\(\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}} \newcommand{\Bconfbin}[1][]{\bar{B}^{\mathrm{conf}}_{#1}} \newcommand{\Bemdbin}[1][]{\bar{B}_{#1}^{\mathrm{EMD}}} \newcommand{\bin}{\mathcal{B}} \newcommand{\Bconft}[1][]{\tilde{B}^{\mathrm{conf}}_{#1}} \newcommand{\fc}{f_c} \newcommand{\fcbin}{\bar{f}_c} \newcommand{\paramphys}[1][]{Θ^{{p}}_{#1}} \newcommand{\paramobs}[1][]{Θ^{ε}_{#1}} \newcommand{\test}{\mathrm{test}} \newcommand{\train}{\mathrm{train}} \newcommand{\synth}{\mathrm{synth}} \newcommand{\rep}{\mathrm{rep}} \newcommand{\MNtrue}{\mathcal{M}^{{p}}_{\text{true}}} \newcommand{\MN}[1][]{\mathcal{M}^{{p}}_{#1}} \newcommand{\MNA}{\mathcal{M}^{{p}}_{Θ_A}} \newcommand{\MNB}{\mathcal{M}^{{p}}_{Θ_B}} \newcommand{\Me}[1][]{\mathcal{M}^ε_{#1}} \newcommand{\Metrue}{\mathcal{M}^ε_{\text{true}}} \newcommand{\Meobs}{\mathcal{M}^ε_{\text{obs}}} \newcommand{\Meh}[1][]{\hat{\mathcal{M}}^ε_{#1}} \newcommand{\MNa}{\mathcal{M}^{\mathcal{N}}_a} \newcommand{\MeA}{\mathcal{M}^ε_A} \newcommand{\MeB}{\mathcal{M}^ε_B} \newcommand{\Ms}{\mathcal{M}^*} \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}}}\)
4.5. Calibration for the neural response model#
As described in our Results, the goal of calibration is twofold. First we want to correlate the value of \(\Bemd{AB;c}\) with the probability \(\Bconf{AB;Ω}\) that, across variations of the data-generating process \(\Mtrue\), model \(A\) has lower risk than model \(B\). We use an epistemic distribution \(Ω\) to describe those variations. Second, we want to ensure that a decision to reject either model based on \(\Bemd{AB}\) is robust: it should hold for any reasonable epistemic distribution \(Ω\) of experimental conditions (and therefore hopefully also for unanticipated experimental variations).
The calibration procedure involves fixing two candidate models and an epistemic distribution \(Ω\). We then sample multiple data-generating models \(\Mtrue\) from \(Ω\), generate a replicate dataset \(\D_\test^\rep\) from each, and compute both \(\Bemd{}\) and \(\Bconf{}\) on that replicate dataset. This process can be repeated for as many epistemic distributions and as many different pairs of candidate models as desired, until we are sufficiently confident in the robustness of \(\Bemd{}\). This means in particular that we cannot expect perfect correlation between \(\Bemd{}\) and \(\Bconf{}\) across all models and conditions.
However we don’t actually require perfect correlation because this is an asymmetrical test: a decision to reject neither model is largely preferred over a decision to reject the wrong one. Therefore what we especially want to ensure is that for any pair of models \(A\), \(B\), and any reasonable epistemic distribution \(Ω\), we have
This is given as either Eq. 2.29 or Eq. 2.30 in the Results.
For the neuron model, we consider four sources of experimental variability: variations in the distribution used to model observation noise \(ξ\), variations in the strength (\(σ_o\)) of observation noise, as well as variations in the strength (\(σ_i\)) and correlation time (\(τ\)) of the external input \(I_{\mathrm{ext}}\). Each epistemic distribution \(Ω\) is therefore described by four distributions over hyperparameters:
Observation noise model
One of Gaussian or Cauchy. The distributions are centered and \(σ_o\) is drawn from the distribution defined below. Note that this is the model used to generate a dataset \(\D_\test^\rep\). The candidate models always evaluate their loss assuming a Gaussian observation model.
Observation noise strength \(σ_o\)
External input strength \(σ_i\)
The parameter \(σ_i\) sets the strength of the input noise such that \(\langle{I_{\mathrm{ext}}^2\rangle} = σ_i^2\).
External input correlation time \(τ\)
The parameter \(τ\) sets the correlation time of the input noise such that \(\langle{I_{\mathrm{ext}}(t)I_{\mathrm{ext}}(t+s)\rangle} = σ_i^2 e^{-s^2/2τ^2}\).
We thus defined 2 statistical distributions for \(ξ\), 2 distributions for \(σ_o\), 2 distributions for \(τ\), and 2 distributions for \(σ_i\). Combined with three model pairs (see Fig. 2.5), this makes a total of \(48 = 3 \times 2 \times 2 \times 2 \times 2\) possible epistemic distributions \(Ω\), each of which can be identified by a tuple such as \(\bigl(\M_A \,\mathtt{vs}\, \M_B\), \(\mathtt{Cauchy}\), \(\mathtt{Low}\,\mathtt{noise}\), \(\mathtt{Strong}\,\mathtt{input}\), \(\mathtt{Long}\,\mathtt{correlation}\bigr)\). Calibration results for each of these conditions are given in Fig. 6.1.
During calibration against an epistemic distribution \(Ω\), drawing a dataset \(\D_\test^\rep\) happens in two steps. First, we randomly draw a vector \(ω\) of epistemic parameters (i.e. hyperparameters) from \(Ω\); for example \(ω = (\underbrace{-0.27 \mathrm{mV}}_{σ_o}, \underbrace{47 \mathrm{ms}}_{τ}, \underbrace{0.003 \mathrm{mV}}_{σ_i})\). Second, we use those parameters to generate the dataset \(\D_\test^\rep\), composed in this case of data points \((t, \tilde{V}^{\LP{}}(t))\). We can then evaluate the loss \(Q\) (given by Eq. 2.5) on those data points. Note that the loss does not depend on the epistemic parameters \(ω\) directly, but in general will involve parameters which are fitted to the simulated data. In short, the vector \(ω\) describes the parameters of a simulated experiment, while \(\D_\test^\rep\) describes a particular outcome of that experiment. In theory we could generate multiple datasets with the same vector \(ω\), but in practice it is more statistically efficient to draw a new experiment for each new dataset.
When choosing epistemic distributions, it is worth remembering that the goal of calibration is to empirically approximate a probability over experimental conditions. Thus choosing a distribution which can generate a large number of conditions – ideally an infinite number – will lead to better estimates. Here the use of continuous distributions for \(σ_o\), \(σ_i\) and \(τ\), and the fact that \(I_{\mathrm{ext}}\) is a continuous process, helps us achieve this goal. Also important is to generate data where the model selection problem is ambiguous, so that as to resolve calibration curves around \(\Bemd{} = 0.5\).
This calibration is an imperfect procedure, and how well it works depends on the quality of the candidate models and the choice of loss function. Here for example, the choice of a pointwise loss makes it sensitive to the timing of spikes; this tends to favour models which produce fewer spikes, since the penalty on a mis-timed spike is high. This is why in Fig. 2.5, in the \(\M_A \,\mathtt{vs}\, \M_D\) comparison, we see a floor on the values of \(\Bconf{AD}\). In short, some models have consistently lower loss even on random data, and so their risk – which is the expectation of their loss – is a priori lower. The bias we see in the \(\M_A \,\mathtt{vs}\, \M_B\) comparison is likely due to a similar effect. (In this case because \(\M_B\) is slightly better than \(\M_A\) on average.)