parameter estimation processes. Our findings therefore underscore the critical
need for careful consideration of the effects of statistical noise in
simulation-based forecasts. If overlooked, these effects can introduce
significant biases and result in overly optimistic forecasts through the
artificial tightening of parameter constraints, thereby exaggerating the
precision of the forecast. The methodology developed in this work enables
researchers to quantify the number of simulations required to effectively
overcome this noise contamination inherent in numerical derivatives. This
quantification is crucial, as the requisite $N_{sims}$ varies significantly
depending on the specific observables and cosmological parameters under study
both in the target and non-target sets. By providing a framework to estimate
these needs, our approach helps ensure that simulation resources are allocated
efficiently, maximizing the fidelity of the resulting cosmological inferences,
and ensuring that Fisher forecasts remain reliable and reflective of the true
uncertainties in cosmological models.

Forecasts using simulations are becoming a common method employed to evaluate
survey strategies and cosmological capabilities, making our findings and our
methods to mitigate noise-induced biases both timely and pertinent for key
cosmological surveys such as DESI, Euclid, LSST, Roman Space Telescope, and
SPHEREx.

Our work was motivated by the effects of selecting between the various
neutrino mass differentiation schemes, (26) - (31), and the findings imply
parallel considerations for the measurement of, and constraints on, other
parameters for which higher order forward differentiation schemes are commonly
used, such as the non-Gaussianity parameter $f_{\mathrm{NL}}$ and the
inflationary scalar-to-tensor ratio $r$. Our results are completely general,
however, and the sensitivity to both the order of the differentiation scheme
and step size means that careful consideration of derivative noise should be
part of simulation-based Fisher forecast analyses for any cosmological
parameters.

Finally, the use of large scale cosmological simulation suites, designed to
model realistic galaxy samples and survey characteristics for forthcoming
observational campaigns, extends beyond Fisher forecasts. These simulations
also underpin emulators used in Monte Carlo Markov Chain (MCMC) analyses,
comparing simulated theories to real data, and in training machine learning
tools for model inference. Characterizing the potential effects of statistical
noise within these simulation-derived tools will also be crucial to exploring
their effectiveness and potential limitations given the extensive
computational resources which underlie their foundations, and should be the
subject of future work. This consideration will enhance the robustness of both
predictive models and the empirical analyses they support.

###### Acknowledgements.

The work of CW and RB is supported by NSF grant AST-2206088, NASA ATP grant
80NSSC18K0695, and NASA ROSES grant 12-EUCLID12-0004. CW would like to thank
Tom Loredo for useful discussions related to analytic marginalization and the
profile likelihood, and John Drew Wilson for useful discussions throughout,
especially related to Appendix B.

## APPENDIX

In Appendix A we consider two physically relevant limits of the derivative
noise distribution. In Appendix B we outline an analytic approach for
estimating the true probability distribution $p^{True}$ and assessing the
quality of agreement of our different approximation schemes, (32) and (46).

##  Appendix A Useful Physical Limits for $\mathscr{F}^{Noise}_{\mu\nu}$

In this appendix, we explore two distinct physical limits relevant for
calculating the probability distribution,
$p\left(\mathscr{F}^{Noise}_{\mu\nu}\right)$, of the statistical noise in the
Fisher matrix, $\mathscr{F}^{Noise}_{\mu\nu}$, for target parameters
$\theta^{\mu}$ and $\theta^{\nu}$. This noise term arises from the process of
calculating finite difference derivatives of observables with respect to the
forecast parameters from a finite number of simulations, and characterizes the
difference between the noise influenced $\overline{\mathscr{F}}_{\mu\nu}$ and
the noise free $\mathscr{F}_{\mu\nu}$. We specifically focus on the
expectation and variance as the key moments for this noise distribution. For
all expressions, we will include the general case for arbitrary parameters
$\theta^{\mu}$ and $\theta^{\nu}$ as well as the diagonal case of the most
relevance to the main text with explicitly
$\theta^{\mu}=\theta^{\nu}=M_{\nu}$.

We first provide some integral equations from the main text that provide
relevant context without having to look back through the paper. Defining the
ideal, noise-free derivative of the observable $\mathcal{O}^{\ i}$ with
respect to parameter $\theta^{A}$, as $d\mathcal{O}^{\
i}_{A}\equiv\frac{\partial\mathcal{O}^{\ i}}{\partial\theta^{A}}$, we can
express the derivative obtained from simulations,
$d\overline{\mathcal{O}}{}^{\ i}_{A}$, in terms of $d\mathcal{O}^{\ i}_{A}$
(eq.(35) in the main text) as

$d\overline{\mathcal{O}}{}^{\ i}_{A}=d\mathcal{O}^{\ i}_{A}+u^{i}_{A}.$ (58)

Here, $u^{i}_{A}$ is a random instance of derivative noise drawn from a
Gaussian distribution with zero mean and variance based on the covariance of
the mean (eq.(34) in the main text),

$\displaystyle\Sigma^{ij}_{AB}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{sims}^{2}}\sum_{n=1}^{N_{sims}}\left(\frac{\partial\mathcal{O}^{\
i}_{n}}{\partial\theta^{A}}-d\overline{\mathcal{O}}{}^{\
i}_{A}\right)\left(\frac{\partial\mathcal{O}^{\
j}_{n}}{\partial\theta^{B}}-d\overline{\mathcal{O}}{}^{\ j}_{B}\right).$ (59)

The contribution from the derivative noise to the marginalized Fisher matrix
for the target parameters can then be written as (eq. (39) in the main text),

$\displaystyle\mathscr{F}^{Noise}_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\overline{\mathscr{F}}_{\mu\nu}-\mathscr{F}_{\mu\nu}$ (60)
$\displaystyle=$ $\displaystyle U_{\mu\nu}+F_{\mu
a}\left(f^{-1}\right)^{ab}F_{b\nu}$ $\displaystyle-\left({F}_{\mu a}+U_{\mu
a}\right)\left(\left(({f}+U)^{-1}\right)^{ab}\right)\left({F}_{b\nu}+U_{b\nu}\right).\
\ \ $

where $\\{\mu,\nu\\}$ indicate target parameters and Latin indices from the
set $\\{a,b,c,d,e,g\\}$, here and below, are used for the set of non-target
parameters which are marginalized over. Latin indices from the set
$\\{i,j,k,l\\}$ are used to indicate observable indices. We emphasize that
indices are only summed on when a matching upper and lower index both appear.
Otherwise, matching pairs of lower indices are not summed over, and simply
function as labels.

In the diagonal case, the mean and variance of the noise contribution to the
marginalized fisher, $\mathscr{F}^{Noise}_{\mu\mu}$, for a specific target
parameter, $\theta^{\mu}$, can then be used to estimate the uncertainty on the
true cosmological constraint, $\sigma_{\mu}^{-2}$, on $\theta^{\mu}$ based on
that measured from the simulations, $\bar{\sigma}_{\mu}^{-2}$, (eq. (41) in
the main text),

$\sigma_{\mu}^{-2}=\left(\bar{\sigma}_{\mu}^{-2}-\left<\mathscr{F}^{Noise}_{\mu\mu}\right>\right)\pm\sqrt{\mathrm{Var\left(\mathscr{F}^{Noise}_{\mu\mu}\right)}}.$
(61)

### Limit $\\#1$: Isolated Noise

We first consider a simple but highly applicable approximation, the isolated
noise limit, which is particularly relevant when $D3$ or $D2_{a}$ is used for
$\theta^{\mu}=M_{\nu}$ differentiation.

In this limit, we assume that noise predominantly affects only the target
parameter derivatives (denoted with Greek indices), meaning for the non-target
parameters, $u^{i}_{a}=0$. This greatly simplifies (60), so that for general
target parameters $\theta^{\mu}$ and $\theta^{\nu}$, we have

$\left<\mathscr{F}^{Noise}_{\mu\nu}\right>\big{|}_{u^{i}_{a}\rightarrow
0}=\Sigma^{ij}_{\mu\nu}C^{-1}_{ij}-d{\mathcal{O}}^{\
i}_{a}C^{-1}_{ij}\Sigma^{jk}_{\mu\nu}C^{-1}_{kl}d{\mathcal{O}}^{l}_{b}({f^{-1}})^{ab}$
(62)

and in the diagonal case of $\theta^{\mu}=\theta^{\nu}=M_{\nu}$, we have

$\left<\mathscr{F}^{Noise}_{M_{\nu}M_{\nu}}\right>\big{|}_{u^{i}_{a}\rightarrow
0}=\Sigma^{ij}_{M_{\nu}M_{\nu}}C^{-1}_{ij}-d{\mathcal{O}}^{\
i}_{a}C^{-1}_{ij}\Sigma^{jk}_{M_{\nu}M_{\nu}}C^{-1}_{kl}d{\mathcal{O}}^{l}_{b}({f^{-1}})^{ab}.$
(63)

where $\Sigma^{ij}_{\mu\nu}$ is defined in (59).

The isolated noise limit also makes calculating
$\mathrm{Var}\left(\mathscr{F}^{Noise}_{\mu\nu}\right)$ analytically
tractable. We start by defining the auxiliary variable:

$Q_{ij}=\big{(}C^{-1}_{ij}-C^{-1}_{ik}d\mathcal{O}^{k}_{a}({f^{-1}})^{ab}d\mathcal{O}^{l}_{b}C^{-1}_{lj}\big{)}$
(64)

so that in the isolated noise limit, for general parameters $\theta^{\mu}$ and
$\theta^{\nu}$ we may compactly write

$\begin{split}\mathrm{Var}\left(\mathscr{F}^{Noise}_{\mu\nu}\right)\big{|}_{u^{i}_{a}\rightarrow
0}&=\left(d\mathcal{O}^{i}_{\mu}d\mathcal{O}^{j}_{\mu}+\frac{1}{2}\Sigma^{ij}_{\mu\mu}\right)\left(Q_{ik}\Sigma^{kl}_{\nu\nu}Q_{lj}\right)\\\
&+\left(2d\mathcal{O}^{i}_{\mu}d\mathcal{O}^{j}_{\nu}+\Sigma^{ij}_{\mu\nu}\right)\left(Q_{ik}\Sigma^{kl}_{\nu\mu}Q_{lj}\right)\\\
&+\left(d\mathcal{O}^{i}_{\nu}d\mathcal{O}^{j}_{\nu}+\frac{1}{2}\Sigma^{ij}_{\nu\nu}\right)\left(Q_{ik}\Sigma^{kl}_{\mu\mu}Q_{lj}\right).\end{split}$
(65)

For the diagonal case with $\theta^{\mu}=\theta^{\nu}=M_{\nu}$ this gives
simply

$\begin{split}\mathrm{Var}\left(\mathscr{F}^{Noise}_{M_{\nu}M_{\nu}}\right)\big{|}_{u^{i}_{a}\rightarrow
0}&=\left(4d\mathcal{O}^{i}_{M_{\nu}}d\mathcal{O}^{j}_{M_{\nu}}+2\Sigma^{ij}_{M_{\nu}M_{\nu}}\right)\left(Q_{ik}\Sigma^{kl}_{M_{\nu}M_{\nu}}Q_{lj}\right).\end{split}$
(66)

Together, these expressions for the mean and variance constitute the isolated
noise Gaussian limit for the general case of $\mathscr{F}^{Noise}_{\mu\nu}$
through equations (62) and (65), and for our specific case study of
$\mathscr{F}^{Noise}_{M_{\nu}M_{\nu}}$ through equations (63) and (66) and are
frequently referred to in the main text due to the relative simplicity of the
above equations.

### Limit $\\#2$: Large $N_{sims}$

As discussed in the main text, there are some scenarios in which the isolated
noise scenario does not serve as a good approximation, for example when using
the $D1_{a}$ derivatives for $M_{\nu}$. In this case, we can consider an
alternative, less simple but still tractable, approximation which provides an
analytic estimate for moments of $\mathscr{F}^{Noise}_{\mu\nu}$ in the limit
of large $N_{sims}$.

In this limit, we assume that all of the $u^{i}_{A}$ terms are small and can
thus be treated perturbatively, which makes calculating the expectation value
of $\mathscr{F}^{Noise}_{\mu\nu}$ analytically tractable. Since
$\Sigma^{ij}_{AB}\rightarrow 0$ as $N_{sims}\rightarrow\infty$, we may treat
$\Sigma^{ij}_{AB}$ as our small parameter when series expanding (60). Notably,
expanding in $\Sigma^{ij}_{AB}$ is equivalent to expanding in $1/N_{sims}$
since all entries of $\Sigma^{ij}_{AB}$ explicitly go as $1/N_{sims}$ as
$N_{sims}\rightarrow\infty$, providing a direct link between the number of
simulations and the perturbation expansion. For general target parameters
$\theta^{\mu}$ and $\theta^{\nu}$, we have to lowest order in $\Sigma$, that:

$\displaystyle\left<\mathscr{F}^{Noise}_{\mu\nu}\right>_{\Sigma^{1}}$
$\displaystyle=\Sigma^{ij}_{\mu\nu}C^{-1}_{ij}-{F}_{\mu
a}({f^{-1}})^{ab}\Sigma^{ij}_{b\nu}C^{-1}_{ij}$
$\displaystyle-\Sigma^{ij}_{\mu a}C^{-1}_{ij}({f^{-1}})^{ab}{F}_{b\nu}$
$\displaystyle-d{\mathcal{O}}^{i}_{a}C^{-1}_{ij}\Sigma^{jk}_{\mu
b}C^{-1}_{kl}d{\mathcal{O}}^{l}_{\nu}({f^{-1}})^{ab}$
$\displaystyle-d{\mathcal{O}}^{i}_{a}C^{-1}_{ij}\Sigma^{jk}_{\mu\nu}C^{-1}_{kl}d{\mathcal{O}}^{l}_{b}({f^{-1}})^{ab}$
$\displaystyle-d{\mathcal{O}}^{i}_{\mu}C^{-1}_{ij}\Sigma^{jk}_{ab}C^{-1}_{kl}d{\mathcal{O}}^{l}_{\nu}({f^{-1}})^{ab}$
$\displaystyle-d{\mathcal{O}}^{i}_{\mu}C^{-1}_{ij}\Sigma^{jk}_{a\nu}C^{-1}_{kl}d{\mathcal{O}}^{l}_{b}({f^{-1}})^{ab}$
$\displaystyle+{F}_{\mu
a}({f^{-1}})^{ab}\left(\Sigma^{ij}_{bc}C^{-1}_{ij}\right)({f^{-1}})^{cd}{F}_{d\nu}$
$\displaystyle+{F}_{\mu
a}({f^{-1}})^{ab}({f^{-1}})^{cd}C^{-1}_{ij}C^{-1}_{kl}$
$\displaystyle\times\Big{[}d{\mathcal{O}}^{i}_{c}\Sigma^{jk}_{bd}d{\mathcal{O}}^{l}_{\nu}+d{\mathcal{O}}^{i}_{d}\Sigma^{jk}_{b\nu}d{\mathcal{O}}^{l}_{c}$
$\displaystyle+d{\mathcal{O}}^{i}_{\nu}\Sigma^{jk}_{cd}d{\mathcal{O}}^{l}_{b}+d{\mathcal{O}}^{i}_{b}\Sigma^{jk}_{c\nu}d{\mathcal{O}}^{l}_{d}\Big{]}$
$\displaystyle+{F}_{\nu
a}({f^{-1}})^{ab}({f^{-1}})^{cd}C^{-1}_{ij}C^{-1}_{kl}$
$\displaystyle\times\Big{[}d{\mathcal{O}}^{i}_{c}\Sigma^{jk}_{bd}d{\mathcal{O}}^{l}_{\mu}+d{\mathcal{O}}^{i}_{d}\Sigma^{jk}_{b\mu}d{\mathcal{O}}^{l}_{c}$
$\displaystyle+d{\mathcal{O}}^{i}_{\mu}\Sigma^{jk}_{cd}d{\mathcal{O}}^{l}_{b}+d{\mathcal{O}}^{i}_{b}\Sigma^{jk}_{c\mu}d{\mathcal{O}}^{l}_{d}\Big{]}$
$\displaystyle-{F}_{\mu
a}({f^{-1}})^{ab}({f^{-1}})^{cd}({f^{-1}})^{eg}{F}_{g\nu}C^{-1}_{ij}C^{-1}_{kl}$
$\displaystyle\times\Big{[}d{\mathcal{O}}^{i}_{b}\Sigma^{jk}_{ce}d{\mathcal{O}}^{l}_{d}+d{\mathcal{O}}^{i}_{b}\Sigma^{jk}_{cd}d{\mathcal{O}}^{l}_{e}$
$\displaystyle+d{\mathcal{O}}^{i}_{c}\Sigma^{jk}_{be}d{\mathcal{O}}^{l}_{d}+d{\mathcal{O}}^{i}_{c}\Sigma^{jk}_{bd}d{\mathcal{O}}^{l}_{e}\Big{]}.$
(67)

where the Latin indices from the set $\\{i,j,k,l\\}$ are used to sum over the
observables.

This expression simplifies in the diagonal case where we set
$\theta^{\mu}=\theta^{\nu}=M_{\nu}$ for concreteness. To lowest order in
$\Sigma$, we have that

$\displaystyle\left<\mathscr{F}^{Noise}_{M_{\nu}M_{\nu}}\right>_{\Sigma^{1}}$
$\displaystyle=\Sigma^{ij}_{M_{\nu}M_{\nu}}C^{-1}_{ij}-2{F}_{M_{\nu}a}({f^{-1}})^{ab}\Sigma^{ij}_{bM_{\nu}}C^{-1}_{ij}$
$\displaystyle-2d{\mathcal{O}}^{i}_{M_{\nu}}C^{-1}_{ij}\Sigma^{jk}_{aM_{\nu}}C^{-1}_{kl}d{\mathcal{O}}^{l}_{b}({f^{-1}})^{ab}$
$\displaystyle-d{\mathcal{O}}^{i}_{a}C^{-1}_{ij}\Sigma^{jk}_{M_{\nu}M_{\nu}}C^{-1}_{kl}d{\mathcal{O}}^{l}_{b}({f^{-1}})^{ab}$
$\displaystyle-d{\mathcal{O}}^{i}_{M_{\nu}}C^{-1}_{ij}\Sigma^{jk}_{ab}C^{-1}_{kl}d{\mathcal{O}}^{l}_{M_{\nu}}({f^{-1}})^{ab}$
$\displaystyle+{F}_{M_{\nu}a}({f^{-1}})^{ab}\left(\Sigma^{ij}_{bc}C^{-1}_{ij}\right)({f^{-1}})^{cd}{F}_{dM_{\nu}}$
$\displaystyle+2{F}_{M_{\nu}a}({f^{-1}})^{ab}({f^{-1}})^{cd}C^{-1}_{ij}C^{-1}_{kl}$
$\displaystyle\times\Big{[}d{\mathcal{O}}^{i}_{c}\Sigma^{jk}_{bd}d{\mathcal{O}}^{l}_{M_{\nu}}+d{\mathcal{O}}^{i}_{d}\Sigma^{jk}_{bM_{\nu}}d{\mathcal{O}}^{l}_{c}$
$\displaystyle+d{\mathcal{O}}^{i}_{M_{\nu}}\Sigma^{jk}_{cd}d{\mathcal{O}}^{l}_{b}+d{\mathcal{O}}^{i}_{b}\Sigma^{jk}_{cM_{\nu}}d{\mathcal{O}}^{l}_{d}\Big{]}$
$\displaystyle-{F}_{M_{\nu}a}({f^{-1}})^{ab}({f^{-1}})^{cd}({f^{-1}})^{eg}{F}_{gM_{\nu}}C^{-1}_{ij}C^{-1}_{kl}$
$\displaystyle\times\Big{[}2d{\mathcal{O}}^{i}_{b}\Sigma^{jk}_{ce}d{\mathcal{O}}^{l}_{d}+d{\mathcal{O}}^{i}_{b}\Sigma^{jk}_{cd}d{\mathcal{O}}^{l}_{e}$
$\displaystyle+d{\mathcal{O}}^{i}_{c}\Sigma^{jk}_{be}d{\mathcal{O}}^{l}_{d}\Big{]}$
(68)

This expression may be used to calculate the expected level of noise to lowest
order in $1/N_{sims}$ present in the measured value of
$\bar{\sigma}^{-2}_{M_{\nu}}$ and is compared to other approaches in Fig. 4.

##  Appendix B Noise Distribution Convergence Tests

In this appendix, we identify when and how we can actually consider our
diagonal noise distributions, $p\left(\mathscr{F}^{Noise}_{\mu\mu}\right)$, as
calculated with either Approx I: $F_{AB}=\overline{F}_{AB}$, or approx II:
$F_{AB}=F^{degen}_{AB}$, as representative of the true noise distribution,
calculated with the true $N_{sims}\rightarrow\infty$ values of the derivatives
$d\mathcal{O}^{i}_{A}$. As in the main text, we refer to three distributions:

$\displaystyle p^{True}$ $\displaystyle\equiv$ $\displaystyle
p\left(\mathscr{F}^{Noise}_{\mu\mu}|\ d\mathcal{O}\right)$ (69)
$\displaystyle\bar{p}$ $\displaystyle\equiv$ $\displaystyle
p\left(\mathscr{F}^{Noise}_{\mu\mu}|\ d\overline{\mathcal{O}}\right)$ (70)
$\displaystyle p^{degen}$ $\displaystyle\equiv$ $\displaystyle
p\left(\mathscr{F}^{Noise}_{\mu\mu}|\ d\mathcal{O}^{degen}\right)$ (71)

where $p^{True}$ refers to the true distribution, $\bar{p}$ refers to the
distribution calculated with Approx. I, and $p^{degen}$ refers to the
calculation with Approx. II. We find that in practice, all of the noise
distributions are highly Gaussian, and thus we may ask the simplified question
of when $\left<\bar{p}\right>$ and $\mathrm{Var}\left(\bar{p}\right)$ or
$\left<p^{degen}\right>$ and $\mathrm{Var}\left(p^{degen}\right)$ are
representative of the true $\left<p^{True}\right>$ and
$\mathrm{Var}\left(p^{True}\right)$.

We begin with the simpler of the two approximations, Approx I:
$F_{AB}=\overline{F}_{AB}$. In accordance with (58), we assume that the mean
derivatives we have from the simulations differ from the true derivatives by a
single unknown instance of the random variable $u^{i}_{A}$. This lets us write

$\bar{p}=p\left(\mathscr{F}^{Noise}_{\mu\mu}|\
d\mathcal{O}^{i}_{A}+\tilde{u}^{i}_{A}\right)$ (72)

where $\tilde{u}^{i}_{A}$ is the actual instance of noise drawn from the
multivariate Gaussian distribution characterized by $\Sigma^{ij}_{AB}$ which
separates the true derivatives $d\mathcal{O}^{\ i}_{A}$ and the derivatives
$d\overline{\mathcal{O}}{}^{\ i}_{A}$ as measured from the simulations.
Ideally, we would like to determine the exact effect that $\tilde{u}^{i}_{A}$
had in order to turn $p^{True}$ into $\bar{p}$ so that we could reverse it, or
at the very least determine if $\bar{p}$ is representative of $p^{True}$ in
terms of mean and variance so that we can reliably use (61).

While a precise reversal is impossible for an individual instance of noise, we
can consider this in statistical context, and reverse the average effects of
noise on $\bar{p}$ to estimate a hypothetical $p^{True}$. We do this by first
adding additional instances of noise to $d\overline{\mathcal{O}}{}^{\ i}_{A}$.
Explicitly, we define the “$n$ times barred” noise distribution
$\overline{p}^{[n]}$ as

$\displaystyle\left.d\overline{\mathcal{O}}^{[n]}\right.^{i}_{A}$
$\displaystyle\equiv$ $\displaystyle d\mathcal{O}^{\ i}_{A}+\sqrt{n}u^{i}_{A}$
(73) $\displaystyle\overline{p}^{[n]}$ $\displaystyle\equiv$ $\displaystyle
p\bigg{(}\mathscr{F}^{Noise}_{\mu\mu}|\ d\overline{\mathcal{O}}^{[n]}\bigg{)}$
(74)

where $u^{i}_{A}$ is sampled from the multivariate Gaussian distribution
characterized by $\Sigma^{ij}_{AB}$. The $\sqrt{n}$ term multiplying
$u^{i}_{A}$ ensures the variance of the new noise term is $n$ times that of a
single instance. Since $u^{i}_{A}$ is a multivariate Gaussian random variable,
$n$ independent instances of $u^{i}_{A}$ added together is statistically
equivalent of one instance of $u^{i}_{A}$ multiplied by $\sqrt{n}$. This
allows for a continuous $n\in\mathbb{R}_{\geq 0}$ rather than just over the
set of positive integers.

From (51) and (74), $p^{True}\equiv\overline{p}^{[0]}$, and from (52),
$\bar{p}$ is one particular instance of $\overline{p}^{[1]}$ if the instance
of $u^{i}_{A}$ drawn in defining $\overline{p}^{[1]}$ was equal to
$\tilde{u}^{i}_{A}$

For fixed $n$, we may calculate a different instance of the distribution
$\overline{p}^{[n]}$ for each instance of the random variable $u^{i}_{A}$, so
that a population of slightly different $\overline{p}^{[n]}$ probability
distributions may be constructed each with its own mean and variance.
Computing the expectation values of these distributions of means and variances
then will give us the average mean, $\mu(n)$ and the average variance,
$\sigma^{2}(n)$, across the set of distributions
$\left\\{\overline{p}^{[n]}\right\\}$ for a given $n$. Both of these
quantities, $\mu(n)$ and $\sigma^{2}(n)$, are continuous and differentiable
functions in $n$ and thus may be Taylor expanded. Given that
$\mu(0)=\left<p^{True}\right>$ and
$\sigma^{2}(0)=\mathrm{Var}\left(p^{True}\right)$, so that through this
approach, the goal of estimating $p^{True}$ may be reformulated as estimating
$\mu(0)=\left<p^{True}\right>$ and
$\sigma^{2}(0)=\mathrm{Var}\left(p^{True}\right)$ from extrapolating
$\\{\mu(n)$} and $\\{\sigma^{2}(n)\\}$ for $n>0$ .

As an example, we may perform this process analytically in the limit of
isolated noise, where the noise in the non-target parameters is negligible,
$u^{i}_{a}=0$, by assumption. In this limit, a random instance of the
probability distribution $\overline{p}^{[n]}$ has as a mean

$\displaystyle\left<\overline{p}^{[n]}\big{|}_{u^{i}_{a}\rightarrow 0}\right>$
$\displaystyle=\left<p^{True}\right>$ (75)

due to the fact that (63) is independent of $d\mathcal{O}^{\ i}_{\mu}$
(substituting out the $M_{\nu}$ index in (63) for a general target parameter
$\theta^{\mu}$), and has as a variance

$\displaystyle\mathrm{Var}\left(\overline{p}^{[n]}\right)\big{|}_{u^{i}_{a}\rightarrow
0}$
$\displaystyle=\mathrm{Var}\left(p^{True}\right)\big{|}_{u^{i}_{a}\rightarrow
0}$ (76)
$\displaystyle+\left(4nu_{\mu}^{i}u_{\mu}^{j}+8\sqrt{n}u_{\mu}^{i}d\mathcal{O}^{j}_{\mu}\right)\left(Q_{ik}\Sigma^{kl}_{\mu\mu}Q_{lj}\right).$
(77)

Taking the expectation value of the above equations across all possible
$\overline{p}^{[n]}$ probability distributions simply amounts to an integral
over the multivariate Gaussian distribution used to define $u^{i}_{\mu}$, so
that in the limit of isolated noise, we have

$\displaystyle\mu(n)\big{|}_{u^{i}_{a}\rightarrow 0}$
$\displaystyle=\left<p^{True}\right>$ (78)

and

$\displaystyle\sigma^{2}(n)|_{u^{i}_{a}\rightarrow
0}=\mathrm{Var}\left(p^{True}\right)\big{|}_{u^{i}_{a}\rightarrow
0}+4n\left(\Sigma^{ij}_{\mu\mu}Q_{ik}\Sigma^{kl}_{\mu\mu}Q_{lj}\right)$ (79)

which are a constant and a linear function of $n$ respectively. While these
expressions become more complicated outside of this limit, all of the above
quantities may still be calculated numerically via sampling.

There is one small issue with the above approach, and that is that in general,
we cannot calculate $d\overline{\mathcal{O}}^{[n]}$ in an unbiased manner due
to the baked-in single instance of noise $\tilde{u}^{i}_{A}$. However, we can
modify the above approach to deal with this fact by using

$\left.d\overline{\mathcal{O}}^{[n]}\right.^{i}_{A}\approx d\mathcal{O}^{\
i}_{A}+\tilde{u}^{i}_{A}+\sqrt{n-1}u^{i}_{A}$ (80)

When (80) is used for $d\overline{\mathcal{O}}^{[n]}$ instead of (73), it is
in general no longer exactly true, but only approximately true, that
$\mu(0)\simeq\left<p^{True}\right>$ and
$\sigma^{2}(0)\simeq\mathrm{Var}\left(p^{True}\right)$. For example, when
using (80) instead of (73) in the limit of isolated noise, 79) becomes

$\displaystyle\sigma^{2}(n)|_{u^{i}_{a}\rightarrow 0}$
$\displaystyle=\mathrm{Var}\left(p^{True}\right)\big{|}_{u^{i}_{a}\rightarrow
0}$
$\displaystyle+4\left((n-1)\Sigma^{ij}_{\mu\mu}+\tilde{u}^{i}_{\mu}\tilde{u}^{j}_{\mu}+2\tilde{u}^{i}_{\mu}d\mathcal{O}^{\
j}_{\mu}\right)\left(Q_{ik}\Sigma^{kl}_{\mu\mu}Q_{lj}\right)$ (81)

so that

$\displaystyle\sigma^{2}(0)|_{u^{i}_{a}\rightarrow 0}$
$\displaystyle=\mathrm{Var}\left(p^{True}\right)\big{|}_{u^{i}_{a}\rightarrow
0}$
$\displaystyle+4\left(\tilde{u}^{i}_{\mu}\tilde{u}^{j}_{\mu}-\Sigma^{ij}_{\mu\mu}+2\tilde{u}^{i}_{\mu}d\mathcal{O}^{\
j}_{\mu}\right)\left(Q_{ik}\Sigma^{kl}_{\mu\mu}Q_{lj}\right)$ (82)

Here, we highlight that on average,
$\left<\tilde{u}^{i}_{\mu}\tilde{u}^{j}_{\mu}\right>=\Sigma^{ij}_{\mu\mu}$ and
$\left<\tilde{u}^{i}_{\mu}d\mathcal{O}^{\ j}_{\mu}\right>=0$ so that even when
approximating (73) with (80), it is still the case that
$\sigma^{2}(0)|_{u^{i}_{a}\rightarrow
0}\simeq\mathrm{Var}\left(p^{True}\right)\big{|}_{u^{i}_{a}\rightarrow 0}$.
Given this, the procedure for verifying numerical convergence is as follows:

  * •

Construct the distributions $\bar{p}$, using Approx. I, and $p^{degen}$, using
Approx. II, via numerical sampling, as in IV.3.

  * •

Construct populations of $\left\\{\overline{p}^{[n]}\right\\}$ using (80) and
(74) across some set of $\\{n\geq 1\\}$ values and from these, calculate
$\mu(n)$ and $\sigma^{2}(n)$ for each $n$ in preparation to Taylor expand.
Care must be taken when sampling and choosing finite difference step sizes in
$n$ so that the derivatives used in the expansion are converged. We use
$\Delta n=0.25$, and require that our sets
$\left\\{\overline{p}^{[n]}\right\\}$ consist of 30000 distributions, each of
which have been sampled 10000 times to determine an accurate measure of
$\mu(n)$ and $\sigma^{2}(n)$ for each $n$.

  * •

Find the lowest ($k^{th}$) order Taylor expansions of $\mu(n)$ and
$\sigma^{2}(n)$ centered at $n=2$, with finite difference derivatives
calculated using points at $n\geq 2$, which can reliably recover
$\mu(1)=\left<\bar{p}\right>$ and
$\sigma^{2}(1)=\mathrm{Var}\left(\bar{p}\right)$. In practice, we restrict
ourselves to $k\leq 2$ order expansions at maximum to avoid over fitting.

  * •

Perform the same $k^{th}$ order Taylor expansion of $\mu(n)$ and
$\sigma^{2}(n)$ centered at $n=1$ to approximate
$\mu(0)=\left<p^{True}\right>$ and
$\sigma^{2}(0)=\mathrm{Var}\left(p^{True}\right)$.

  * •

Compare the reconstructed Gaussian approximation for $p^{True}$, using
$\mu(0)$ and $\sigma^{2}(0)$, to $\bar{p}$ and $p^{degen}$ to assess how
representative they each are.

Figure 9 shows the results of this process to estimate $\mu(0)$ and
$\sigma^{2}(0)$ for target parameter $M_{\nu}$ using the $D1_{a}$
differentiation scheme and observable $P_{halos}(k)$. In accordance with Fig.
4, we can see that $p\left(\mathscr{F}^{Noise}_{M_{\nu}M_{\nu}}\right)$ for
this differentiation scheme and observable is not within the physical limit of
isolated noise, and thus we should expect both $\mu(n)$ and $\sigma^{2}(n)$ to
vary as a continuous function of $n$.

We find a first-order and second-order Taylor expansion can be used to
reliably reproduce the $\sigma^{2}(n)$ and $\mu(n)$ functions respectively,
and use these to extrapolate to $n=0$, calculating $\mu(0)\simeq
0.39\mathrm{eV}^{-2}$, a small increase to the $\bar{p}$ value (Approx. I) of
$\mu(1)\simeq 0.33\mathrm{eV}^{-2}$, and $\sigma^{2}(0)=8.8\times
10^{-3}\mathrm{eV}^{-2}$ instead of the $\bar{p}$ value of
$\sigma^{2}(1)=1.5\times 10^{-2}\mathrm{eV}^{-2}$. If we use the hypothetical
$p^{True}$ defined by these Taylor expanded quantities to modify the
confidence limit prediction of $\sigma^{-2}_{M_{\nu},D1_{a}}$ from (61), we
find now that $\sigma^{-2}_{M_{\nu},D1_{a}}=0.72\pm 0.09\mathrm{eV}^{-2}$,
which is slightly different than the $\bar{p}$ interval of
$\sigma^{-2}_{M_{\nu},D1_{a}}=0.77\pm 0.12\mathrm{eV}^{-2}$ quoted in Table 1
of the main text.

Figure 9: Taylor expansion approach to estimating statistics for $p^{[0]}$ for
$P_{halos}$ for $D1_{a}$ shown in Fig. 10. $\mu(n)$ [red, left axis] and
$\sigma^{2}(n)$ [blue, right axis] are shown evaluated using (80) and (74) via
numerical sampling from $n=1$ to $n=3$ with $\Delta n=0.25$, and extended down
to $n=0$ via Taylor expansion.  Figure 10: Gaussian approximations for
$\bar{p}$ [light blue, solid], and $p^{degen}$ [purple, solid], the average
$\overline{p}^{[2]}$ [orange, solid], the average $\overline{p}^{[3]}$ [green,
solid] and the $1^{st}$ order Taylor approximations for
$\overline{p}^{[1]}\simeq\bar{p}$ [blue, dashed] and $\overline{p}^{[0]}\simeq
p^{True}$ [black, dashed] using $P_{Halos}(k)$ as the observable for target
parameter $M_{\nu}$ with the $D3$ [left], $D2_{a}$ [center] or $D1_{a}$
[right] differentiation schemes.

Figure 10 shows the results of applying this process to check all noise
distributions previously shown in Fig. 4. Focusing first on the $D1_{a}$
differentiation scheme, we can see the combination of $\mu(n)$ and
$\sigma^{2}(n)$ shown in Fig. 9 represented as actual probability
distributions for $n\in\\{3,2,1,0\\}$. As n decreases from 3 down to 0, the
distributions become narrower with the mean slightly but steadily increasing,
just as shown in Fig. 9. It is this small difference between $\bar{p}$ and our
hypothetical $p^{True}=p^{[0]}$ which slightly modifies our confidence
interval for $\sigma^{-2}_{M_{\nu},D1_{a}}$ as discussed above.

For $D3$ and $D2_{a}$ differentiation schemes, we find that $p^{degen}$ is
more representative of our hypothetical $p^{True}$ than $\bar{p}$. Since
according to Fig. 4 these noise distributions are all well within the physical
limit of isolated noise, this can be understood analytically through (78) and
(79), where $\mu(n)$ is independent of $n$ and $\sigma^{2}(n)$ is linear in
$n$. Thus, while $\left<\bar{p}\right>$ and $\left<p^{degen}\right>$ will both
agree with $\mu(0)=\left<p^{True}\right>$, for the variances we have

$\displaystyle\mathrm{Var}\left(p^{degen}\right)|_{u^{i}_{a}\rightarrow
0}=\mathrm{Var}\left(p^{True}\right)\big{|}_{u^{i}_{a}\rightarrow
0}-2\left(d\mathcal{O}^{i}_{\mu}d\mathcal{O}^{j}_{\mu}Q_{ik}\Sigma^{kl}_{\nu\nu}Q_{lj}\right)$
(83)

and (dropping subdominant terms linear in $\tilde{u}^{i}_{\mu}$)

$\displaystyle\mathrm{Var}\left(\bar{p}\right)|_{u^{i}_{a}\rightarrow
0}\simeq\mathrm{Var}\left(p^{True}\right)\big{|}_{u^{i}_{a}\rightarrow
0}+4\left(\tilde{u}^{i}_{\mu}\tilde{u}^{j}_{\mu}Q_{ik}\Sigma^{kl}_{\nu\nu}Q_{lj}\right)$
(84)

so that now

$\mathrm{Var}\left(p^{degen}\right)|_{u^{i}_{a}\rightarrow
0}<\mathrm{Var}\left(p^{True}\right)|_{u^{i}_{a}\rightarrow
0}\lesssim\mathrm{Var}\left(\bar{p}\right)|_{u^{i}_{a}\rightarrow 0}$ (85)

When noise within the target parameters is large, it is often the case that
the term separating $\mathrm{Var}\left(p^{degen}\right)|_{u^{i}_{a}\rightarrow
0}$ and $\mathrm{Var}\left(p^{True}\right)|_{u^{i}_{a}\rightarrow 0}$ is much
smaller than that which separates
$\mathrm{Var}\left(\bar{p}\right)|_{u^{i}_{a}\rightarrow 0}$ and
$\mathrm{Var}\left(p^{True}\right)\big{|}_{u^{i}_{a}\rightarrow 0}$, so that
$p^{degen}$ is more representative of $p^{True}$ than $\bar{p}$ is. For
$D2_{a}$, this correspondence is almost exact between $p^{degen}$ and the
reconstructed $p^{True}$.

Finally, we note that even in cases when noise is isolated to a single
parameter which is not within the target parameter set, we find that
$p^{degen}$ may still be highly representative of a hypothetical $p^{True}$ if
the assumptions of (46) are relaxed so that the noisiest parameter derivative
as defined by $\Sigma^{ij}_{A,A}C^{-1}_{ij}/\overline{F}_{A,A}$ is
reconstructed as a degenerate linear combination of the others. This
application of Approx. II to the noisiest parameter, rather than the target
parameter, allows the degeneracy’s between the true parameter derivatives to
be more faithfully represented in scenarios in which noise is almost
exclusively concentrated within a single non-target parameter. This is
demonstrated in Fig. 7 showing predictions for
$p\left(\mathscr{F}^{Noise}_{\mu\mu}\right)$ on $\Omega_{m}$ and $\sigma_{8}$
when reconstructing the noisy D3 $M_{\nu}$ derivative as a degenerate linear
combination of the other parameters.

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