1}\int_{\theta_1}^{\theta_2}\mathcal L(\theta)\,d\theta.}
$$
Thus evidence is the average <likelihood> over the allowed parameter range, not its maximum and not the unnormalized area alone. Interpret the supplied measured distribution as proportional to a common <likelihood>, $\mathcal L(\theta)=C\exp[-|100u-\theta|/(50u)]$, where the common constant $C$ cancels in model comparisons. If that distribution were instead a posterior calculated using a nonuniform reference prior, it would first need to be divided by that prior. No such alternative prior is specified here.
\b[Evidence calculation.] Put $x=\theta/u$. The three integrals, divided by the same $C$, are
$$
\begin{aligned}
Z_A/C&=\frac1{10}\int_{50}^{60}e^{(x-100)/50}dx
=5(e^{-0.8}-e^{-1})=0.407247615,\\
Z_B/C&=\frac1{20}\left[\int_{90}^{100}e^{(x-100)/50}dx+
\int_{100}^{110}e^{-(x-100)/50}dx\right]
=5(1-e^{-0.2})=0.906346235,\\
Z_C/C&=\frac1{100}\left[\int_{50}^{100}e^{(x-100)/50}dx+
\int_{100}^{150}e^{-(x-100)/50}dx\right]
=1-e^{-1}=0.632120559.
\end{aligned}
$$
Consequently the three pairwise <Bayes factors> may be given as
$$
\boxed{\frac{Z_B}{Z_A}=2.22554,\qquad
\frac{Z_B}{Z_C}=1.43382,\qquad
\frac{Z_C}{Z_A}=1.55218.}
$$
\b[B has the largest evidence.] Model A is narrow but lies away from the <likelihood> peak. Model C contains that peak but spreads much of its normalized prior over less well-fitting values. Model B concentrates its parameter prior near the peak and achieves the best average. Width is therefore not a universal penalty by itself: changing an interval can also improve its location. At fixed centre near a localized <likelihood> peak, extending the interval into poorly fitting values dilutes the evidence.
\Image[/past-exam-of-the-frontiers-of-observational-astrophysics-course-of-the-university-of-cambridge/2015/paper-1-evidence-averages.png]
{title=Original likelihood averages over the three top-hat prior intervals; bars show evidence and separately prior-weighted model probabilities}
\b[Between-model prior information.] The supplied $R_A:R_B:R_C=1:1:0.3$ are model prior odds, distinct from each model's uniform parameter prior. They multiply the evidences to give posterior model weights. The corresponding posterior odds are
$$
\boxed{\frac{p(B\mid\mathbf d)}{p(A\mid\mathbf d)}=2.22554,\qquad
\frac{p(B\mid\mathbf d)}{p(C\mid\mathbf d)}=4.77940,\qquad
\frac{p(A\mid\mathbf d)}{p(C\mid\mathbf d)}=2.14752.}
$$
Normalizing these weights gives $p(A\mid\mathbf d)=0.270915$, $p(B\mid\mathbf d)=0.602933$ and $p(C\mid\mathbf d)=0.126152$. \b[B is preferred with either equal model priors or the stated theoretical prior odds.] Multiplying by $R$ changes posterior odds; it does not redefine the <Bayes factors>.
\b[Euclidean <luminosity> selection.] In a flux-limited catalogue, a source with isotropic <luminosity> $L$ is detected out to
$$
r_{\max}=\sqrt{\frac{L}{4\pi f_{\min}}},\qquad
V_{\max}\propto r_{\max}^3\propto L^{3/2}.
$$
A homogeneous Euclidean population with intrinsic <luminosity function> $\varphi(L)$ is sampled approximately as $dN_{\rm det}/dL\propto\varphi(L)L^{3/2}$. Brighter objects occupy a much larger accessible volume and are overrepresented; at a fixed large distance the faint tail is missing entirely. This is <Malmquist bias>. Using the selected population as if it were representative can bias <luminosity> <calibrations> or inferred correlations with $\theta$. A selection function, a volume-limited sample, or appropriate accessible-volume weighting can address the sampling effect, subject to completeness assumptions.
The Euclidean derivation concerns flux-limited luminous objects or selected tracers associated with the structures. A <CMB> temperature map is a diffuse surface-brightness field on a cosmological last-scattering surface: it is not automatically a Euclidean catalogue of identical inverse-square point sources. For a particular CMB-structure catalogue, its actual detection statistic and selection function must be specified before applying this luminosity-weighting law quantitatively.
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