SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where in the final line we have defined  $\mathcal{F}(\eta) \coloneqq \mathrm{KL}(\eta ||\pi)$ . Now, under the assumption that  $\pi$  is strongly log-concave, [43, Proof of Theorem 3] guarantees the existence of a mirror map  $\phi : \mathcal{X} \to \mathbb{R} \cup \{\infty\}$  such that the mirrored target  $\nu = (\nabla \phi)_{\#} \pi$  is also strongly log-concave. It follows, in particular, that  $\mathcal{F}(\eta) \coloneqq \mathrm{KL}(\eta ||\pi)$  is geodesically convex [e.g., 2, Chapter 9]. Thus, arguing as in the proof of [81, Proposition 3.3], with  $w_0 = 1$ , it follows that
$$
\begin{array}{l} \frac {1}{T} \sum_ {t = 1} ^ {T} \mathcal {F} (\eta_ {t}) - \mathcal {F} (\nu) \leq \frac {L}{T} \left[ 1 + \int_ {\mathbb {R} ^ {d}} | | y | | \sqrt {T \ln (1 + 9 6 K ^ {2} T ^ {2} | | y | | ^ {2})} \nu (\mathrm {d} y) \right. \\ \left. + \int_ {\mathbb {R} ^ {d}} | | y | | \sqrt {T \ln (1 + 9 6 T ^ {2} \| y \| ^ {2})} \eta_ {0} (\mathrm {d} y) \right]. \tag {106} \\ \end{array}
$$
Finally, once more using the definition of the mirror map and in particular the fact that  $\nu = (\nabla \phi)_{\#}\pi$  and  $\eta_0 = (\nabla \phi)_{\#}\eta_0$ , we can rewrite the RHS in (106) as the RHS in (102). This completes the proof.
D Other Mirrored Coin Sampling Algorithms
In this section, we provide some other mirrored coin sampling algorithms besides Coin MSVGD. Recall, from Sec. 4.2, the general form of the mirrored coin sampling algorithm. Let  $x_0 \sim \mu_0 \in \mathcal{P}_2(\mathcal{X})$ , and  $y_0 = \nabla \phi(x_0) \in \mathbb{R}^d$ . Then, for  $t \in \mathbb{N}$ , update
$$
y _ {t} = y _ {0} + \frac {\sum_ {s = 1} ^ {t - 1} c _ {\eta_ {s}} (y _ {s})}{t} \left(1 + \sum_ {s = 1} ^ {t - 1} \langle c _ {\eta_ {s}} (y _ {s}), y _ {s} - y _ {0} \rangle\right), x _ {t} = \nabla \phi^ {*} (x _ {t}), \tag {107}
$$
where  $c_{\eta_t}(y_t) = -\nabla_{W_2}\mathcal{F}(\eta_t)(y_t)$ , or some variant thereof. As noted in Sec. 4.2, for different choices of  $\mathcal{F}$  and  $(c_{\eta_t})_{t\in \mathbb{N}_0}$ , one obtains learning-rate free analogues of different mirrored ParVI algorithms.
In Sec. 4.2, we provided the Coin MSVGD algorithm (Alg. 2), which corresponds to  $\mathcal{F} = \mathrm{KL}(\cdot |\nu)$  and  $(c_{\eta_t})_{t\in \mathbb{N}_0} = (-P_{\eta_t,k_\phi}\nabla \log (\frac{\eta_t}{\nu}))_{t\in \mathbb{N}_0}$ . This can be viewed as the coin sampling analogue of MSVGD (Sec. 3.2.1 and App. B.2). We now provide two more mirrored coin sampling algorithms, which correspond to the coin sampling analogues of MLAWGD (App. B.3) and MKSDD (App. B.4).
D.1 Coin MLAWGD
Algorithm 6 Coin MLAWGD
input: target density  $\pi$ , kernel  $k_{\mathcal{L}_{\nu}}$ , mirror map  $\phi$ , initial particles  $(x_0^i)_{i=1}^N \sim \mu_0$ .
initialise: for  $i\in [N],y_0^i = \nabla \phi (x_0^i)$
For  $i\in [N]$
$$
y _ {t} ^ {i} = y _ {0} ^ {i} - \frac {\sum_ {s = 1} ^ {t - 1} \frac {1}{N} \sum_ {j = 1} ^ {N} \nabla_ {y ^ {i}} k _ {\mathcal {L} _ {\nu}} \left(y _ {s} ^ {i} , y _ {s} ^ {j}\right)}{t} \left(1 - \sum_ {s = 1} ^ {t - 1} \langle \frac {1}{N} \sum_ {j = 1} ^ {N} \nabla_ {y ^ {i}} k _ {\mathcal {L} _ {\nu}} \left(y _ {s} ^ {i} , y _ {s} ^ {j}\right) , y _ {s} ^ {i} - y _ {0} ^ {i} \rangle\right)
$$
$$
x _ {t} ^ {i} = \nabla \phi^ {*} (y _ {t} ^ {i}).
$$
where  $k_{\mathcal{L}_{\nu}}$  is the mirrored LAWGD kernel defined in (81).
D.2 Coin MKSDD
Algorithm 7 Coin MKSDD
input: target density  $\pi$ , kernel  $k$ , mirror map  $\phi$ , initial particles  $(x_0^i)_{i=1}^N \sim \mu_0$ .
$$
y _ {t} ^ {i} = y _ {0} ^ {i} - \frac {\sum_ {s = 1} ^ {t - 1} \frac {1}{N ^ {2}} \sum_ {j = 1} ^ {N} \nabla_ {y ^ {i}} k _ {\nu , \phi} (y _ {s} ^ {j} , y _ {s} ^ {i})}{t} \left(1 - \sum_ {s = 1} ^ {t - 1} \langle \frac {1}{N ^ {2}} \sum_ {j = 1} ^ {N} \nabla_ {y ^ {i}} k _ {\nu , \phi} (y _ {s} ^ {j} , y _ {s} ^ {i}), y _ {s} ^ {i} - y _ {0} ^ {i} \rangle\right)
$$
where  $k_{\nu ,\phi}$  is the mirrored Stein kernel defined in (90).
E Adaptive Coin MSVGD
In Sec. 4.1 - 4.2, we described a betting strategy which is valid under the assumption that the sequence of outcomes  $(c_{t})_{t\in \mathbb{N}}\coloneqq (c_{\eta_{t}}(y_{t}))_{t\in \mathbb{N}}$  are bounded by one, in the sense that  $||c_{\eta_t}(y_t^i)||\leq 1$  for all  $t\in [T]$  and for all  $i\in [N]$  [see also 81, Sec. 3.4]. In practice, this may not be the case. If, instead,  $||c_{\eta_t}(y_t^i)||\leq L$  for some constant  $L$ , then one can simply replace  $c_{\eta_t}(y_t)$  by its normalised version, namely  $\hat{c}_{\eta_t}(y_t) = c_{\eta_t}(y_t) / L$  in Alg. 2.
If, on the other hand, the constant  $L$  is unknown in advance, then one can use a coordinate-wise empirical estimate  $L_{t,j}^{i}$  which is adaptively updated as the algorithm runs, following [72] and later [81, App. D].<sup>3</sup> We provide full details of this version of Coin MSVGD in Alg. 8. One can also obtain an analogous version of Coin MIED; we omit the full details here in the interest of brevity.
Finally, when we use a coin sampling algorithm to perform inference for a (fairness) Bayesian neural network (see Sec. 6.3), we alter the denominator of the betting fraction in Alg. 8 such that it is at least  $100L_{t,j}^{i}$ . Thus, the update equation in Alg. 8 (or, analogously, the update equation in the adaptive version of Coin MIED) now reads as
$$
y _ {t, j} ^ {i} = y _ {0, j} ^ {i} + \frac {\sum_ {s = 1} ^ {t - 1} c _ {s , j} ^ {i}}{\max \left(G _ {t , j} ^ {i} + L _ {t , j} ^ {i} , 1 0 0 L _ {t , j} ^ {i}\right)} \left(1 + \frac {R _ {t , j} ^ {i}}{L _ {t , j} ^ {i}}\right). \tag {108}
$$
This modification, recommended in [72, Sec. 6], has the effect of restricting the value of the particles in the first few iterations. We should emphasise that these adaptive algorithms, which in practice we use in all experiments, are still free of any hyperparameters which need to be tuned by hand.
3Here,  $i\in [N]$  indexes the particle, and  $j\in [d]$  indexes the dimension (see Alg. 8).
Algorithm 8 Adaptive Coin MSVGD
initialise: for  $i\in [N]$ $y_0^i = \nabla \phi (x_0^i)$  ; for  $i\in [N]$  and  $\bar{j}\in [d],L_{0,j}^{i} = 0,G_{0,j}^{i} = 0,R_{0,j}^{i} = 0.$
$$
c _ {t - 1} ^ {i} = - P _ {\hat {\eta} _ {t}, k _ {\phi}} \nabla \log \left(\frac {\hat {\eta} _ {t - 1}}{\nu}\right) \left(y _ {t - 1} ^ {i}\right), \hat {\eta} _ {t - 1} = \frac {1}{N} \sum_ {k = 1} ^ {N} \delta_ {y _ {t - 1} ^ {k}} \quad (\text {c o m p u t e S V G D g r a d i e n t})
$$
For  $i\in [N],j\in [d]$
$$
L _ {t, j} ^ {i} = \max  \left(L _ {t - 1, j} ^ {i}, \left| c _ {t - 1, j} ^ {i} \right|\right) \quad (\text {u p d a t e m a x . o b s e r v e d s c a l e})
$$
$$
G _ {t, j} ^ {i} = G _ {t - 1, j} ^ {i} + \left| c _ {t - 1, j} ^ {i} \right| \quad (\text {u p d a t e s u m o f a b s . v a l u e o f g r a d i e n t s})
$$
$$
R _ {t, j} ^ {i} = \max  \left(R _ {t - 1, j} ^ {i} + \left\langle c _ {t - 1, j} ^ {i}, y _ {t - 1, j} ^ {i} - y _ {0, j} ^ {i} \right\rangle , 0\right) \quad \text {(u p d a t e t o t a l r e w a r d)}
$$
$$
y _ {t, j} ^ {i} = y _ {0, j} ^ {i} + \frac {\sum_ {s = 1} ^ {t - 1} c _ {s , j} ^ {i}}{G _ {t , j} ^ {i} + L _ {t , j} ^ {i}} \left(1 + \frac {R _ {t , j} ^ {i}}{L _ {t , j} ^ {i}}\right). \tag {update the particles}
$$
For  $i\in [N]$ $x_{t}^{i} = \nabla \phi^{*}(y_{t}^{i})$
output:  $(x_{T}^{i})_{i = 1}^{N}$
Computational Cost. It is worth noting that, in terms of computational and memory cost, the adaptive Coin MSVGD algorithm is no worse than MSVGD when the latter is paired, as is common, with a method such as Adagrad [31], RMSProp [92], or Adam [48]. The computational cost per iteration is  $O(N^2)$ , due to the kernelised gradient. Meanwhile, in terms of memory requirements, it is necessary to keep track of the maximum observed absolute value of the gradients (component-wise), the sum of the absolute value of the gradients (component-wise), and the total reward (component-wise). This is similar to, e.g., Adam, which keeps track of exponential moving averages of the gradient and the (component-wise) squared gradient [48].
F Additional Experimental Details
In this section, we provide additional details relevant to the experiments carried out in Sec. 6. We implement all methods using Python 3, PyTorch, and TensorFlow. We use the existing implementations of MSVGD and SVMD [82] and MIED [60] to run these methods. We perform all experiments using a MacBook Pro 16" (2021) laptop with Apple M1 Pro chip and 16GB of RAM.
For the experiments in Sec. 6.1 and 6.2, we use the inverse multiquadric (IMQ) kernel  $k(x,x^{\prime}) = (1 + \{||x - x^{\prime}||_{2}^{2} / h^{2})^{-\frac{1}{2}}$  due to its favourable convergence control properties (e.g., [38]). Unless otherwise stated, the bandwidth  $h$  is determined using the median heuristic [36, 63]. For the experiment in Sec. 6.3, following [60], we use the  $s$ -Riesz family of mollifipliers  $\{\phi_{\epsilon}^{s}\}$  with  $s = d + 10^{-4}$  and  $\epsilon = 10^{-8}$ .
F.1 Simplex Targets
For both experiments in Sec. 6.1, we initialise all methods with i.i.d samples from a Dirichlet(5) distribution and use the entropic mirror map
$$
\phi (x) = \sum_ {k = 1} ^ {d} x _ {k} \log x _ {k} + \left(1 - \sum_ {k = 1} ^ {d} x _ {k}\right) \log \left(1 - \sum_ {k = 1} ^ {d} x _ {k}\right). \tag {109}
$$
We run all methods for  $T = 500$  iterations using  $N = 50$  particles. In the interest of a fair comparison, we run all of the learning-rate dependent methods (MSVGD, SVMD, projected SVGD) using RMSProp [92], an extension of the Adagrad algorithm [31], to adapt the learning rates.
Sparse Dirichlet Posterior. We first consider the sparse Dirichlet posterior in [73], which is given by
$$
\pi (x) = \frac {1}{Z} \prod_ {k = 1} ^ {d + 1} x _ {k} ^ {n _ {k} + \alpha_ {k} - 1}, \quad x \in \operatorname {i n t} (\Delta), \tag {110}
$$
for parameters  $\alpha_{1},\ldots ,\alpha_{d + 1} > 0$  and observations  $n_1,\dots ,n_{d + 1}$ , where  $n_k$  is the number of observations of category  $k$ . We consider the sparse regime, in which most of the  $n_k$  are zero and the remaining  $n_k$  are large, e.g., of order  $O(d)$ . In particular, as in [82], we set  $\alpha_{k} = 0.1$  for  $k\in \{1,\dots ,20\}$  and use count data  $n_1 = 90$ ,  $n_2 = n_3 = 5$ , and  $n_j = 0$  for  $j > 3$ . This target is used to replicate the multimodal sparse conditionals that arise in Latent Dirichlet Allocation (LDA) [9]. While this distribution is not log-concave, the dual-target is strictly log-concave under the entropic mirror map.
Quadratic Target. We next consider the quadratic simplex target introduced in [1]. In this case, the target takes the form
$$
\pi (x) = \frac {1}{Z} \exp \left(- \frac {1}{2 \sigma^ {2}} x ^ {\top} A x\right), \tag {111}
$$
where  $A \in \mathbb{R}^{d \times d}$  is a symmetric positive semidefinite matrix with all entries bounded in magnitude by 1. Following [82], we set  $\sigma = 0.01$  and generate  $A$  by normalising products of random matrices with i.i.d. elements drawn from Unif[-1, 1].
F.2 Confidence Intervals for Post Selection Inference
Given data  $(X,y)\in \mathbb{R}^{n\times p}\times \mathbb{R}^n$ , the randomised Lasso is defined as the solution of [89, Sec. 3.1]
$$
\hat {\beta} = \underset {\beta \in \mathbb {R} ^ {p}} {\arg \min } \left[ \frac {1}{2} | | y - X \beta | | _ {2} ^ {2} + \lambda | | \beta | | _ {1} - \omega^ {\top} \beta + \frac {\varepsilon}{2} | | \beta | | _ {2} ^ {2} \right], \tag {112}
$$
where  $\lambda$  is the  $\ell_1$  penalty parameter,  $\varepsilon \in \mathbb{R}_+$  is a ridge term, and  $\omega \in \mathbb{R}^p$  is an auxiliary random vector drawn from some distribution  $\mathbb{G}$  with density  $g$ . We choose  $\mathbb{G}$  to be a zero-mean independent Gaussian distribution. The scale of this distribution, and the parameter  $\varepsilon$ , are both determined automatically by the randomized Lasso function in the selective Inference package. The distribution of interest has density proportional to (e.g., [90, Sec. 4.2])
$$
\hat {g} \left(\hat {\beta} _ {E}, \hat {z} _ {- E}\right) \propto g \left(\varepsilon \left(\hat {\beta} _ {E}, 0\right) ^ {\top} - X ^ {\top} \left(y - X _ {E} \hat {\beta} _ {E}\right) + \lambda \left(\hat {z} _ {E}, \hat {z} _ {- E}\right) ^ {\top}\right), \tag {113}
$$
with support  $\mathcal{D} = \{(\hat{\beta}_E, \hat{z}_{-E}) : \mathrm{sign}(\hat{\beta}_E) = \hat{z}_E, ||\hat{z}_{-E}||_{\infty} \leq 1\}$ . In all experiments, following [89], we integrate out  $\hat{z}_{-E}$  analytically and reparameterise  $\hat{\beta}_E$  as  $\hat{z}_E \odot |\hat{\beta}_E|$ . This yields a log-concave density for  $|\hat{\beta}_E|$ , with domain given by the non-negative orthant in  $\mathbb{R}^q$ . We then use the mirror map
$$
\phi (x) = \sum_ {k = 1} ^ {d} \left(x _ {k} \log x _ {k} - x _ {k}\right). \tag {114}
$$
Synthetic Example. We consider the simulation in [79, Sec. 5.3], with  $n = 100$  and  $p = 40$ . The design matrix  $X$  is generated from an equi-correlated model. In particular, for  $i = 1, \dots, n$ , the rows  $x_{i}^{\mathrm{i.i.d.}} \sim \mathcal{N}(0, \Sigma)$ , where  $\Sigma_{ij} = \delta_{i,j} + \rho (1 - \delta_{i,j})$ , with  $\rho = 0.3$ . We also normalise the columns of  $X$  to have norm 1. Meanwhile, the response is distributed as  $y \sim \mathcal{N}(0, I_n)$ , independent of  $X$ . That is the true parameter  $\beta = 0$ . We set  $\varepsilon = \frac{\hat{\sigma}^2}{\sqrt{n}}$ , where  $\hat{\sigma}$  denotes the empirical standard deviation of the response. We set  $\lambda$  using the default value returned by theoretical lambda in the selectiveInference package, multiplied by a factor of  $0.7n$ . This adjustment reduces the  $\ell_1$  regularization, ensuring that a reasonable subset of the  $p = 40$  features are selected.
We run each algorithm for  $T = 1000$  iterations. For Coin MSVGD, MSVGD, and SVMD, we use  $N = 50$  particles, and generate  $N_{\mathrm{total}}$  samples by aggregating the particles from  $N_{\mathrm{total}} / N$  independent runs. For the norejection method, we generate  $N_{\mathrm{total}}$  samples by running the algorithm for  $N_{\mathrm{total}}$  iterations after burn-in. Finally, following [82], in Fig. 3 the error bars for the actual coverage levels correspond to  $95\%$  Wilson intervals [101].
HIV-1 Drug Resistance. We use the vitro measurement of log-fold change under 3TC as the response and include mutations that appear more than 10 times in the dataset as predictors. The resulting dataset consists of  $n = 633$  cases and  $p = 91$  predictors. In the selection stage, we apply the randomised Lasso, setting the regularisation parameter to the theoretical value in selectiveInference, multiplied by a factor of  $n$ . We implement SVMD and MSVGD using RMSProp [92], with an initial learning rate of  $\gamma = 0.01$  following [82]. We implement each method for  $T = 2000$  iterations, using  $N = 50$  particles.
F.3 Fairness Bayesian Neural Network
We use a train-test split of  $80\% / 20\%$ , as in [60, 65]. We implement the methods in [60, 65] using their source code, using the default hyperparameters. We run all algorithms for  $T = 2000$  iterations, and using  $N = 50$  particles. Similar to [60], we found that one of the algorithms (Control + SVGD) in [64] got stuck after initialisation, with accuracy close to 0.75 and did not improve during training. We thus omit the results for this method in Fig. 5.
G Additional Numerical Results
G.1 Simplex Targets
In Fig. 6 and Fig. 7, we provide additional results for the sparse Dirichlet posterior [73] and the quadratic simplex target [1] considered in Sec. 6.1. In particular, in Fig. 6a and Fig. 7a, we plot the energy distance to a set of ground truth samples either obtained i.i.d. (sparse Dirichlet target) or using NUTS [42] (quadratic target) versus the number of iterations for Coin MSVGD, MSVGD, SVMD, and projected versions of Coin SVGD and SVGD. Meanwhile, in Fig. 6b and Fig. 7b, we compare the performance of Coin MSVGD with the performance of MSVGD, for multiple choices of the learning rate.
As observed in Sec. 6.1, for the sparse Dirichlet posterior, Coin MSVGD converges more rapidly than MSVGD and SVMD, even for well-chosen values of the learning rate (Fig. 6). Meanwhile, for the quadratic target, Coin MSVGD generally converges more rapidly than MSVGD but not as fast as SVMD [82], which leverages the log-concavity of the target in the primal space (Fig. 7). In both cases, for poorly chosen values of the learning rate, MSVGD (e.g.,  $\gamma = 5 \times 10^{-1}$  in Fig. 6b and  $\gamma = 1 \times 10^{-4}$ ,  $\gamma = 1 \times 10^{-3}$ , or  $\gamma = 5 \times 10^{-1}$  in Fig. 7b) entirely fails to converge to the true target.
Results for Different Numbers of Particles. In Fig. 8 and Fig. 9, we provide additional results for the two simplex targets, now varying the number of particles  $N$ . The results are similar to those reported in Sec. 6.1. In particular, Coin MSVGD has comparable performance to MSVGD and SVMD with optimal learning rates, but significantly outperforms both algorithms for sub-optimal learning rates. In addition, as expected, the posterior approximation quality of Coin MSVGD, as measured by the energy distance, improves as a function of the number of particles  $N$  (Fig. 8d and Fig. 9d).
G.2 Sampling from a Uniform Distribution
In this section, we consider the task of sampling from the uniform distribution on the unit square, namely  $\mathrm{Unif}[-1,1]^2$ . For MSVGD and Coin MSVGD, we once again use the entropic mirror map. We also now use the RBF kernel, with a fixed bandwidth of  $h = 0.01$ . For MIED and Coin MIED, following [60], we reparameterise the particles using tanh to eliminate the constraint. We show results for several choices of mollifiers, in each casing choosing the smallest  $\epsilon$  which guarantees the optimisation remains stable. We initialise all methods with particles drawn i.i.d. from  $\mathcal{U}[-0.5,0.5]^2$ . For both methods, in the interest of a fair comparison, we adapt the learning rates using RMSProp [92]. We use  $N = 100$  particles and run each experiment for  $T = 250$  iterations. We then assess the quality of the posterior approximations by computing the energy distance to a set of 1000 samples drawn i.i.d. from the true target distribution.
The results for this experiment are shown in Fig. 10. Similar to our previous synthetic examples, Coin MSVGD is competitive with the optimal performance of MSVGD. Meanwhile, for sub-optimal choices of the learning rate, the coin sampling algorithm is clearly preferable. The same is also true for Coin MIED which, depending on the choice of mollifier, obtains comparable or superior performance to MIED with the optimal choice of learning rate.
G.3 Post-Selection Inference
In this section, we provide additional results for the post-selection inference examples considered in Sec. 6.2. In Fig. 11, we once again plot the coverage of the CIs obtained using Coin MSVGD, MSVGD, SVMD, and the norejection algorithm in selectiveInference [91], as we vary either the nominal coverage or the total number of samples. Now, however, we consider two sub-optimal choices of the learning rate for MSVGD and SVMD. In particular, we now use an initial learning rate of  $\gamma = 0.001$  (Fig. 11a - 11b) or  $\gamma = 0.1$  (Fig. 11c - 11d) for RMSProp [92], rather than  $\gamma = 0.01$  as in Fig. 3 (see Sec. 6.2).
In this example, the use of an adaptive kernel (e.g., with the bandwidth chosen according to the median rule), can cause the samples from MSVGD to collapse [60].
In this case, we see that the performance of both MSVGD and SVMD deteriorates. In particular, they both now obtain significantly lower coverage than Coin MSVGD. As noted in the main text, this increased coverage is of particular importance for smaller sample sizes (see Fig. 11b and Fig. 11c), and for greater nominal coverage levels (see Fig. 11a and 11c), where the standard CIs generally undercover.
Finally, in Fig. 12, we perform a similar experiment for the real-data example involving the HIV-1 drug resistance dataset. In particular, we compute  $90\%$  CIs for the subset of mutations selected by the randomised Lasso, using 5000 samples and the same five methods as before: 'unadjusted' (the unadjusted CIs), 'standard' (the method in selectiveInference), MSVGD, SVMD, and Coin MSVGD. Now, however, we use sub-optimal choices of the initial learning rate in RMSProp, namely,  $\gamma = 0.001$  (Fig. 12a) and  $\gamma = 1.0$  (Fig. 12b), compared to  $\gamma = 0.01$  in the main paper.
Similar to the synthetic example, the impact of using a sub-optimal choice of learning rate is clear. In particular, CIs obtained using MSVGD and SVMD now differ substantially from those obtained by Coin MSVGD and the standard approach. This means, in particular, that several mutations are now incorrectly flagged as significant by MSVGD and SVMD (e.g., mutation P41L, P62V, P70R, or P215Y in Fig. 12b), when in reality there is insufficient evidence after conditioning on the selection event.