<figure><img src="image_69.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_71.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_72.png" /><figcaption>(d)</figcaption></figure>

**Figure O.** Birth-hybridisation model with speciation rate 20 and hybridisation rate 1 (mean number of reticulations close to zero) and a normal prior with mean 0.1 and standard deviation of 0.01 on the origin height. We plot the simulated networks (orange) against the sampled networks (blue) summarising the networks under: (a) Number of reticulations (b) Time until first reticulation (c) Height of the network (d) Length of the network.

## 8.1.2 Results

In the first experiment the sampler converged to the specified prior in all three cases (for both origin height priors) based on the computed summary statistics (see Figs O-Q and Figs R-T). The convergence of the sampler in all cases is a good indication that the implemented moves worked well enough. The ESS for the sampled networks given the normal prior on the network origin were: 1001 for the first case (mean number of reticulation close to zero); 844 for the second case (mean number of reticulations close to one); 1001 for the third case (mean number of reticulations close to two). The ESS for the sampled networks given the exponential prior on the network origin were: 872 for the first case; 955 for the second case; 838 for the third case.

Note that the normal and exponential priors on the origin height permit to describe different knowledge on the expected number of reticulations, see Figs Q(a) and T(a).
<figure><img src="image_73.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_74.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_75.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_76.png" /><figcaption>(d)</figcaption></figure>

**Figure P.** Birth-hybridisation model with speciation rate 20 and hybridisation rate 2 (mean number of reticulations close to one) and normal prior with mean 0.1 and standard deviation of 0.01 on the origin height. We plot the simulated networks (orange) against the sampled networks (blue) summarising the networks under: (a) Number of reticulations (b) Time until first reticulation (c) Height of the network (d) Length of the network.
<figure><img src="image_77.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_78.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_79.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_80.png" /><figcaption>(d)</figcaption></figure>

**Figure Q.** Birth-hybridisation model with speciation rate 20 and hybridisation rate 3 (mean number of reticulations close to two) and normal prior with mean 0.1 and standard deviation of 0.01 on the origin height. We plot the simulated networks (orange) against the sampled networks (blue) summarising the networks under: (a) Number of reticulations (b) Time until first reticulation (c) Height of the network (d) Length of the network.
<figure><img src="image_81.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_82.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_83.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_84.png" /><figcaption>(d)</figcaption></figure>

**Figure R.** Birth-hybridisation model with speciation rate 20 and hybridisation rate 1 (mean number of reticulations close to zero) and an exponential prior with mean 0.1 on the origin height. We plot the simulated networks (orange) against the sampled networks (blue) summarising the networks under: (a) Number of reticulations (b) Time until first reticulation (c) Height of the network (d) Length of the network.
<figure><img src="image_85.png" /><figcaption>**Figure S.** Birth-hybridisation model with speciation rate 20 and hybridisation rate 2 (mean number of reticulations close to one) and an exponential prior with mean 0.1 on the origin height. We plot the simulated networks (orange) against the sampled networks (blue) summarising the networks under: (a) Number of reticulations (b) Time until first reticulation (c) Height of the network (d) Length of the network.</figcaption></figure>
<figure><img src="image_86.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_87.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_88.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_89.png" /><figcaption>(d)</figcaption></figure>

**Figure T.** Birth-hybridisation model with speciation rate 20 and hybridisation rate 3 (mean number of reticulations close to two) and an exponential prior with mean 0.1 on the origin height. We plot the simulated networks (orange) against the sampled networks (blue) summarising the networks under: (a) Number of reticulations (b) Time until first reticulation (c) Height of the network (d) Length of the network.
<table><thead><tr><th>#Chain</th><th>Network prior</th><th>Mean reticulations</th><th>Pop size prior</th></tr></thead><tbody><tr><td>1,2</td><td>BH(20,1)</td><td>0.371</td><td>Γ(1, 200)</td></tr><tr><td>3,4</td><td>BH(20,1)</td><td>0.371</td><td>Γ(1, 20)</td></tr><tr><td>5,6</td><td>BH(20,1)</td><td>0.371</td><td>Γ(1, 1000)</td></tr><tr><td>7,8</td><td>BH(20,1)</td><td>0.371</td><td>Γ(1, 2000)</td></tr><tr><td>9,10</td><td>BH(20,2)</td><td>0.861</td><td>Γ(1, 200)</td></tr><tr><td>11,12</td><td>BH(20,2)</td><td>0.861</td><td>Γ(1, 20)</td></tr><tr><td>13,14</td><td>BH(20,2)</td><td>0.861</td><td>Γ(1, 1000)</td></tr><tr><td>15,16</td><td>BH(20,2)</td><td>0.861</td><td>Γ(1, 2000)</td></tr><tr><td>17,18</td><td>BH(20,3)</td><td>2.265</td><td>Γ(1, 200)</td></tr><tr><td>19,20</td><td>BH(20,3)</td><td>2.265</td><td>Γ(1, 20)</td></tr><tr><td>21,22</td><td>BH(20,3)</td><td>2.265</td><td>Γ(1, 1000)</td></tr><tr><td>23,24</td><td>BH(20,3)</td><td>2.265</td><td>Γ(1, 2000)</td></tr></tbody></table>

**Table G.** BH(birth rate, hybridisation rate) refers to the birth-hybridisation process of Zhang et al. with the specified birth and hybridisation rates. For data simulated with network A, only chains 1,2,3,4,9,10,11,12,17,18,19,20 were run. We indicate the mean number of reticulation for the Birth-Hybridization model given an exponential prior with mean 0.1 on network origin. Note that we only used the exponential prior in the experiment in Section 8.2.

## 8.2 Experiments on 10,000 simulated sites

### 8.2.1 Protocol

In the second experiment we assess how population size priors and network priors influence SNAPPNET's inferences, in particular the rate of convergence and sampling efficiency of the MCMC sampler. Recall that the network prior specifies a hybridization rate, whereas the prior on population sizes affects the probability of coalescence, and therefore that of ILS. Thus, these two priors have an important role in determining the relative probability of hybridization and ILS as causes of incongruent (non-tree-like) signals in the data.

We simulated 10,000 SNPs for network A and network B under the multispecies network coalescent using SIMSNAPPNET. For each of these two simulated SNP datasets, we ran 12 (for network A) or 24 (for network B) MCMC chains, for 500,000 iterations each. See Table G for details on the priors specified for each chain. In this experiment we only use the exponential prior with mean 0.1 on the network origin.

Briefly, as in the experiment of Sec. 8.1, we specified a network prior using the birth-hybridisation model of Zhang et al. [7]. Again, we fixed the birth rate to 20 for all MCMC chains and chose a hybridisation rate so that the mean number of reticulations is close to zero, one or two. Furthermore we specified either a 'correct' or 'incorrect' prior on population size ('correct' implies the mean of the prior distribution corresponds to the population size parameter used to simulate the SNP dataset). The 'correct' population size prior on each
branch was specified as Γ(1, 200). The 'incorrect' population size prior on each branch was specified as Γ(1, 20). For network B we considered two additional incorrect population size priors, namely Γ(1, 1000) and Γ(1, 2000). Note that the rest of the priors of the model used the default SNAPPNET settings. In order to assess convergence we ran two MCMC chains for each prior setting (as specified in Table G). We randomly drew initial networks and population sizes for each MCMC chain from the prior distribution. Also note that, here we do not impose any upper bound on the number of reticulations in the sampled networks.

### 8.2.2 Results for network A

We summarize results for data simulated under network A in Fig U (MCMC chains with correct population size priors) and Fig V (MCMC chains with incorrect population size priors). We also give detailed summary statistics in Table H (MCMC chains with correct population size priors) and Table I (MCMC chains with incorrect population size priors). We note that all chains with correct population size priors converged to the correct topology, network height and network length (see Figs U(c) and U(d)). We assume convergence for network topology since there was only one unique topology for each posterior distribution of the chains with correct population size priors. In each case the unique topology matched up with the topology of network A. Furthermore in Fig U(b) all chains have similar prior distributions. This could be due to the topology of network A that is very unlikely under all the specified birth-hybridization model priors (similar to sampling from a flat prior). We also note a much lower ESS under the model prior with reticulation mean close to zero (see ESS in Table H).

Chains with incorrect population size priors also converged to the correct topology. Similar to correct priors, there was only one unique topology for all chains. However the chains did not converge to the correct network height or network length. This is not unexpected since the length of a branch and its associated population size are correlated (see Bryant et al. [2] for more detail). Furthermore the ESS for chains with incorrect population size priors is significantly lower than chains with correct population size priors (see ESS in Table H and Table I). There is also a difference in ESS between chains with different topology priors. In this case ESS is highest when the mean number of reticulations on the network topology prior is close to one and lowest when mean number of reticulations for the network topology prior is close to two. This seems to suggest that specifying a prior with correct mean number of reticulations can improve sampling efficiency.

<table><caption>**Table H.** MCMC summary statistics for network A (correct population size priors)</caption><thead><tr><th>Posterior</th><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><td>mean</td><td>-16124.9199</td><td>-16123.2496</td><td>-16123.8596</td></tr><tr><td>stdev</td><td>4.7795</td><td>4.9024</td><td>5.0388</td></tr><tr><td>median</td><td>-16124.601</td><td>-16122.7308</td><td>-16123.2257</td></tr></tbody></table>
<table><caption>Table I. MCMC summary statistics for network A (incorrect priors)</caption><tr><td>95% HPD Interval</td><td>[-16134.1, -16116.4]</td><td>[-16133.4, -16114.9]</td><td>[-16133.0, -16114.7]</td></tr><tr><td>Auto-correlation time</td><td>4537.3247</td><td>1336.3834</td><td>1454.4489</td></tr><tr><td>Effective sample size</td><td>198.5752</td><td>674.2077</td><td>619.4786</td></tr><tr><td><strong>Network height</strong></td><td>0</td><td>1</td><td>2</td></tr><tr><td>mean</td><td>0.076</td><td>0.0769</td><td>0.0767</td></tr><tr><td>stdev</td><td>5.12E-03</td><td>4.09E-03</td><td>4.34E-03</td></tr><tr><td>median</td><td>0.0768</td><td>0.0772</td><td>0.0774</td></tr><tr><td>95% HPD Interval</td><td>[0.0647, 0.0854]</td><td>[0.0686, 0.0844]</td><td>[0.0682, 0.0847]</td></tr><tr><td>Auto-correlation time</td><td>10376.3331</td><td>3552.1422</td><td>3800.8964</td></tr><tr><td>Effective sample size</td><td>86.8322</td><td>253.6498</td><td>237.0493</td></tr><tr><td><strong>Network length</strong></td><td>0</td><td>1</td><td>2</td></tr><tr><td>mean</td><td>0.213</td><td>0.2144</td><td>0.2137</td></tr><tr><td>stdev</td><td>0.0124</td><td>9.93E-03</td><td>0.0107</td></tr><tr><td>variance</td><td>1.54E-04</td><td>9.87E-05</td><td>1.14E-04</td></tr><tr><td>95% HPD Interval</td><td>[0.1882, 0.2369]</td><td>[0.1967, 0.2357]</td><td>[0.1934, 0.2344]</td></tr><tr><td>Auto-correlation time</td><td>7533.4457</td><td>3113.3204</td><td>3818.75</td></tr><tr><td>Effective sample size</td><td>119.6</td><td>289.4016</td><td>235.9411</td></tr></table>

<table><thead><tr><th>Posterior</th><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><td>mean</td><td>-15953.476</td><td>-15917.2941</td><td>-15917.7676</td></tr><tr><td>stdev</td><td>5.1764</td><td>7.0277</td><td>6.7969</td></tr><tr><td>median</td><td>-15952.9488</td><td>-15916.1712</td><td>-15917.1603</td></tr><tr><td>95% HPD Interval</td><td>[-15962.4, -15943.6]</td><td>[-15932.0, -15905.4]</td><td>[-15930.3, -15905.5]</td></tr><tr><td>Auto-correlation time</td><td>2395.3008</td><td>2164.8158</td><td>3002.7331</td></tr><tr><td>Effective sample size</td><td>167.4111</td><td>185.2352</td><td>133.545</td></tr><tr><td><strong>Network height</strong></td><td>0</td><td>1</td><td>2</td></tr><tr><td>mean</td><td>0.0548</td><td>0.0588</td><td>0.047</td></tr><tr><td>stdev</td><td>9.30E-03</td><td>0.0121</td><td>0.0112</td></tr><tr><td>median</td><td>0.0548</td><td>0.06</td><td>0.0476</td></tr><tr><td>95% HPD Interval</td><td>[0.0394, 0.073]</td><td>[0.0383, 0.0791]</td><td>[0.025, 0.0662]</td></tr><tr><td>Auto-correlation time</td><td>16248.1423</td><td>1.00E+05</td><td>34725.0867</td></tr><tr><td>Effective sample size</td><td>24.6797</td><td>4.0042</td><td>11.5478</td></tr><tr><td><strong>Network length</strong></td><td>0</td><td>1</td><td>2</td></tr><tr><td>mean</td><td>0.1655</td><td>0.1783</td><td>0.156</td></tr><tr><td>stdev</td><td>0.0206</td><td>0.0252</td><td>0.0255</td></tr><tr><td>median</td><td>0.1649</td><td>0.1805</td><td>0.158</td></tr><tr><td>95% HPD Interval</td><td>[0.1264, 0.2024]</td><td>[0.1331, 0.2263]</td><td>[0.1064, 0.201]</td></tr><tr><td>Auto-correlation time</td><td>11802.0222</td><td>74214.3724</td><td>27882.5047</td></tr><tr><td>Effective sample size</td><td>33.9772</td><td>5.4033</td><td>14.3818</td></tr></tbody></table>
<figure><img src="image_90.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_91.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_92.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_93.png" /><figcaption>(d)</figcaption></figure>

**Figure U.** Summary distributions of all chains with correct population size priors (chain numbers 1,2,9,10,17,18) given data simulated from network A. We summarize the MCMC chains by combining them, that is: Chains 1 and 2 are indicated by the blue line (mean reticulations close to zero); Chains 9 and 10 are indicated by the orange line (mean reticulations close to one); Chains 17 and 18 are indicated by the green line (mean reticulations close to two); We plot the following distributions (a) Likelihood (b) Prior (c) Network height (d) Network length. Note that network height and network length used to simulate data are indicated by red lines.
<figure><img src="image_94.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_95.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_96.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_97.png" /><figcaption>(d)</figcaption></figure>

**Figure V.** Summary distributions of all chains with incorrect population size priors Gamma(1,20) (chain numbers 3,4,11,12,19,20) given data simulated from network A. We summarize the MCMC chains by combining them, that is: Chains 3 and 4 are indicated by the blue line (mean reticulations close to zero); Chains 11 and 12 are indicated by the orange line (mean reticulations close to one); Chains 19 and 20 are indicated by the green line (mean reticulations close to two); We plot the following distributions (a) Likelihood (b) Prior (c) Network height (d) Network length. Note that network height and network length used to simulate data are indicated by red lines.
### 8.2.3 Results for network B

We summarize results for data simulated under network B in Fig W (MCMC chains with correct population size priors), Fig X (MCMC chains with incorrect population size priors $\Gamma(1, 20)$), Fig Y (MCMC chains with incorrect population size priors $\Gamma(1, 1000)$) and Fig Z (MCMC chains with incorrect population size priors $\Gamma(1, 2000)$). We also give detailed summary statistics in Table J (MCMC chains with correct population size priors), Table K (MCMC chains with incorrect population size priors $\Gamma(1, 20)$), Table L (MCMC chains with incorrect population size priors $\Gamma(1, 1000)$) and Table M (MCMC chains with incorrect population size priors $\Gamma(1, 2000)$). We note that all chains with correct population size priors converged to the correct topology (posterior distribution contained only one network topology), network height and network length (see Figs W(c) and W(d)). Chains with incorrect population size priors also converged to the correct topology in most cases except for two cases: $\{\text{BH}(20,1),\Gamma(1, 1000)\}$ and $\{\text{BH}(20,1),\Gamma(1, 2000)\}$. Therefore we were able to recover the correct topology 83.33% of the time. This is consistent with results in the simulation study of the main text. There is also a difference in ESS between chains with different topology priors. However in this case it is not clear how the prior affects the sampling efficiency (see ESS of Posterior distribution in Table J, Table K, Table L and Table M).

<table><caption>**Table J.** MCMC summary statistics for Network B (correct population size priors)</caption><thead><tr><th>Posterior</th><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><td>mean</td><td>-16693.3667</td><td>-16692.247</td><td>-16690.8344</td></tr><tr><td>stdev</td><td>10.9267</td><td>11.2221</td><td>7.5266</td></tr><tr><td>median</td><td>-16692.4881</td><td>-16691.493</td><td>-16690.2388</td></tr><tr><td>95% HPD Interval</td><td>[-16706., -16678.2155]</td><td>[-16705.0995, -1667389]</td><td>[-16705., -16676.7232]</td></tr><tr><td>Auto-correlation time</td><td>1000</td><td>1032</td><td>1138.8635</td></tr><tr><td>Effective sample size</td><td>501</td><td>489</td><td>439.9122</td></tr><tr><td>Network height</td><td>0</td><td>1</td><td>2</td></tr><tr><td>mean</td><td>0.0793</td><td>0.0796</td><td>0.0797</td></tr><tr><td>stdev</td><td>4.08E-03</td><td>3.71E-03</td><td>3.37E-03</td></tr><tr><td>median</td><td>0.0799</td><td>0.0797</td><td>0.08</td></tr><tr><td>95% HPD Interval</td><td>[0.0714, 0.086]</td><td>[0.072, 0.0858]</td><td>[0.0729, 0.0852]</td></tr><tr><td>Auto-correlation time</td><td>4047.2001</td><td>2842.2081</td><td>2047.1454</td></tr><tr><td>Effective sample size</td><td>123.7893</td><td>176.2714</td><td>244.731</td></tr><tr><td>Network length</td><td>0</td><td>1</td><td>2</td></tr><tr><td>mean</td><td>0.2421</td><td>0.2412</td><td>0.2412</td></tr><tr><td>stdev</td><td>0.0106</td><td>0.0106</td><td>9.82E-03</td></tr><tr><td>median</td><td>0.2431</td><td>0.2417</td><td>0.2418</td></tr><tr><td>95% HPD Interval</td><td>[0.2223, 0.2621]</td><td>[0.2229, 0.2625]</td><td>[0.2193, 0.2588]</td></tr><tr><td>Auto-correlation time</td><td>3262.683</td><td>2788.3608</td><td>1948.3473</td></tr><tr><td>Effective sample size</td><td>153.5546</td><td>179.6755</td><td>257.141</td></tr></tbody></table>
<figure><img src="image_98.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_99.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_100.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_101.png" /><figcaption>(d)</figcaption></figure>

**Figure W.** Summary distributions of all chains with correct population size priors (chain numbers 1,2,9,10,17,18 given data simulated under network B. We summarize the MCMC chains by combining them, that is: Chains 1 and 2 are indicated by the blue line (mean reticulations close to zero); Chains 9 and 10 are indicated by the orange line (mean reticulations close to one); Chains 17 and 18 are indicated by the green line (mean reticulations close to two); We plot the following distributions (a) Likelihood (b) Prior (c) Network height (d) Network length. Note that network height and network length used to simulate data are indicated by red lines.
<figure><img src="image_102.png" /></figure>

(a)

<figure><img src="image_103.png" /></figure>

(b)

<figure><img src="image_104.png" /></figure>

(c)

<figure><img src="image_105.png" /></figure>

(d)

**Figure X.** Summary distributions of all chains with incorrect population size priors (chain numbers 3,4,7,8,11,12) given data simulated from network B. We summarize the MCMC chains by combining them, that is: Chains 3 and 4 are indicated by blue line (mean reticulations close to zero); Chains 7 and 8 are indicated by orange line (mean reticulations close to one); Chains 11 and 12 are indicated by green line (mean reticulations close to two); We plot the following distributions (a) Likelihood (b) Prior (c) Network height (d) Network length. Note that network height and network length used to simulate data are indicated by red lines.
<figure><img src="image_106.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_107.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_108.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_109.png" /><figcaption>(d)</figcaption></figure>

**Figure Y.** In this figure we plot summary distributions of all chains with incorrect population size priors Gamma(1,20) (chain numbers 5,6,13,14,21,22) given data simulated from Network B. We summarize the MCMC chains by combining them, that is: Chains 5 and 6 are indicated by blue line (mean reticulations close to zero); Chains 13 and 14 are indicated by orange line (mean reticulations close to one); Chains 21 and 22 are indicated by green line (mean reticulations close to two); We plot the following distributions (a) Likelihood (b) Prior (c) Network height (d) Network length. Note that network height and network length used to simulate data are indicated by red lines.
<figure><img src="image_110.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_111.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_112.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_113.png" /><figcaption>(d)</figcaption></figure>

**Figure Z.** In this figure we plot summary distributions of all chains with incorrect population size priors (chain numbers 7,8,15,16,23,24) given data simulated from network B. We summarize the MCMC chains by combining them, that is: Chain 7 and 8 are indicated by blue line (mean reticulations close to zero); Chain 15 and 16 is indicated by orange line (mean reticulations close to one); Chain 23 and 24 are indicated by green line (mean reticulations close to two); We plot the following distributions (a) Likelihood (b) Prior (c) Network height (d) Network length. Note that network height and network length used to simulate data are indicated by red lines.
<table><caption>**Table K.** MCMC summary statistics for Network B (incorrect population size priors Gamma(1,20))</caption><thead><tr><th></th><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><th colspan="4">Posterior</th></tr><tr><td>mean</td><td>-16632.0842</td><td>-16630.1919</td><td>-16629.0074</td></tr><tr><td>stdev</td><td>8.0517</td><td>8.0343</td><td>8.3013</td></tr><tr><td>median</td><td>-16631.0958</td><td>-16629.7991</td><td>-16627.8702</td></tr><tr><td>95% HPD Interval</td><td>[-16647.6, -16617.2]</td><td>[-16646.3, -16615.2]</td><td>[-16644.3, -16613.8]</td></tr><tr><td>Auto-correlation time</td><td>3035.0395</td><td>2302.8325</td><td>1819.3308</td></tr><tr><td>Effective sample size</td><td>132.1235</td><td>174.1334</td><td>220.4107</td></tr><tr><th colspan="4">Network height</th></tr><tr><td>mean</td><td>0.0639</td><td>0.0568</td><td>0.0578</td></tr><tr><td>stdev</td><td>0.011</td><td>0.0119</td><td>0.0117</td></tr><tr><td>median</td><td>0.0645</td><td>0.0558</td><td>0.0571</td></tr><tr><td>95% HPD Interval</td><td>[0.044, 0.0838]</td><td>[0.039, 0.083]</td><td>[0.0392, 0.0789]</td></tr><tr><td>Auto-correlation time</td><td>25480.1793</td><td>59294.2576</td><td>31555.2738</td></tr><tr><td>Effective sample size</td><td>15.7377</td><td>6.7629</td><td>12.7079</td></tr><tr><th colspan="4">Network length</th></tr><tr><td>mean</td><td>0.1958</td><td>0.1826</td><td>0.1831</td></tr><tr><td>stdev</td><td>0.024</td><td>0.0253</td><td>0.0255</td></tr><tr><td>median</td><td>0.196</td><td>0.1836</td><td>0.1827</td></tr><tr><td>95% HPD Interval</td><td>[0.1543, 0.2412]</td><td>[0.1398, 0.2311]</td><td>[0.1378, 0.2273]</td></tr><tr><td>Auto-correlation time</td><td>19208.3139</td><td>36324.2679</td><td>23849.9552</td></tr><tr><td>Effective sample size</td><td>20.8764</td><td>11.0395</td><td>16.8134</td></tr></tbody></table>

<table><caption>**Table L.** MCMC summary statistics for Network B (incorrect population size priors Gamma(1,1000))</caption><thead><tr><th></th><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><th colspan="4">Posterior</th></tr><tr><td>mean</td><td>-16836.9572</td><td>-16693.995</td><td>-16722.2755</td></tr><tr><td>stdev</td><td>6.9406</td><td>7.7519</td><td>7.5566</td></tr><tr><td>median</td><td>-16836.2496</td><td>-16693.5673</td><td>-16722.1075</td></tr><tr><td>95% HPD Interval</td><td>[-16850.7, -16824.8]</td><td>[-16709.2, -16679.9]</td><td>[-16738.3, -16709.7]</td></tr><tr><td>auto-correlation time (ACT)</td><td>1665.166</td><td>1018.3201</td><td>1230.9875</td></tr><tr><td>effective sample size (ESS)</td><td>240.8168</td><td>393.7858</td><td>325.7547</td></tr><tr><th colspan="4">Network height</th></tr><tr><td>mean</td><td>0.0726</td><td>0.0807</td><td>0.0811</td></tr><tr><td>stdev</td><td>1.88E-03</td><td>2.13E-03</td><td>2.20E-03</td></tr><tr><td>median</td><td>0.0724</td><td>0.0808</td><td>0.0811</td></tr><tr><td>95% HPD Interval</td><td>[0.0695, 0.0764]</td><td>[0.0768, 0.0847]</td><td>[0.0774, 0.0854]</td></tr><tr><td>auto-correlation time (ACT)</td><td>1000</td><td>1199.9843</td><td>1160.6579</td></tr><tr><td>effective sample size (ESS)</td><td>401</td><td>334.171</td><td>345.4937</td></tr><tr><th colspan="4">Network length</th></tr><tr><td>mean</td><td>0.2375</td><td>0.2511</td><td>0.2781</td></tr><tr><td>stdev</td><td>5.38E-03</td><td>7.50E-03</td><td>7.67E-03</td></tr><tr><td>median</td><td>0.2371</td><td>0.2509</td><td>0.2784</td></tr><tr><td>95% HPD Interval</td><td>[0.228, 0.2492]</td><td>[0.2374, 0.2664]</td><td>[0.2633, 0.2928]</td></tr></tbody></table>
<table><caption>**Table M.** MCMC summary statistics for Network B (incorrect population size priors Gamma(1,2000))</caption><tr><td>auto-correlation time (ACT)</td><td>1000</td><td>1221.2116</td><td>1413.5422</td></tr><tr><td>effective sample size (ESS)</td><td>401</td><td>328.3624</td><td>283.6845</td></tr></table>

<table><thead><tr><th></th><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><th colspan="4">Posterior</th></tr><tr><td>mean</td><td>-16852.5586</td><td>-16715.134</td><td>-16713.4537</td></tr><tr><td>stdev</td><td>6.4594</td><td>7.6484</td><td>7.6353</td></tr><tr><td>95% HPD Interval</td><td>[-16864.9, -16841.0]</td><td>[-16731.4, -16701.6]</td><td>[-16727.7, -16698.8]</td></tr><tr><td>auto-correlation time (ACT)</td><td>1216.2537</td><td>1762.3287</td><td>1120.84</td></tr><tr><td>effective sample size (ESS)</td><td>329.7009</td><td>227.5398</td><td>357.7674</td></tr><tr><th colspan="4">Network height</th></tr><tr><td>mean</td><td>0.0729</td><td>0.0813</td><td>0.0811</td></tr><tr><td>stdev</td><td>1.92E-03</td><td>2.25E-03</td><td>2.10E-03</td></tr><tr><td>95% HPD Interval</td><td>[0.0697, 0.0769]</td><td>[0.0767, 0.0855]</td><td>[0.0771, 0.0852]</td></tr><tr><td>auto-correlation time (ACT)</td><td>1455.1062</td><td>1073.7888</td><td>1425.0583</td></tr><tr><td>effective sample size (ESS)</td><td>275.5813</td><td>373.444</td><td>281.392</td></tr><tr><th colspan="4">Network length</th></tr><tr><td>mean</td><td>0.2386</td><td>0.2539</td><td>0.2524</td></tr><tr><td>stdev</td><td>5.61E-03</td><td>7.39E-03</td><td>6.74E-03</td></tr><tr><td>variance</td><td>3.14E-05</td><td>5.46E-05</td><td>4.55E-05</td></tr><tr><td>95% HPD Interval</td><td>[0.2284, 0.2493]</td><td>[0.239, 0.2668]</td><td>[0.239, 0.2653]</td></tr><tr><td>auto-correlation time (ACT)</td><td>1519.2302</td><td>1462.2703</td><td>1020.1866</td></tr><tr><td>effective sample size (ESS)</td><td>263.9495</td><td>274.2311</td><td>393.0654</td></tr></tbody></table>
### 8.2.4 Operator acceptance rates

To better understand the behavior of the MCMC sampler, we inspect the acceptance rates for the 5 operators acting on the network topology (*AddReticulation*, *DeleteReticulation*, *FlipReticulation*, *RelocateBranch*, *RelocateBranch-Narrow*), the 4 operators updating branch lengths (*NodeSlider*, *NodeUniform*, *NetworkMultiplier*, *OriginMultiplier*) and the 2 operators updating population sizes (*ChangeGamma*, *ChangeAllGamma*).

We summarize the acceptance rates for network B in Table N, Table O and Table P. Each table focuses on a different population size prior, while averaging across the topology priors.

We observe that MCMC moves that update topology have a much lower acceptance rate than MCMC moves that update branch lengths and population sizes. *FlipReticulation* moves, which flip the direction of a reticulation branch, are the least likely to be accepted. There is no clear difference in the acceptance rates between different population size priors. More work is needed to determine what the proposal weights should be in order to optimally sample from the posterior distribution.

<table><caption>**Table N.** MCMC acceptance rates for Network B (correct population size priors).</caption><thead><tr><th>Id</th><th>Pr_accept</th><th>Pr_proposed</th><th>Pr_accept</th></tr></thead><tbody><tr><td><strong>Topology moves</strong></td><td></td><td></td><td></td></tr><tr><td>AddReticulation</td><td>1.43E-04</td><td>2.32E-02</td><td>3.31E-06</td></tr><tr><td>DeleteReticulation</td><td>4.55E-05</td><td>2.32E-02</td><td>1.06E-06</td></tr><tr><td>FlipReticulation</td><td>8.05E-06</td><td>2.35E-02</td><td>1.89E-07</td></tr><tr><td>RelocateBranch</td><td>3.07E-02</td><td>2.34E-02</td><td>7.20E-04</td></tr><tr><td>RelocateBranchNarrow</td><td>1.81E-03</td><td>2.33E-02</td><td>4.21E-05</td></tr><tr><td><strong>Branch length</strong></td><td></td><td></td><td></td></tr><tr><td>NodeSlider</td><td>5.29E-01</td><td>2.32E-02</td><td>1.23E-02</td></tr><tr><td>NodeUniform</td><td>2.73E-01</td><td>2.32E-02</td><td>6.33E-03</td></tr><tr><td>NetworkMultiplier</td><td>2.92E-01</td><td>1.15E-02</td><td>3.36E-03</td></tr><tr><td>OriginMultiplier</td><td>7.50E-01</td><td>1.18E-02</td><td>8.86E-03</td></tr><tr><td><strong>Population size</strong></td><td></td><td></td><td></td></tr><tr><td>ChangeGamma</td><td>3.16E-01</td><td>3.49E-01</td><td>1.10E-01</td></tr><tr><td>ChangeAllGamma</td><td>2.80E-01</td><td>3.48E-01</td><td>9.75E-02</td></tr></tbody></table>

<table><caption>**Table O.** MCMC acceptance rates for Network B (incorrect population size priors $\Gamma(1, 1000)$).</caption><thead><tr><th>Id</th><th>Pr_accept</th><th>Pr_proposed</th><th>Pr_accept</th></tr></thead><tbody><tr><td><strong>Topology moves</strong></td><td></td><td></td><td></td></tr><tr><td>AddReticulation</td><td>1.52E-04</td><td>2.33E-02</td><td>3.55E-06</td></tr><tr><td>DeleteReticulation</td><td>6.72E-05</td><td>2.32E-02</td><td>1.56E-06</td></tr><tr><td>FlipReticulation</td><td>1.44E-05</td><td>2.33E-02</td><td>3.33E-07</td></tr></tbody></table>
<table><caption>**Table P.** MCMC acceptance rates for Network B (incorrect population size priors $\Gamma(1, 2000)$).</caption><tr><td>RelocateBranch</td><td>2.59E-02</td><td>2.33E-02</td><td>6.02E-04</td></tr><tr><td>RelocateBranchNarrow</td><td>6.01E-04</td><td>2.34E-02</td><td>1.41E-05</td></tr><tr><td><strong>Branch length</strong></td><td></td><td></td><td></td></tr><tr><td>NodeSlider</td><td>5.20E-01</td><td>2.33E-02</td><td>1.21E-02</td></tr><tr><td>NodeUniform</td><td>2.68E-01</td><td>2.32E-02</td><td>6.22E-03</td></tr><tr><td>NetworkMultiplier</td><td>2.58E-01</td><td>1.15E-02</td><td>2.97E-03</td></tr><tr><td>OriginMultiplier</td><td>7.42E-01</td><td>1.17E-02</td><td>8.67E-03</td></tr><tr><td><strong>Population size</strong></td><td></td><td></td><td></td></tr><tr><td>ChangeGamma</td><td>3.36E-01</td><td>3.49E-01</td><td>1.17E-01</td></tr><tr><td>ChangeAllGamma</td><td>3.15E-01</td><td>3.49E-01</td><td>1.10E-01</td></tr></table>

<table><thead><tr><th>Id</th><th>Pr_accept</th><th>Pr_proposed</th><th>Pr_proposed</th><th>Pr_accept</th></tr></thead><tbody><tr><td><strong>Topology moves</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>AddReticulation</td><td>1.14E-04</td><td></td><td>2.32E-02</td><td>2.64E-06</td></tr><tr><td>DeleteReticulation</td><td>3.25E-05</td><td></td><td>2.32E-02</td><td>7.54E-07</td></tr><tr><td>FlipReticulation</td><td>2.43E-05</td><td></td><td>2.33E-02</td><td>5.68E-07</td></tr><tr><td>RelocateBranch</td><td>2.71E-02</td><td></td><td>2.33E-02</td><td>6.31E-04</td></tr><tr><td>RelocateBranchNarrow</td><td>5.49E-04</td><td></td><td>2.33E-02</td><td>1.28E-05</td></tr><tr><td><strong>Branch length</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>NodeSlider</td><td>5.11E-01</td><td></td><td>2.33E-02</td><td>1.19E-02</td></tr><tr><td>NodeUniform</td><td>2.53E-01</td><td></td><td>2.32E-02</td><td>5.86E-03</td></tr><tr><td>NetworkMultiplier</td><td>2.65E-01</td><td></td><td>1.15E-02</td><td>3.04E-03</td></tr><tr><td>OriginMultiplier</td><td>7.46E-01</td><td></td><td>1.18E-02</td><td>8.77E-03</td></tr><tr><td><strong>Population size</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>ChangeGamma</td><td>3.69E-01</td><td></td><td>3.49E-01</td><td>1.29E-01</td></tr><tr><td>ChangeAllGamma</td><td>3.80E-01</td><td></td><td>3.49E-01</td><td>1.32E-01</td></tr></tbody></table>
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