\(\require{mathtools} \newcommand{\notag}{} \newcommand{\tag}{} \newcommand{\label}[1]{} \newcommand{\sfrac}[2]{#1/#2} \newcommand{\bm}[1]{\boldsymbol{#1}} \newcommand{\num}[1]{#1} \newcommand{\qty}[2]{#1\,#2} \renewenvironment{align} {\begin{aligned}} {\end{aligned}} \renewenvironment{alignat} {\begin{alignedat}} {\end{alignedat}} \newcommand{\pdfmspace}[1]{} % Ignore PDF-only spacing commands \newcommand{\htmlmspace}[1]{\mspace{#1}} % Ignore PDF-only spacing commands \newcommand{\scaleto}[2]{#1} % Allow to use scaleto from scalerel package \newcommand{\RR}{\mathbb R} \newcommand{\NN}{\mathbb N} \newcommand{\PP}{\mathbb P} \newcommand{\EE}{\mathbb E} \newcommand{\XX}{\mathbb X} \newcommand{\ZZ}{\mathbb Z} \newcommand{\QQ}{\mathbb Q} \newcommand{\fF}{\mathcal F} \newcommand{\dD}{\mathcal D} \newcommand{\lL}{\mathcal L} \newcommand{\gG}{\mathcal G} \newcommand{\hH}{\mathcal H} \newcommand{\nN}{\mathcal N} \newcommand{\pP}{\mathcal P} \newcommand{\BB}{\mathbb B} \newcommand{\Exp}{\operatorname{Exp}} \newcommand{\Binomial}{\operatorname{Binomial}} \newcommand{\Poisson}{\operatorname{Poisson}} \newcommand{\linop}{\mathcal{L}(\mathbb{B})} \newcommand{\linopell}{\mathcal{L}(\ell_1)} \DeclareMathOperator{\trace}{trace} \DeclareMathOperator{\Var}{Var} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\col}{col} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\gt}{>} \definecolor{highlight-blue}{RGB}{0,123,255} % definition, theorem, proposition \definecolor{highlight-yellow}{RGB}{255,193,7} % lemma, conjecture, example \definecolor{highlight-orange}{RGB}{253,126,20} % criterion, corollary, property \definecolor{highlight-red}{RGB}{220,53,69} % criterion \newcommand{\logL}{\ell} \newcommand{\eE}{\mathcal{E}} \newcommand{\oO}{\mathcal{O}} \newcommand{\defeq}{\stackrel{\mathrm{def}}{=}} \newcommand{\Bspec}{\mathcal{B}} % Spectral radiance \newcommand{\X}{\mathcal{X}} % X space \newcommand{\Y}{\mathcal{Y}} % Y space \newcommand{\M}{\mathcal{M}} % Model \newcommand{\Tspace}{\mathcal{T}} \newcommand{\Vspace}{\mathcal{V}} \newcommand{\Mtrue}{\mathcal{M}_{\mathrm{true}}} \newcommand{\MP}{\M_{\mathrm{P}}} \newcommand{\MRJ}{\M_{\mathrm{RJ}}} \newcommand{\qproc}{\mathfrak{Q}} \newcommand{\D}{\mathcal{D}} % Data (true or generic) \newcommand{\Dt}{\tilde{\mathcal{D}}} \newcommand{\Phit}{\widetilde{\Phi}} \newcommand{\Phis}{\Phi^*} \newcommand{\qt}{\tilde{q}} \newcommand{\qs}{q^*} \newcommand{\qh}{\hat{q}} \newcommand{\AB}[1]{\mathtt{AB}~\mathtt{#1}} \newcommand{\LP}[1]{\mathtt{LP}~\mathtt{#1}} \newcommand{\NML}{\mathrm{NML}} \newcommand{\iI}{\mathcal{I}} \newcommand{\true}{\mathrm{true}} \newcommand{\dist}{D} \newcommand{\Mtheo}[1]{\mathcal{M}_{#1}} % Model (theoretical model); index: param set \newcommand{\DL}[1][L]{\mathcal{D}^{(#1)}} % Data (RV or generic) \newcommand{\DLp}[1][L]{\mathcal{D}^{(#1')}} % Data (RV or generic) \newcommand{\DtL}[1][L]{\tilde{\mathcal{D}}^{(#1)}} % Data (RV or generic) \newcommand{\DpL}[1][L]{{\mathcal{D}'}^{(#1)}} % Data (RV or generic) \newcommand{\Dobs}[1][]{\mathcal{D}_{\mathrm{obs}}^{#1}} % Data (observed) \newcommand{\calibset}{\mathcal{C}} \newcommand{\N}{\mathcal{N}} % Normal distribution \newcommand{\Z}{\mathcal{Z}} % Partition function \newcommand{\VV}{\mathbb{V}} % Variance \newcommand{\T}{\mathsf{T}} % Transpose \newcommand{\EMD}{\mathrm{EMD}} \newcommand{\dEMD}{d_{\mathrm{EMD}}} \newcommand{\dEMDtilde}{\tilde{d}_{\mathrm{EMD}}} \newcommand{\dEMDsafe}{d_{\mathrm{EMD}}^{\text{(safe)}}} \newcommand{\e}{ε} % Model confusion threshold \newcommand{\falsifythreshold}{ε} \newcommand{\bayes}[1][]{B_{#1}} \newcommand{\bayesthresh}[1][]{B_{0}} \newcommand{\bayesm}[1][]{B^{\mathcal{M}}_{#1}} \newcommand{\bayesl}[1][]{B^l_{#1}} \newcommand{\bayesphys}[1][]{B^{{p}}_{#1}} \newcommand{\Bconf}[1]{B^{\mathrm{epis}}_{#1}} \newcommand{\Bemd}[1]{B^{\mathrm{EMD}}_{#1}} 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\newcommand{\MsA}{\mathcal{M}^*_A} \newcommand{\MsB}{\mathcal{M}^*_B} \newcommand{\Msa}{\mathcal{M}^*_a} \newcommand{\MsAz}{\mathcal{M}^*_{A,z}} \newcommand{\MsBz}{\mathcal{M}^*_{B,z}} \newcommand{\Msaz}{\mathcal{M}^*_{a,z}} \newcommand{\MeAz}{\mathcal{M}^ε_{A,z}} \newcommand{\MeBz}{\mathcal{M}^ε_{B,z}} \newcommand{\Meaz}{\mathcal{M}^ε_{a,z}} \newcommand{\zo}{z^{0}} \renewcommand{\lL}[2][]{\mathcal{L}_{#1|{#2}}} % likelihood \newcommand{\Lavg}[2][]{\mathcal{L}^{/#2}_{#1}} % Geometric average of likelihood \newcommand{\lLphys}[2][]{\mathcal{L}^{{p}}_{#1|#2}} \newcommand{\Lavgphys}[2][]{\mathcal{L}^{{p}/#2}_{#1}} % Geometric average of likelihood \newcommand{\lLL}[3][]{\mathcal{L}^{(#3)}_{#1|#2}} \newcommand{\lLphysL}[3][]{\mathcal{L}^{{p},(#3)}_{#1|#2}} \newcommand{\lnL}[2][]{l_{#1|#2}} % Per-sample log likelihood \newcommand{\lnLt}[2][]{\widetilde{l}_{#1|#2}} \newcommand{\lnLtt}{\widetilde{l}} % Used only in path_sampling \newcommand{\lnLh}[1][]{\hat{l}_{#1}} \newcommand{\lnLphys}[2][]{l^{{p}}_{#1|#2}} \newcommand{\lnLphysL}[3][]{l^{{p},(#3)}_{#1|#2}} \newcommand{\Elmu}[2][1]{μ_{{#2}}^{(#1)}} \newcommand{\Elmuh}[2][1]{\hat{μ}_{{#2}}^{(#1)}} \newcommand{\Elsig}[2][1]{Σ_{{#2}}^{(#1)}} \newcommand{\Elsigh}[2][1]{\hat{Σ}_{{#2}}^{(#1)}} \newcommand{\pathP}{\mathop{{p}}} % Path-sampling process (generic) \newcommand{\pathPhb}{\mathop{{p}}_{\mathrm{Beta}}} % Path-sampling process (hierarchical beta) \newcommand{\interval}{\mathcal{I}} \newcommand{\Phiset}[1]{\{\Phi\}^{\small (#1)}} \newcommand{\Phipart}[1]{\{\mathcal{I}_Φ\}^{\small (#1)}} \newcommand{\qhset}[1]{\{\qh\}^{\small (#1)}} \newcommand{\Dqpart}[1]{\{Δ\qh_{2^{#1}}\}} \newcommand{\LsAzl}{\mathcal{L}_{\smash{{}^{\,*}_A},z,L}} \newcommand{\LsBzl}{\mathcal{L}_{\smash{{}^{\,*}_B},z,L}} \newcommand{\lsA}{l_{\smash{{}^{\,*}_A}}} \newcommand{\lsB}{l_{\smash{{}^{\,*}_B}}} \newcommand{\lsAz}{l_{\smash{{}^{\,*}_A},z}} \newcommand{\lsAzj}{l_{\smash{{}^{\,*}_A},z_j}} \newcommand{\lsAzo}{l_{\smash{{}^{\,*}_A},z^0}} \newcommand{\leAz}{l_{\smash{{}^{\,ε}_A},z}} \newcommand{\lsAez}{l_{\smash{{}^{*ε}_A},z}} \newcommand{\lsBz}{l_{\smash{{}^{\,*}_B},z}} \newcommand{\lsBzj}{l_{\smash{{}^{\,*}_B},z_j}} \newcommand{\lsBzo}{l_{\smash{{}^{\,*}_B},z^0}} \newcommand{\leBz}{l_{\smash{{}^{\,ε}_B},z}} \newcommand{\lsBez}{l_{\smash{{}^{*ε}_B},z}} \newcommand{\LaszL}{\mathcal{L}_{\smash{{}^{*}_a},z,L}} \newcommand{\lasz}{l_{\smash{{}^{*}_a},z}} \newcommand{\laszj}{l_{\smash{{}^{*}_a},z_j}} \newcommand{\laszo}{l_{\smash{{}^{*}_a},z^0}} \newcommand{\laez}{l_{\smash{{}^{ε}_a},z}} \newcommand{\lasez}{l_{\smash{{}^{*ε}_a},z}} \newcommand{\lhatasz}{\hat{l}_{\smash{{}^{*}_a},z}} \newcommand{\pasz}{p_{\smash{{}^{*}_a},z}} \newcommand{\paez}{p_{\smash{{}^{ε}_a},z}} \newcommand{\pasez}{p_{\smash{{}^{*ε}_a},z}} \newcommand{\phatsaz}{\hat{p}_{\smash{{}^{*}_a},z}} \newcommand{\phateaz}{\hat{p}_{\smash{{}^{ε}_a},z}} \newcommand{\phatseaz}{\hat{p}_{\smash{{}^{*ε}_a},z}} \newcommand{\Phil}[2][]{Φ_{#1|#2}} % Φ_{\la} \newcommand{\Philt}[2][]{\widetilde{Φ}_{#1|#2}} % Φ_{\la} \newcommand{\Philhat}[2][]{\hat{Φ}_{#1|#2}} % Φ_{\la} \newcommand{\Philsaz}{Φ_{\smash{{}^{*}_a},z}} % Φ_{\lasz} \newcommand{\Phileaz}{Φ_{\smash{{}^{ε}_a},z}} % Φ_{\laez} \newcommand{\Philseaz}{Φ_{\smash{{}^{*ε}_a},z}} % Φ_{\lasez} \newcommand{\mus}[1][1]{μ^{(#1)}_*} \newcommand{\musA}[1][1]{μ^{(#1)}_{\smash{{}^{\,*}_A}}} \newcommand{\SigsA}[1][1]{Σ^{(#1)}_{\smash{{}^{\,*}_A}}} \newcommand{\musB}[1][1]{μ^{(#1)}_{\smash{{}^{\,*}_B}}} \newcommand{\SigsB}[1][1]{Σ^{(#1)}_{\smash{{}^{\,*}_B}}} \newcommand{\musa}[1][1]{μ^{(#1)}_{\smash{{}^{*}_a}}} \newcommand{\Sigsa}[1][1]{Σ^{(#1)}_{\smash{{}^{*}_a}}} \newcommand{\Msah}{{\color{highlight-red}\mathcal{M}^{*}_a}} \newcommand{\Msazh}{{\color{highlight-red}\mathcal{M}^{*}_{a,z}}} \newcommand{\Meah}{{\color{highlight-blue}\mathcal{M}^{ε}_a}} \newcommand{\Meazh}{{\color{highlight-blue}\mathcal{M}^{ε}_{a,z}}} \newcommand{\lsazh}{{\color{highlight-red}l_{\smash{{}^{*}_a},z}}} \newcommand{\leazh}{{\color{highlight-blue}l_{\smash{{}^{ε}_a},z}}} \newcommand{\lseazh}{{\color{highlight-orange}l_{\smash{{}^{*ε}_a},z}}} \newcommand{\Philsazh}{{\color{highlight-red}Φ_{\smash{{}^{*}_a},z}}} % Φ_{\lasz} \newcommand{\Phileazh}{{\color{highlight-blue}Φ_{\smash{{}^{ε}_a},z}}} % Φ_{\laez} \newcommand{\Philseazh}{{\color{highlight-orange}Φ_{\smash{{}^{*ε}_a},z}}} % Φ_{\lasez} \newcommand{\emdstd}{\tilde{σ}} \DeclareMathOperator{\Mvar}{Mvar} \DeclareMathOperator{\AIC}{AIC} \DeclareMathOperator{\epll}{epll} \DeclareMathOperator{\elpd}{elpd} \DeclareMathOperator{\MDL}{MDL} \DeclareMathOperator{\comp}{COMP} \DeclareMathOperator{\Lognorm}{Lognorm} \DeclareMathOperator{\erf}{erf} \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator{\Image}{Image} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\SE}{SE} % standard error \DeclareMathOperator{\Unif}{Unif} \DeclareMathOperator{\Poisson}{Poisson} \DeclareMathOperator{\SkewNormal}{SkewNormal} \DeclareMathOperator{\TruncNormal}{TruncNormal} \DeclareMathOperator{\Exponential}{Exponential} \DeclareMathOperator{\exGaussian}{exGaussian} \DeclareMathOperator{\IG}{IG} \DeclareMathOperator{\NIG}{NIG} \DeclareMathOperator{\Gammadist}{Gamma} \DeclareMathOperator{\Lognormal}{Lognormal} \DeclareMathOperator{\Beta}{Beta} \newcommand{\sinf}{{s_{\infty}}}\)
4.1. Poisson noise model for black body radiation observations#
In Eq. 2.33, we used a Poisson counting process to simulate the observation noise for recordings of a black body’s radiance. This is a more realistic model than Gaussian noise for this system, while still being simple enough to serve our illustration.
The physical motivation is as follows. We assume that data are recorded with a spectrometer which physically separates photons of different wavelengths and measures their intensity with a CCD array. We further assume for simplicity that wavelengths are integrated in bins of equal width, such that the values of \(λ\) are sampled uniformly (the case with non-uniform bins is less concise but otherwise equivalent). We also assume that the device uses a fixed time window to integrate fluxes, such that what it detects are effective photon counts. The average number of counts is proportional to the radiance, but also to physical parameters of the sensor (including size, integration window and sensitivity) which we collect into the factor \(s\); the units of \(s\) are \({\mathrm{m}^{2} \cdot \mathrm{nm} \cdot \mathrm{photons} \cdot \mathrm{sr}} / {\mathrm{kW}}\), such that \(s \Bspec_a(λ; T)\) is a number of photons. Since the photons are independent, the recorded number of photons will be random and follow a Poisson distribution. This leads to the following data-generating process \(\Mtrue\):
Here \(\Bspec_0\) captures the effect of dark currents and random photon losses on the radiance measurement. Recall that a random variable \(k\) following a distribution \(\Poisson(μ)\) has probability mass function \(P(k) = \frac{μ^k e^{-μ}}{k!}\).
Note that we divide by \(s\) so that \(\Bspec\) also has dimensions of radiance and is comparable with the models \(\Bspec_{\mathrm{P}}\) and \(\Bspec_{\mathrm{RJ}}\) defined in Eq. 2.31 and Eq. 2.32.