Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime

We derive general bounds on the probability that the empirical first-passage time $\overline{\tau}_n\equiv \sum_{i=1}^n\tau_i/n$ of a reversible ergodic Markov process inferred from a sample of $n$ independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct non-asymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Computer simulations often especially suffer from insufficient sampling, which leads to substantial errors in inferred rates [133][134][135][136] and, in the worst case, erroneous conclusions (see discussion in [118,137]).Even extensive computing resources may result in only a few independent estimates spread over many orders of magnitude, rendering uncertainty quantification challenging and not amenable to standard error analysis [121].
Constructing reliable confidence intervals is a fundamental challenge in statistical inference, and many prevalent methods only hold when n → ∞.The applicability of such asymptotic results in a finite-sample setting is, by FIG. 1. Deviations of empirical first-passage times from the true mean and model systems.(a) Schematic probability density of empirical first-passage time τ n inferred from a sample of n realizations of an ergodic reversible Markov process.The tail probability that the estimate τ n deviates from the true mean ⟨τ ⟩ by more or equal than t upwards P(τn ≥ ⟨τ ⟩ + t) or downwards P(τn ≤ ⟨τ ⟩ − t) is shown in green and blue, respectively.(b) Brownian molecular search process in a ddimensional domain (here d = 2) with outer radius R and target radius a. Discrete-state Markov jump models of protein folding for (c) a toy protein and (d) experimentally inferred model of calmodulin [129].Transitions between states obey detailed balance and absorbing targets are colored red.definition, problematic.In particular, Central-Limit-and bootstrapping-based methods [138] may easily underestimate the uncertainty for small n and fail to guarantee coverage of the confidence level [121,[139][140][141][142][143][144].
Conversely, Bayesian methods (e.g.[145]), despite not relying on asymptotic arguments, must be treated with care, as estimates and their uncertainties are sensitive to, dependent on, and potentially biased by, the specification of the prior distribution, especially in the small-sample setting [120,146] (see [128,134,[147][148][149] for kinetic inference).Moreover, prior-dependent uncertainty estimates seem to remain, even in the asymptotic limit, an elusive problem (see extended discussion in [150]).
There is thus a pressing need for understanding fluctuations of inferred empirical first-passage times, a rigorous error control, and reliable non-asymptotic error estima-arXiv:2301.08732v4[cond-mat.stat-mech]19 Oct 2023 tion in the small-sample regime.These are fundamental unsolved problems of statistical kinetics and are essential for the analysis of experimental and simulation data.
Here, we present general bounds on fluctuations of empirical first-passage times that allow a rigorous uncertainty quantification (e.g. using confidence intervals with guaranteed coverage probabilities for all sample sizes) under minimal assumptions.We prove non-asymptotic lower (L) and upper (U) bounds on the deviation probability P(τ n ≥ ⟨τ ⟩ + t) and P(τ n ≤ ⟨τ ⟩ − t) (see Fig. 1a), i.e., the probability that the empirical first-passage time inferred from a sample of n ≥ 1 realizations of an ergodic reversible Markov process, τ n , deviates from the true mean ⟨τ ⟩ by more than t in either direction, (1) the upper bounds U ± n (t) corresponding to so-called concentration inequalities [151].The most conservative version of the derived upper bounds is independent of any details about the underlying dynamics.We use the bounds U ± n (t) to quantify the uncertainty of the inferred sample mean τ n in a general setting and under minimal assumptions, for all n ≥ 1.We further derive general lower (M) and upper (M) bounds on the expected minimum τ − n = min i∈ [1,n] τ i and maximum τ + n = max i∈ [1,n] τ i of n realizations, i.e.
controlling the uncertainty of first-passage times even when multiple time-scales are involved, rendering ⟨τ ⟩ an a priori insufficient statistic.The validity and sharpness of bounds are demonstrated by means of spatially confined Brownian search processes in dimensions 1 and 3 (Fig. 1b), and discrete-state Markov jump models of protein folding for a toy protein [45,134,152,153] (Fig. 1c) and the experimentally inferred model of calmodulin [129] (Fig. 1d).We conclude with a discussion of the practical implications of the results and further research directions.Setup.-Weconsider time-homogeneous Markov processes x t on a continuous or discrete state-space Ω with generator L corresponding to a Markov rate-matrix or an effectively one-dimensional Fokker-Planck operator.Let the transition probability density to find x t at x at time t given that it evolved from x 0 be p t (x|x 0 ) ≡ e Lt δ x0 (x) where δ x0 (x) denotes the Dirac or Kronecker delta for continuous and discrete state-spaces, respectively.We assume the process to be ergodic lim t→∞ p t (x|x 0 ) = p eq (x), where p eq (x) ≡ e −φ(x) denotes the equilibrium probability density and φ(x) the generalized potential in units of thermal energy k B T [154].We assume that L obeys detailed balance [155] and is either (i) bounded, (ii) Ω is finite with reflecting boundary ∂Ω, or (iii) Ω is infinite but φ(x) sufficiently confining (see [156]).Each of the conditions (i)-(iii) ensures that the spectrum of L is discrete [157].
We are interested in the first-passage time to a target a when x t=0 is drawn from a density p 0 (x) and focus on the setting p 0 (x) = peq (x), since x 0 usually cannot be controlled experimentally (see e.g., [38,39,61,62,65,134,158,159]), and the tilde denotes that the absorbing state is excluded (see Appendix Ia for details).
For completeness we also provide results for general initial conditions p 0 (x) in Appendix Ib and [150] that require more precise conditions on φ(x).The probability density of τ for such processes has the generic form [44,45] with first-passage rates µ k > 0 and (not necessarily positive) spectral weights If x 0 is drawn from the equilibrium density, peq (x), we have ℘ a (t|p eq ) ≡ Ω\a ℘ a (t|x 0 )p eq (x 0 )dx 0 [160] which renders all weights non-negative, w k ≡ Ω\a w x0 k peq dx 0 ≥ 0 (see proof in [150]).We henceforth abbreviate S a (t|p eq ) ≡ S a (t).
To exemplify the need for uncertainty bounds in Eq. ( 1) we show in Fig. 2a-d that the probability that τ n − ⟨τ ⟩ lies within a desired range of say ± 10% of the longest firstpassage time scale µ −1 ) is low even for n ≈ 50 for all models in Fig. 1b-d.This inherent intrinsic noise-floor of the inferred observable τ n for any n is embodied in, and can be explicitly demonstrated by, the existence of lower bounds L ± n (see Appendix II).Cramér-Chernoff bounds.-Totackle this issue we now prove upper bounds using the Cramér-Chernoff approach.
Illustration of deviation bounds.-Theupper U ± n (t) and lower L ± n (t) bounds on P(±[τ n − ⟨τ ⟩] ≥ t) in Eqs.(7) and (15), respectively, are examplified in Fig. 2e-h (see black and red lines) for the model systems shown in Fig. 1b-d.Note that to illustrate all bounds we formally let t → −t for the left tails L − n (t) and U − n (t), such that their support is on [−⟨τ ⟩, ∞).Deviation probabilities are in turn expressed as P(sgn(t)δτ n ≥ |t|) where sgn(x) denotes the signum function and δτ n = τ n − ⟨τ ⟩.
To assess the quality of our bounds we scale probabilities P 1/n such that L ± n (t) and U ± n (t) collapse onto a master curve for all n (see inset Fig. 2f).Symbols denote empirical deviation probabilities obtained by sampling τ n for different n (see [150] for details), which approach the upper bound as n increases.For n = 1 right-tail deviations are close to L + 1 (t) even for w 1 ≤ 1 [168].As expected the model-free bound U ± n (t; 2) (yellow) holds universally but is generally more conservative.Notably, it remains remarkably good even for C ≳ 1.3 (Fig. 2e-g).
Uncertainty quantification.-Thebounds ( 7) and ( 11) provide the elusive systematic framework to rigorously quantify the uncertainty of the estimate τ n for any, and especially for small, sample sizes.In particular, they allow the construction of "with high probability" guarantees such as confidence intervals, which-unlike traditional confidence intervals in statistics-are not only asymptotically correct but hold for any n.Concentration-based guarantees do further not require specifying a prior belief as in the Bayesian context.Setting U ± n (t ± α± ; C) = α ± for chosen acceptable left-and right-tail error probabilities α ± (with α + + α − < 1), we get an implicit definition of the confidence interval [−t − α− , t + α+ ] at confidence level (or "coverage probability") 1 − (α − + α + ) in the form stating that with probability of at least 1 − α the sample mean τ n lies within [⟨τ ⟩ − t − α− , ⟨τ ⟩ + t + α+ ].Confidence intervals are closely related to, and can be used for, statistical significance tests [169,170], and beyond that provide quantitative bounds on statistical uncertainty.Two-sided central confidence intervals for δτ n as a function of n for a confidence level of α = 0.1 and models systems in Fig. 1b-d are shown (rescaled to a master scaling) in Fig. 3a (for a detailed discussion see [150]).
In particular, we may now also answer the practical question: How many realizations are required to achieve a desired accuracy with a specified probability?To ensure with probability of at least 1 − α that δτ n * ∈ [−t − α− , t + α+ ] one needs a minimal sample size n * defined via The number of samples n * required to guarantee that τ n * falls within a symmetric interval of length ∆t = 0.2/µ 1 , (i.e. ) with probability of at least 1 − α is shown in Fig. 3b for several values of C (intersections with the dashed line yield n * guaranteeing a coverage of at least 90%).Fig. 3c depicts the complementary symmetric interval ∆t covering the range of δτ n for a given n with probability of at least 90%.Note that hundreds to thousands of samples may be required to ensure an accuracy of ±0.1/µ 1 with a 90% confidence, which is seemingly not met in experiments [118][119][120][121][122][123][124].
Eqs. (12)(13) can be solved for t ± α± and n * , respectively, using standard root-finding methods (see [150]) and constitute our second main result.They allow for rigorous error control in kinetic inference in the small-n regime.Using Eq. (11) we can further construct system-independent but more conservative universal confidence intervals (see yellow line in Fig. 3b,c).Interestingly, even when C ≈ 1 the universal bound remains reasonably tight, only for C ≪ 1 differences become substantial.
Bounding extreme deviations.-Finally,we show that ⟨τ ⟩ controls the range of inferred τ i in any sample of n independent realizations, i.e., it sharply bounds the average minimum τ − n and maximum τ + n deviations from the mean m ± n ≡ τ ± n − ⟨τ ⟩.As our third main results we prove (see sketch of proof and saturation conditions in Appendix III and [150] for details) two-sided bounds where Remarkably, the tight bounds on µ 1 ⟨m ± n ⟩ are completely specified by the dimensionless parameter µ 1 ⟨τ ⟩.This motivates the inference of τ n in a general setting, even when ⟨τ ⟩ is a priori not representative, since via the bounds (14) (shown in Fig. 3d-e) ⟨τ ⟩ controls the range of estimates.The bounds (14) provide insight about the slowest and fastest out of n first-passage times, and are thus relevant in the "few encounter" limit [44,46,47,171].
Conclusion.-Leveraging spectral analysis and the framework of concentration inequalities we derived gen-eral upper and lower bounds on the probability that the empirical first-passage time τ n inferred from n independent realizations deviates from the true mean ⟨τ ⟩ by any given amount.Using these bounds we constructed nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus go beyond Central-Limitand bootstrapping-based methods which fail for small n.The results require minimal input and in particular do not require any prior belief as in the Bayesian approach that is known to be problematic and likely underestimates the uncertainty in sub-sampling settings.Our concentration-based results and bounds on extreme deviations allow for rigorous, model-free error control and reliable error estimation, which is essential for the analysis of experimental and simulation data.They may further be extended to non-ergodic and irreversible dynamics.
Acknowledgments.-Financial support from Studienstiftung des Deutschen Volkes (to R.B.) and the German Research Foundation (DFG) through the Emmy Noether Program GO 2762/1-2 (to A.G.) is gratefully acknowledged.
Appendix Ia: Equilibrium initial conditions.-Wemainly consider that the initial value x 0 of the firstpassage process is drawn from a "quasi" stationary equilibrium density peq (x).However, compared to standard relaxation processes with "true" stationary density p eq (x), an appropriate definition of peq (x) for absorption (i.e., first-passage) processes is more subtle as the absorbing target a has to be accounted for.In discrete state-space we have peq (x k̸ =a ) ≡ p eq (x k )/ k̸ =a p eq (x k ), i.e., the quasistationary density peq (x) is obtained via renormalization of p eq (x) by excluding the target a.On the contrary, in the continuous state-space setting, the absorbing state a has nominally zero measure such that trivially peq (x) = p eq (x) remains unchanged.
Appendix Ib: Arbitrary initial conditions.-Unlessthe state space Ω is both discrete and finite, an extension to arbitrary initial conditions, where x 0 is drawn from a general density p 0 (x), requires some additional assumptions about the generalized potential φ(x).In particular, when p 0 (x) ̸ = peq (x), φ(x) must be sufficiently confining to ensure a "nice" asymptotic growth of eigenvalues ν k of L of the relaxation dynamics [157], i.e., lim k→∞ ν k = bk β with β > 1/2 and 0 < b < ∞.The latter condition is automatically satisfied when Ω is finite, since regular Sturm-Liouville problems display Weyl asymptotics with β = 2 [172].The condition is in fact satisfied by most physically relevant processes with discrete spectra, including the (Sturm-Liouville irregular) Ornstein-Uhlenbeck or Rayleigh process [173] with β = 1.
Manifestations of general initial conditions p 0 (x 0 ) ̸ = peq (x 0 ) are then fully accounted for by simply replacing 2 .Additional details, such as a proof of convergence of the sum i w i 1 wi>0 /µ 2 i , are given in [150].
Appendix II: Lower bounds on deviation probabilities.-Thereexists a "noise floor" in the estimate τ n for any n.Since µ k ≤ µ k+1 and the weights are non-negative w k ≥ 0 [150] and normalized [44,45], the survival probability obeys the bound w 1 e −µ1t ≤ S a (t) ≤ e −µ1t .Using that ) n such that we arrive at the lower bounds We remark that analogous results are also obtained for upper bounds (see [150]) which, however, are weaker than those derived above in Eqs. ( 7) and ( 11) with the Cramér-Chernoff approach and concurrently require more information about the dynamics.
Appendix III: Sketch of proof of extreme-deviation bounds and their saturation.-Inessence, the proofs of "squeeze" bounds M ± n ≤ ⟨m ± n ⟩ ≤ M ± n rely on bounding the average value of the minimum τ − n ≡ min i∈ [1,n] τ i or maximum τ + n ≡ max i∈ [1,n] τ i out of n first-passage times.As a first step recall that τ i 's are i.i.d., such that we may write ⟨τ dt solely in terms of the survival probability S a (t), that is, This enables us to fully exploit the mathematical structure of the spectral decomposition of S a (t), allowing us to ultimately arrive at Eq. ( 14) by bounding the respective integrands in Eq. ( 16).
To briefly outline the key ideas, bounds M − n and M + n follow by recognizing (since ℘ a (t|p eq ) is monotonic in t) that ⟨τ + n ⟩ is maximal and ⟨τ − n ⟩ is minimal, respectively, whenever the contribution w 1 /µ 1 of the longest first-passage time-scale is maximal.For any fixed value of µ 1 ⟨τ ⟩ ∈ (0, 1], this condition is generally met in the presence of a spectral gap µ 1 /µ k → 0, ∀k > 1, for which one may utilize in Eq. ( 16) the ansatz The announced bounds are consequently obtained after some straightforward calculations and become saturated (i.e. the inequality becomes an equality) in systems with the above stated spectral gap [150].
The main idea behind proving the bound M + n relies on the inequality S a (t) ≥ w k+ e −µ k + t , were k + ≡ argmin k w k > 0 denotes the smallest k for which the corresponding weight w k is strictly positive.Saturation occurs when w k+ → 1.
The remaining bound M − n follows directly from upper bounding the convex function x n , ∀n ≥ 1, x ≥ 0, by means of Jensen's inequality and is saturated when w k → 1 for some k and for degenerate systems with identical first-passage eigenvalues µ i = µ, ∀i for some µ > 0.

S1. ISSUES WITH BAYESIAN INFERENCE IN THE SMALL-SAMPLE REGIME
Here we continue and extend the discussion on issues arising in Bayesian inference in the small-sample regime and conclude with difficulties that may even arise in the asymptotic large-sample limit.Bayesian methods (see e.g.[1]) do not rely on asymptotic arguments and are therefore often (in general erroneously [2,3]) believed to readily alleviate the small-sample problem.Bayesian estimates are highly sensitive to, dependent on, and potentially biased by, the specification of the prior distribution, especially in the small-sample setting [1,[4][5][6].Due to the prior dependence of estimates and their uncertainties, Bayesian methods must be treated with care when applied to small samples [7,8] (see [9][10][11][12][13] specifically for kinetic inference) and can perform worse than asymptotic frequentist methods [7].
Moreover, so-called "credible intervals"-the Bayesian analogue to confidence intervals-have a nominally different meaning, as they treat the estimated parameter as a random variable.Bayesian posterior intervals are similarly affected by limited sampling [14], i.e. the constructed uncertainty estimates and their quality are sensitive to the choice of prior probability [2,3] and may likely underestimate the true uncertainty and thus fail to provide trustworthy confidence intervals [11,15].

S2. SPECTRAL REPRESENTATION AND PREPARATORY LEMMAS
In this section we provide additional background on the spectral analysis of first-passage problems and some auxiliary Lemmas.In particular, we prove that for equilibrium initial conditions all spectral first-passage weights w k (p eq ) are non-negative and that for general initial conditions p 0 (x 0 ) the sum of positive spectral weights is always bounded.

A. Spectral representation and general results
First, we recall some general results using the spectral representation of first-passage processes (for more details see e.g.[37,38]).As stated in the Letter, we consider time-homogeneous Markov processes x t on a continuous or discrete state-space Ω with (forward) generator L corresponding to a Markov rate-matrix or an effectively one-dimensional Fokker-Planck operator.Let the transition probability density to find x t at x at time t given that it evolved from x 0 be p t (x|x 0 ) ≡ e Lt δ x0 (x) where δ x0 (x) denotes the Dirac or Kronecker delta for continuous and discrete state-spaces, respectively.We assume the process to be ergodic lim t→∞ p t (x|x 0 ) = p eq (x), where p eq (x) ≡ e −φ(x) denotes the equilibrium probability density and φ(x) the corresponding generalized potential in units of thermal energy k B T .We assume that L obeys detailed balance, such that it is self-adjoint in the left eigenspace with respect to a scalar product weighted by e −φ(x) and the operator e φ(x)/2 Le −φ(x)/2 is self-adjoint with respect to a flat measure.
We assume that L is either (i) bounded, (ii) Ω is finite with reflecting boundary ∂Ω, or that (iii) Ω is infinite but φ(x) is sufficiently confining (precisely, we require that φ(x) satisfies the Poincaré inequality, i.e. lim |x|→∞ (|∇φ . Each of the conditions (i)-(iii) ensures that the eigenvalue spectrum of L is discrete.The relaxation eigenvalue problem (for the inner product (•|•) defined with respect to a flat Lebesgue measure) reads x) , ν 0 = 0 and ν k≥1 > 0. The first-passage time to a target a for x t=0 drawn from a density p 0 (x 0 ) is defined as τ = inf t [ t |x t = a, p 0 (x 0 )].We will use ⟨•⟩ to denote an average over all first-passage paths {x t ′ } 0≤t ′ ≤τ , i.e. those that hit a only once.The first-passage time density to a, ℘ a (t|x 0 ) = ⟨δ(t − τ [{x t ′ }])⟩ to reach the absorbing target at x = a, starting initially from x 0 , has the general spectral representation where µ k is the k-th first-passage rate and w k (x 0 ) its corresponding first-passage weight [37,38].In similar fashion the survival probability is expressed as We note that in contrast to the relaxation eigenvalues ν k , the first-passage rates µ k = µ k (a) depend in the location of the absorbing target.Moreover, for any target location a the interlacing theorem holds [37,38] : where equality occurs iff w k (x 0 ) = 0, i.e. for a where Ψ R k (a) = 0. Laplace transforming the spectral expansion of the first-passage time density (S1)-according to f (s) ≡ e −st f (t) dt with f being a generic function locally integrable on t ∈ [0, ∞)-yields The first-passage weights are then obtained by using the residue theorem to invert the Laplace transformed renewal theorem [37][38][39] where ṗ(a, s|a) = ∂ s p(a, s|a) is taken at s = −µ k and {ν l , Ψ R l , Ψ L l } are the corresponding relaxation eigenmodes [37,38].The weights satisfy k≥1 w k (x 0 ) = 1 and the first non-zero weight is strictly positive w 1 (x 0 ) > 0.Moreover, the relaxation eigenvalues ν 0 = 0 and all ν k>0 ≥ 0 are real as a result of detailed balance.

B. Lemma 1: All weights are non-negative for equilibrium initial conditions
In the Letter we focus on equilibrium initial conditions, that is we assume that x 0 is drawn from the invariant measure, p eq (x 0 ), which in the particular case of diffusion processes is assumed to have a reflecting boundary at a (i.e.we focus on the one-sided first-passage process).We further introduce the non-negative modified spectral weights wk (x 0 ) ≡ w k (x 0 )θ(sgn[w k (x 0 )]) and now prove that for a normalized equilibrium probability density of initial conditions p 0 (x 0 ) that excludes the target-i.e.peq (x 0 ) ≡ p eq (x 0 )[1 − δ a (x 0 )]/(1|p eq (x 0 )[1 − δ a (x 0 )]) where δ a (x 0 ) is the Dirac measure (note that (1|p eq ) = 1)-all weights w k are rendered non-negative.We thus have wk (p eq ) = w k (p eq ) ≥ 0, ∀k.

C. Lemma 2: Sum of positive weights is bounded from above
For the sake of completeness we present supplementary results for general initial conditions p 0 (x 0 ).Recall from the Letter that we require some additional conditions on φ(x) or Ω in this more general setting.
Recall further that the m-th moment of τ is given by ⟨τ , where equality holds when p 0 = peq (since in this case all w k ≥ 0, i.e., wk (p eq ) = w k (p eq ) as discussed before).
To prove this consider w max ≡ max k≥k * wk (p 0 ) such that w max /µ 2+n k ≥ w k (p 0 )/µ 2+n k , ∀k.Let the smallest k for which the asymptotic scaling holds be k * then we may split the summation as k≥1 = Because the first term is nominally finite we only need to prove convergence of the second sum, which we do by means of the integral test.We define a function We now sum over all k ≥ k * to obtain, using where the last integral converges because 1−α(2+n) < 0, ∀n ≥ 0, which in turn proves convergence of k≥1 wk (p 0 )/µ 2 k .

S3. EXTREME VALUE BOUNDS AND COMPARISON WITH CRAM ÉR-CHERNOFF BOUNDS
In the Letter we derive lower bounds L ± n (t) on the deviation probability P(τ n − ⟨τ ⟩ ≥ t) and P(⟨τ⟩ − τ n ≥ t) by utilizing extremal events, i.e., we consider the maximal and minimal first-passage time in a sample of n ≥ 1 i.i.d.realizations.In this section we derive analogous upper bounds building on the same ideas.

A. Extreme value bounds for the sample mean τ n
Recall that for the reversible Markov dynamics considered the equilibrium survival probability S a (t|p eq ) ≡ S a (t) in its spectral representation (S2) obeys (S7) For the upper bound we use µ k ≤ µ k+1 and that k>0 w k = 1 are normalized, whereas the lower bound follows since w k ≥ 0, ∀k, as we consider equilibrium initial conditions throughout.Moreover, from extreme value theory it follows where we introduce τ + n ≡ max i∈ [1,n] τ i and τ − n ≡ min i∈ [1,n] τ i , respectively.Clearly, since τ − n ≤ τ n ≤ τ + n we may write Using Eq. (S8) in combination with Eq. (S7) we directly arrive at the lower bounds L ± n (t) (see Eq. ( 4) in the Letter) Introduced considerations are, however, not restricted to only lower bounds such that we can further leverage bounds on the equilibrium survival probability (S7) to analogously obtain corresponding upper bounds as As we will illustrate next, the upper bounds for the sample mean (S11) are much weaker than those derived with the Cramér-Chernoff approach (Eq.( 7) in the Letter) and require more information about the dynamics.

B. Comparison of Cramér-Chernoff vs Extreme value Bounds
In this section we directly compare the concentration-based upper bounds U ± n (t) (see Eq. ( 7) in the Letter) that are obtained with the Cramér-Chernoff approach, with the upper bounds (S11) which are based on extreme value considerations in analogy to the lower bounds L ± n (t).Similar to Fig. 2e-h of the Letter we now exemplify and compare both upper bounds in Fig. S1 for the model systems shown in Fig. 1b-d.
In Fig. S1a-d we equivalently express re-scaled deviation probabilities P 1/n (sgn(t)δτ n ≥ |t|) in a single panel, i.e., for the left tail we formally let t → −t such that t as shown now has support in [−⟨τ ⟩, ∞) and sgn(x) = ±1 for ±x > 0 and sgn(0) = 0 denotes the signum function.Empirical deviation probabilities (symbols) as a function of t are computed from statistics obtained by sampling τ n for different fixed n values.Extreme value lower bounds L ± n (t) (S10) for both tails are depicted in red.Here we now focus on comparing the upper bounds.Concentration inequalities U ± n (t; C) (Eq.( 7)) are again depicted as black lines whereas the corresponding extreme value upper bounds are represented as dashed/dotted lines where the respective coloring indicates the number of realizations n.Note, that the concentration bounds (and the lower bounds) collapse onto a single master curve due to the employed scaling P 1/n , whereas the extreme value upper bounds do not due to their different functional form (compare Eq. (S11)).Evidently, while for n = 1 the extreme value bounds remains close to the actual deviation probability, already for n = 3 they become considerably less tight and overshoot heavily for all considered models.Moreover, extreme value upper bounds become increasingly weak (even trivial at times) as n increases, therefore highlighting that Cramér-Chernoff-type bounds are vastly more suitable.
Motivated by the discussion above we next want to gain more quantitative insights for which sample sizes n the Cramér-Chernoff approach becomes more favorable.For this purpose we introduce a quality factor Q ∈ [0, ∞) that is informally defined as

Q ≡
Extreme value upper bound Cramér-Chernoff-type upper bound .(S12) A value Q > 1 therefore indicates that the Cramér-Chernoff bound is tighter and Q < 1 suggests that the extreme value bound should be favored, respectively.In Fig. S1e-h we illustrate the quality factor Q as a function of sample size n for different fixed dimensionless deviation values µ 1 t (star symbols in Fig. S1a-d).Remarkably for all model systems considered-which span a large range of possible C values-the Cramér-Chernoff approach is already superior even in the small-sample regime n ≲ 4.Moreover, we can further study the particular n * , for which one would reach Q = 1, as a function of some desired deviation µ 1 t relative to the longest time scale 1/µ 1 .Note, that again for the left tail we let t → −t (see discussion above).As depicted in Fig. S1i-l for our model systems, n * (blue) generally is found to be well below n = 8, i.e., even for most small sample sizes the derived Cramér-Chernoff-type bounds can be considered to be the better choice, especially when considering large µ 1 t (i.e.large deviations).Lastly, one could ask the question why the extreme value upper bound is so "weak" when n increases even just slightly.To answer this question we recall that-since we are interested in deviations of the sample mean τ n around ⟨τ ⟩-we bound the sample mean with the minimal and maximal first-passage time according to τ − n ≤ τ n ≤ τ + n which is further used, in combination with bounds on the survival probability (S7), to derive corresponding upper bounds (S11).Clearly, as n increases we expect this bound to become increasingly loose as by larger sample sizes we increase the chances of sampling rare first-passage times, i.e., maximal and minimal first-passage time that strongly deviate from the (sample) mean-this also explain why bounds (S10) and (S11) are only particularly tight for n = 1 as here τ − n = τ 1 = τ + n .In contrast, the Cramér-Chernoff method requires a much more delicate mathematical analysis involving bounds of the moment generating function.The Cramér-Chernoff-type bound has the additional advantage that it can be further used to universally bound deviation probabilities where no specific information about the underlying system is required (see Eq. ( 11) in the Letter).Moreover, even the version of Cramér-Chernoff bounds U ± n (t; C) that require input of one system-dependent constant C still require less information about the dynamics since extreme value upper bounds (S11) partly also require knowledge about the first-passage weight w 1 and ⟨τ ⟩ itself.

S4. COMPLETE PROOF OF CONCENTRATION INEQUALITIES AND THEIR ASYMPTOTICS
In this section we provide various additional details on the upper bounds U ± n (t; C) (Eq.( 7) of the Letter).In particular, we prove the required bounds on the cumulant generating function, compute their corresponding Cramér transform, and give further information about the large-sample limit n → ∞, as well as the model-free version of the bounds.Note that while the following proofs build on the mathematical structure encoded in the spectral decomposition, the derived bounds do not require knowledge of the full first-passage time spectrum {w i , µ i }.

A. Theorem 1: Cramér-Chernoff bound for the right tail τ ≥ ⟨τ ⟩
We begin with the right tail, i.e. upwards deviations such that τ ≥ ⟨τ ⟩, and start by proving a bound for the moment generating function of the deviation of the first-passage time τ from the mean ⟨τ ⟩.Using the spectral representation (S1) and the inequality x ≤ e x−1 , ∀x ∈ R, we find for all λ < µ k .Moreover, for |λ| < µ 1 we may further expand the sum k>0 w k 1−λ/µ k = m≥0 λ m k>0 w k /µ m k using the geometric series.Recall that the moments are given by ⟨τ m ⟩ = m! k>0 w k /µ m k , such that we obtain Since µ 1 ≤ µ k>1 and all first-passage weights w k are positive (due to equilibrium initial conditions) we find Introducing ψ δτ (λ) ≡ ln⟨e λδτ ⟩, with δτ = τ − ⟨τ ⟩ for the right tail, we immediately identify the upper bound which concludes the derivation of the upper expression in Eq. ( 6) of the Letter.Note that we further have introduced the dimensionless quantities t ≡ µ 1 t, C = µ 2 1 ⟨τ 2 ⟩, and λ = λ/µ 1 in the last step.In the case of general initial conditions p 0 (x 0 ) ̸ = peq (x 0 ) we must simply replace C → 2 i w i 1 wi>0 (µ 1 /µ i ) 2 (see Lemma 2).

C. Behavior of upper bounds U ± n (t) for large sample sizes
Here, we present some further remarks about the limit of large sample sizes.Asymptotically as n → ∞, U ± n (t) is substantial only for t/C ≪ 1.For the right tail bound h + (u) we immediately find that for u ≪ 1 we can Taylor expand ). Consequently we directly obtain h + (u) = u 2 /2 − O(u 3 ), i.e., the upper tail is sub-Gaussian for small deviations and will converge to a Gaussian as n → ∞.For the left tail we furthermore have arcosh(1 + x) = ln(1 + x + x(x + 2)) and thus lim x→∞ arcosh(1 + x) = ln(2x) − 1/(2x) 2 .As a result it follows that A lengthy but straightforward calculation subsequently reveals that lim u→0 Λ(u We therefore have that lim u→0 h − (u) = u 2 /2 − O(u 4 ), i.e., both tails are sub-Gaussian for t/C ≪ 1 with C ≡ µ 2 1 ⟨τ 2 ⟩.

D. Proof of bounds on C and model-free concentration inequalities
Notably, system details only enter the Cramér transforms (S17) and (S27) (and consequently upper bounds on the deviation probability due to Chernoff's inequality) in the form of a system-specific constant C ≡ µ 2  1 ⟨τ 2 ⟩.Note that here we only allow for equilibrium initial conditions.Recalling that the moments of the first-passage time τ are expressed as where we have used that w k are non-negative, normalized, and µ 1 ≤ µ k>1 .Consequently, by Eq. (S31), we immediately find that the system-constant itself is bounded 0 ≤ 2w 1 ≤ C ≤ 2. Note that analogous considerations can be used to more generally obtain 0 ≤ m!w 1 ≤ µ m 1 ⟨τ m ⟩ ≤ m! for the m-th moment, i.e., for m = 1 we find the chain of inequalities 0 ≤ w 1 ≤ µ 1 ⟨τ ⟩ ≤ 1.
The fact that C ∈ (0, 2] can now be further leveraged to arrive at the model-free bounds (Eq.( 11) in the Letter) which require no information about the underlying system.Recall the upper bounds of the cumulant generating function ϕ δτ ( λ; C) and their corresponding Cramér transform ϕ * δτ ( t; C), i.e., and hence U ± n (t; C) ≤ U ± n (t; 2) which completes the derivation of Eq. ( 11) in the Letter.

S5. MODEL SYSTEMS AND DETAILS ON NUMERICAL METHODS
In the Letter we exemplify our results by considering a Brownian molecular search process in dimensions d = 1 and d = 3, as well as discrete-state Markov-jump models of protein folding for a 8-state toy protein and the experimentally inferred model of calmodulin (compare Fig. 1b-d).In this section we present further details on the model systems and their numerical treatment.

A. Continuous-time discrete-state Markov jump process
As illustrative discrete-state continuous-time Markov-jump models of protein folding we consider a simple 8-state toy protein [11,38] and further use the experimentally inferred folding network of the cellular calcium sensor protein calmodulin [42].Since we consider equilibrium initial conditions, proteins start from an initial state drawn from the equilibrium density peq (x 0 )-note that the tilde denotes that the absorbing target is excluded-from which they search the native state a (here a = (1, 1, 1) for the 8-state model and a = F 1234 for calmodulin; cf.Fig. 1b-d).Arrows in the networks denote possible transitions, e.g. a transition from state i to state j that occurs with the corresponding rate L ji .We consider reversible dynamics, i.e., the resulting transition matrix L of the relaxation process satisfies detailed balance p eq,j /p eq,i = L ji /L ij = exp(F i − F j ) and transitions rates are connected to the free energy of the states F i [43].
We recall that the first-passage time density ℘ a (t) can be evaluated by using the spectral representation (S1).To this end we set up the modified transition matrix, adopting in this section the Dirac bra-ket notation, La = L − |a⟩⟨a| where |a⟩ ≡ (0, . . ., 0, 1, 0, . ..) ⊺ defines a vector with all entries zero expect at the a-th position of the absorbing state where it equals one.This effectively removes all transitions that correspond to jumps leaving the absorbing state a. Next, we carry out an eigendecomposition of La and determine the eigenvalues µ k , right eigenvectors |ϕ R ⟩, and left eigenvectors ⟨ϕ L |.We subsequently use obtained eigenmodes to compute the first-passage weights [37,38]), and recall that µ k and w k determine the moments according to ⟨τ m ⟩ = m! k>0 w k /µ m k .Corresponding relevant parameters of the Markov jump models are listed in Tab.I. Next we give further details on how matching transition rates are constructed.For the 8-state toy protein model we randomly generate a free energy level F i for each state i ∈ {1, 2, 3, 4, 5, 6, 7, 8} with F i uniformly distributed within the interval 0 ≤ F i ≤ 10.Transition rates that satisfy detailed balance are then obtained using the ansatz and where ∆F i ≡ F i − F j and thus ln(L ji /L ij ) = ∆F i = F i − F j .Obtained individual transition rates are listed in Tab.II.In the experimental setup a constant external force f , a so-called pretension, is applied to the calmodulin protein via optical tweezers [42].Folding and unfolding processes are observed at different pretensions ranging from 6 pN to 13 pN and corresponding force-dependent transition rates k i→j (f ) = L ji (f ) between two conformational states i and j are measured.Note that i, j ∈ {Unfold, F 12 , F 123 , F 23 , F 34 , F 1234 } and we further map states according to Unfold ↔ 1, F 12 ↔ 2, F 123 ↔ 3, F 23 ↔ 4, F 34 ↔ 5, and F 1234 ↔ 6 for convenience.For our purposes we choose, without loss of generality, a pretension of f = 9 pN and obtain the corresponding measured transitions rates from Fig. S8 in the Supplementary Material of [42].Clearly, experimental transitions rates are accompanied with measurement uncertainties which is reflected in slight "deviations" from a mathematically precise definition of detailed balance.To mitigate this issue, and to ensure that transition rates precisely obey detailed balance k i→j p eq,i = k j→i p eq,j , we further have to slightly adjust the rates.
First, we compute the invariant density p eq from the experimental rates and obtain a corresponding free energy level F i = − ln(p eq,i ).Next, we use the ansatz (S33), i.e., L ji = A i exp(∆F i /2) and L ij = A i exp(−∆F i /2) where we introduce a constant A i .Finally, A i 's are chosen such that resulting transition rates fall within experimental error bars in Ref. [42].Obtained transition rates are listed in Table III.We also test our theory for Markov processes on a continuous state-space.More precisely, we consider the spatially confined diffusive search of a Brownian particle in a d-dimensional unit sphere with a reflecting boundary at R = 1 and a perfectly absorbing spherical target of radius 0 < a < 1, here a = 0.1, in the center (compare Fig. 1b).The closest distance of the particle to the surface of the absorbing sphere at time t is a confined Bessel process (see e.g.[38,44,45]) which time evolution obeys the Itô equation where dW t is the increment of a Wiener process (i.e.Gaussian white noise) with ⟨dW t ⟩ = 0 and ⟨dW t dW t ′ ⟩ = δ(t − t ′ )dt, and we have set, without loss of generality, D = 1.The general case with any 0 < D < ∞ and a sphere of radius R is covered by expressing time in units of R 2 /D.For d = 1 Eq.(S34) reduces to a 1 dimensional Brownian motion which has the equilibrium first-passage weights and matching first-passage eigenvalues are obtained as Moreover, for d = 3 the first-passage time probability density of the Bessel process can be evaluated exactly and has the equilibrium weights with the first-passage eigenvalues µ k being the solutions of the transcendental equation can be solved analytically using Newton's series [38].Relevant parameters for the spatially confined Brownian search process with a = 0.1 are listed in Tab.IV.Note that the mean for d = 3 and arbitrary x 0 can be obtained directly [46] as ⟨τ a (x 0 )⟩ = [1/a − 1/x + a 2 /2 − x 2 /2]/3 which integrated over all starting positions, i.e. equilibrium initial conditions, gives the global mean first-passage time . (S37) Finally we provide some further details on the sampling method used to obtain the statistics of (i) the first-passage time τ , (ii) the sample-mean τ n ≡ i τ i /n, and (iii) the expected maximal and minimal deviation from the mean ⟨τ ± n − ⟨τ ⟩⟩ for a sample with a fixed number n of independent realizations for all considered models.We recall that after determining the first-passage eigenvalues µ k and first-passage weights w k , the first-passage time density ℘ a (t) (S1) and survival probability S a (t) (S2) are fully characterized.To now sample the random variable τ , i.e. individual realizations of the first-passage process, we employ the so-called inversion sampling method [47].This method allows us to generate independent samples of τ from ℘ a (t) given its cumulative distribution function (CDF) which is directly related to the survival probability according to 1 − S a (t).Note that for discrete-state dynamics the number of states M is finite, i.e. k = 1, . . ., M , and therefore Eq. (S2) (and hence the CDF) is a finite sum.In contrast, for continuous-state dynamics we formally have M = ∞, meaning that sums are here not finite.For the following numerical evaluation of the spatially confined Brownian search process we therefore truncate the sum after M = 1000 terms.The first-passage time densities ℘ a (t) obtained via inversion sampling (symbols) for all considered models are shown in Fig. S4a-d and corroborated by the corresponding analytical result (S1) (dashed black line).
For Fig. 2a-d in the Letter empirical probabilities that τ n − ⟨τ ⟩ lies within a desired range of ± 10% of the longest first-passage time scale µ −1 ), are computed using statistics of the sample mean τ n by fixing n, i.e., the number of individual realizations the average is taken over.In particular, we have n ∈ {1, 2, 3, 5, 10, 20, 30, 40, 50, 75, 100, 150, 200, 300, 400, 500}.Subsequently, for each individual fixed n the sample mean τ n itself is sampled a total of N = 10 6 times.That is, we first draw n first-passage times τ , compute τ n by averaging over the drawn n realizations, and finally repeat this step N = 10 6 times to obtain statistics of τ n for all n values introduced above.Probability densities of the sample mean are shown in Fig. S4e-h for n ∈ {3, 5, 10, 20} and all model systems.Corresponding true mean first-passage times ⟨τ ⟩ are highlighted in grey.
In Fig. 2e-h of the Letter the probabilities to deviate more than t in either direction, P(±[τ n − ⟨τ ⟩] ≥ t), are computed from analogous statistics of the sample mean τ n .Since we also consider empirical probabilities for rare events with large deviations (i.e.large µ 1 t) we however require substantially more statistics of τ n .To this end we now have N = 10 7 for n ∈ {1, 3} and N = 10 11 for n ∈ {5, 10, 20}.In addition it should be further noted that we re-scale obtained probabilities according to P 1/n .To compute an empirical deviation probability where e.g.P 1/20 = 0.1 one would be thus required to sample rare events that occur with a probability of ≃ 10 −20 .
In Fig. 3a of the Letter each data point corresponds to the relative error µ 1 (τ n − ⟨τ ⟩) (note that µ 1 and ⟨τ ⟩ are different for each model) where the sample mean τ n is again obtained by first fixing n and then sampling n first-passage times τ according to the inversion sampling method and subsequently taking the average.
In Fig. 3d-e in the Letter and Figs.S6 and S7 below we show the expected deviation of the maximum τ + n and minimum τ − n in a sample of n realizations from the mean first-passage time ⟨τ ⟩ as a function of µ 1 ⟨τ ⟩.To sample values over the entire range of possible values µ 1 ⟨τ ⟩ ∈ [0, 1] we generate 10000 random folding landscapes for the Markov jump process network of Calmodulin and the 8-state toy protein model, respectively.Each data point shown corresponds to one particular realization of the Markov jump process for the given network topology where transitions rates obeying detailed balance are constructed using the methods described in Sec.S5 A. The free energy levels F i are now uniformly distributed within the interval 0 ≤ F i ≤ 20.For each random realization of a folding landscape (i.e., data point) we draw a fixed number of independent first-passage times n ∈ {3, 5, 10, 20} using inversion sampling to determine the minimum τ − n and maximum τ + n , receptively.To compute the associated averages ⟨τ ± n ⟩ we acquire minimal and maximal τ ± n 's for a total of 10 7 independent draws of n realizations for each of the 10000 respective random folding landscapes.The same procedure is applied to the confined diffusion process where for d = 3 we vary the target radius a ∈ [0.05, 0.95].Note that in the case of d = 1 we have no parameter to vary, i.e., there is only one data point.Confidence intervals are practically useful as they answer questions such as e.g.: How many realizations are required to achieve a desired accuracy with a specified probability?Or: For a given number of realizations a desired accuracy is achieved with at least what probability?
In the case of symmetric confidence intervals t + α+ = t − α− (see Fig. S5 red lines) the interval endpoints are implicitly defined via the last line of Eq. (S38) which is easily solved using standard root-finding procedures like the bi-section method [49].The same holds true for other interval choices, however, when specifying the error probabilities α ± directly-as done for e.g.two-sided central intervals (α ± = α/2) or one-sided intervals-it suffices to solve Eq. (S39) with the respective α ± .Hereby, the lower confidence limit t − α− is again easily obtained using standard root-finding methods.Notably, the upper confidence limit t + α+ can now be solved analytically.To show this we consider U + n (t + α+ ; C) = α + , i.e., we identify the t + α+ that solves The roots are identified as and we identify t + α+ = t 2 as the relevant solution.Having obtained an explicit expression for t + α+ further allows us to re-insert it into the left-hand side of Eq. (S39), i.e., we find that with a probability of at least 1 The required number of realizations n * to ensure with a probability of at least 1 − α that δτ n is found within some interval [−t − α− , t + α+ ] (e.g.symmetric interval in Fig. 3b) is analogous identified according to Eq. ( 13) in the Letter which once again is readily solved via e.g. the bisection method.Moreover, in the case of one-sided intervals one immediately finds the corresponding analytical expression where n * denotes the required number to ensure that ±δτ n ≤ t with at least 1 − α ± .

S7. CONTROLLING UNCERTAINTY OF FIRST-PASSAGE TIMES WHEN ⟨τ ⟩ IS AN INSUFFICIENT STATISTIC
Note that whenever µ 1 ⟨τ ⟩ substantially differs from 1, or essentially equivalently C becomes substantially smaller than 2, many time-scales enter the problem and ℘(t) tends to largely differ from an exponential.In turn, ⟨τ ⟩ a priori becomes an insufficient statistic.However, as we prove below the expected range of inferred τ in general, and thus even in cases when ⟨τ ⟩ is an insufficient statistic, turns out to be sharply bounded from above and below by functions of µ 1 ⟨τ ⟩ only.We therefore ultimately seek for bounds on µ 1 (⟨τ ± n − ⟨τ ⟩) as a function of µ 1 ⟨τ ⟩.Therefore, even when ⟨τ ⟩ is an a priori insufficient statistic it is useful and insightful to infer the empirical first-passage time τ n and control its uncertainty, as it in turn sets sharp upper bounds on the range of inferred first passage times.

S8. PROOF OF BOUNDS ON EXTREME DEVIATIONS FROM ⟨τ ⟩
In this section we prove lower (M ± n ) and upper (M ± n ) bounds on the expected deviation of the maximum and minimum from the mean in a sample of n i.i.d.realizations of τ .That is, we consider the average ⟨m ± n ⟩ of the random quantity m ± n ≡ τ ± n − ⟨τ ⟩ where τ + n ≡ max i∈ [1,n] τ i and τ − n ≡ min i∈ [1,n] τ i , respectively.Note that here from we drop the subscript a in the survival probability S a (t) → S(t) for ease of notation.We now prove the upper bound on the expected maximal deviation from the mean in sample of n realizations and we therefore examine ⟨m + n ⟩ at any given fixed value of µ 1 ⟨τ ⟩.Because the first-passage density at equilibrium ℘(t) is monotonically decaying with t and normalized (i.e.all w k ≥ 0 and k w k = 1; see Eq. (S6)), the maximum ⟨τ + n ⟩ will obviously be maximized at fixed µ 1 ⟨τ ⟩ when w 1 -the weight of the longest first-passage time-scale-is maximal (see Eq. (S48)).
As a first step we therefore consider how the constraint µ 1 ⟨τ ⟩ = const.(note that µ 1 ⟨τ ⟩ ∈ (0, 1]) affects the first-passage spectrum {w k , µ k }, i.e.where the first term embodies the spectral gap and is defined as a distribution.With the above ansatz ⟨τ + n ⟩ is maximal (and ⟨τ − n ⟩ is minimal; see Sec.S8 D for the corresponding lower bound) under the constraint µ 1 ⟨τ ⟩ = const..We thus continue with Eq. (S53) to prove the upper bound on ⟨τ + n ⟩.Note that in the following calculation we omit lim ε→0 for the moment and later explicitly state when we take the limit.First we may write, which completes the proof of the upper bound M + n in Eq. ( 14) in the Letter.We remark that announced bounds are saturated (i.e., the inequality becomes an equality) in systems with the above stated spectral gap.The validity and sharpness of the upper bound (solid lines) is illustrated for different model systems and several values of n in Fig. S6.

FIG. S1 .
FIG. S1.Comparison between Cramér-Chernoff-type upper bounds U ± n (t; C) and extreme value upper bounds for a spatially confined Brownian search process in dimensions (a,e,i) d = 1 and (b,f,j) d = 3, and discrete-state Markov jump processes for (c,d,k) the inferred model of calmodulin and (d,h,l) a 8-state toy protein.(a-d) Scaled probabilities P 1/n (sgn(t)δτ n ≥ |t|) that the sample mean τ n inferred from n ≥ 1 realizations deviations from ⟨τ ⟩ by more than t in either direction.Right tail areas are shown for t > 0 and left for t < 0, respectively.Cramér-Chernoff upper bounds U ± n (t; C) are displayed as black and extreme value upper bounds as dashed lines, respectively.Corresponding lower bounds L ± n (t) are depicted as red lines and symbols denote scaled empirical deviation probabilities obtained from the statistics of τ n for different n. (e-h) Quality factor Q as a function of n for different fixed relative deviations µ1t (see star symbols (a-d)).(i-l) Sample size n * (blue) for which both upper bounds are equal, i.e., Q = 1, as a function of re-scaled deviations.
FIG. 2. Deviation probabilities and corresponding bounds for a spatially confined Brownian search process in (a,e) d = 1 and (b,f) d = 3 dimensions, and Markov-jump models of protein folding for (c,g) the experimentally inferred model of calmodulin and (d,h) the toy protein.(a-d) Probability that δτ n = τ n − ⟨τ ⟩ lies within a range of ±10% of the longest time-scale 1/µ1, bounds are depicted as red and black lines, respectively, and the model-free upper bound U ± n (t; 2) as the dashed yellow line.Symbols denote corresponding scaled empirical deviation probabilities as a function of t and are sampled for different n.

TABLE I .
Parameters for the Markov jump models for the 8-state toy protein and the inferred model of calmodulin.Listed are values for the first-passage eigenvalues µ k , first-passage weights w k , and the first ⟨τ ⟩ and second moment ⟨τ 2 ⟩.

TABLE II .
Transition rates for the 8-state toy protein model obtained via the ansatz described in the main text.transition rate ki→j transition rate ki→j transition rate ki→j transition rate ki→j

TABLE III .
Transition rates of the Markov jump model for the calmodulin protein.Rates are extracted from the Supplemental Material of Ref.[42]and modified such that they obey detailed balance precisely according to the maintext.transition rate ki→j transition rate ki→j transition rate ki→j

TABLE IV .
Parameters for the spatially confined Brownian molecular search process in dimensions 1 and 3. Listed are values for the first 5 first-passage eigenvalues µ k , first-passage weights w k , and the first ⟨τ ⟩ and second moment ⟨τ 2 ⟩, respectively a .For the numerical evaluation of ⟨τ ⟩ and ⟨τ 2 ⟩ as listed we truncate the sum after M = 1000 terms. a