Commit c1c8067f authored by Gilles Kratzer's avatar Gilles Kratzer
Browse files

update before CRAN

parent 163b3a8a
Pipeline #2479 passed with stage
in 2 seconds
No preview for this file type
This diff is collapsed.
......@@ -133,6 +133,7 @@
<p class="section-desc"></p>
<ul>
<li><a href="Articles/convergence.html">mcmcabn: a structural MCMC sampler for DAGs learned from observed systemic datasets</a></li>
<li><a href="mcmcabn.html">mcmcabn: a structural MCMC sampler for DAGs learned from observed systemic datasets</a></li>
</ul>
</div>
......
......@@ -226,57 +226,57 @@
<a class="sourceLine" id="cb9-5" data-line-number="5"><span class="co">#&gt; Thinning: 99</span></a>
<a class="sourceLine" id="cb9-6" data-line-number="6"><span class="co">#&gt; </span></a>
<a class="sourceLine" id="cb9-7" data-line-number="7"><span class="co">#&gt; Maximum score: -11151</span></a>
<a class="sourceLine" id="cb9-8" data-line-number="8"><span class="co">#&gt; Empirical mean: -11323</span></a>
<a class="sourceLine" id="cb9-9" data-line-number="9"><span class="co">#&gt; Empirical standard deviation: 337</span></a>
<a class="sourceLine" id="cb9-8" data-line-number="8"><span class="co">#&gt; Empirical mean: -11165</span></a>
<a class="sourceLine" id="cb9-9" data-line-number="9"><span class="co">#&gt; Empirical standard deviation: 7.41</span></a>
<a class="sourceLine" id="cb9-10" data-line-number="10"><span class="co">#&gt; Quantiles of the posterior network score:</span></a>
<a class="sourceLine" id="cb9-11" data-line-number="11"><span class="co">#&gt; 0.025 0.25 0.5 0.75 0.975</span></a>
<a class="sourceLine" id="cb9-12" data-line-number="12"><span class="co">#&gt; BN score -12350 -11279 -11181 -11156 -11151</span></a>
<a class="sourceLine" id="cb9-12" data-line-number="12"><span class="co">#&gt; BN score -11189 -11168 -11164 -11160 -11154</span></a>
<a class="sourceLine" id="cb9-13" data-line-number="13"><span class="co">#&gt; </span></a>
<a class="sourceLine" id="cb9-14" data-line-number="14"><span class="co">#&gt; </span></a>
<a class="sourceLine" id="cb9-15" data-line-number="15"><span class="co">#&gt; Global acceptance rate: 0.164</span></a>
<a class="sourceLine" id="cb9-15" data-line-number="15"><span class="co">#&gt; Global acceptance rate: 0.205</span></a>
<a class="sourceLine" id="cb9-16" data-line-number="16"><span class="co">#&gt; Accepted Rejected</span></a>
<a class="sourceLine" id="cb9-17" data-line-number="17"><span class="co">#&gt; MBR 10 27</span></a>
<a class="sourceLine" id="cb9-18" data-line-number="18"><span class="co">#&gt; MC3 144 795</span></a>
<a class="sourceLine" id="cb9-19" data-line-number="19"><span class="co">#&gt; REV 10 15</span></a>
<a class="sourceLine" id="cb9-17" data-line-number="17"><span class="co">#&gt; MBR 9 24</span></a>
<a class="sourceLine" id="cb9-18" data-line-number="18"><span class="co">#&gt; MC3 186 752</span></a>
<a class="sourceLine" id="cb9-19" data-line-number="19"><span class="co">#&gt; REV 10 20</span></a>
<a class="sourceLine" id="cb9-20" data-line-number="20"><span class="co">#&gt; </span></a>
<a class="sourceLine" id="cb9-21" data-line-number="21"><span class="co">#&gt; </span></a>
<a class="sourceLine" id="cb9-22" data-line-number="22"><span class="co">#&gt; Sample size adjusted for autocorrelation: 326</span></a>
<a class="sourceLine" id="cb9-22" data-line-number="22"><span class="co">#&gt; Sample size adjusted for autocorrelation: 51.6</span></a>
<a class="sourceLine" id="cb9-23" data-line-number="23"><span class="co">#&gt; </span></a>
<a class="sourceLine" id="cb9-24" data-line-number="24"><span class="co">#&gt; Autocorrelations by lag:</span></a>
<a class="sourceLine" id="cb9-25" data-line-number="25"><span class="co">#&gt; 0 1 2 3 4 5 6 7 8 9 10</span></a>
<a class="sourceLine" id="cb9-26" data-line-number="26"><span class="co">#&gt; acf 1 0.434 0.214 0.155 0.109 0.0889 0.0713 0.0129 0.0195 -0.026 0.0185</span></a></code></pre></div>
<a class="sourceLine" id="cb9-25" data-line-number="25"><span class="co">#&gt; 0 1 2 3 4 5 6 7 8 9 10</span></a>
<a class="sourceLine" id="cb9-26" data-line-number="26"><span class="co">#&gt; acf 1 0.568 0.562 0.523 0.478 0.459 0.457 0.419 0.402 0.37 0.335</span></a></code></pre></div>
<p>This function return some MCMC diagnostics and descriptive metrics.</p>
<p>The main advantage of computing MCMC samples from the posterior is that we can structuraly query them. Hence we account for the uncertainty in the structures through Bayesian model averaging. In practice, we compute the probabilities of the arcs in the DAG or the probability of a given part of a structure. Below is displayed the individual frequencies of the arcs.</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb10-1" data-line-number="1"><span class="kw"><a href="../reference/query.html">query</a></span>(<span class="dt">mcmcabn =</span> mcmc.out.asia)</a>
<a class="sourceLine" id="cb10-2" data-line-number="2"><span class="co">#&gt; Asia Smoking Tuberculosis LungCancer Bronchitis Either</span></a>
<a class="sourceLine" id="cb10-3" data-line-number="3"><span class="co">#&gt; Asia 0.00000 0.00799 0.017 0.00599 0.00799 0.00799</span></a>
<a class="sourceLine" id="cb10-4" data-line-number="4"><span class="co">#&gt; Smoking 0.01499 0.00000 0.033 0.41658 0.43457 0.09990</span></a>
<a class="sourceLine" id="cb10-5" data-line-number="5"><span class="co">#&gt; Tuberculosis 0.02498 0.03197 0.000 0.09990 0.00699 0.13886</span></a>
<a class="sourceLine" id="cb10-6" data-line-number="6"><span class="co">#&gt; LungCancer 0.01299 0.39760 0.158 0.00000 0.05794 0.19680</span></a>
<a class="sourceLine" id="cb10-7" data-line-number="7"><span class="co">#&gt; Bronchitis 0.00999 0.48452 0.019 0.10290 0.00000 0.08891</span></a>
<a class="sourceLine" id="cb10-8" data-line-number="8"><span class="co">#&gt; Either 0.02098 0.08591 0.752 0.74326 0.01898 0.00000</span></a>
<a class="sourceLine" id="cb10-9" data-line-number="9"><span class="co">#&gt; XRay 0.00999 0.02398 0.341 0.28871 0.00599 0.63137</span></a>
<a class="sourceLine" id="cb10-10" data-line-number="10"><span class="co">#&gt; Dyspnea 0.01099 0.09690 0.188 0.29570 0.67133 0.40959</span></a>
<a class="sourceLine" id="cb10-11" data-line-number="11"><span class="co">#&gt; XRay Dyspnea</span></a>
<a class="sourceLine" id="cb10-12" data-line-number="12"><span class="co">#&gt; Asia 0.00899 0.00599</span></a>
<a class="sourceLine" id="cb10-13" data-line-number="13"><span class="co">#&gt; Smoking 0.03297 0.06593</span></a>
<a class="sourceLine" id="cb10-14" data-line-number="14"><span class="co">#&gt; Tuberculosis 0.04895 0.20579</span></a>
<a class="sourceLine" id="cb10-15" data-line-number="15"><span class="co">#&gt; LungCancer 0.05794 0.13287</span></a>
<a class="sourceLine" id="cb10-16" data-line-number="16"><span class="co">#&gt; Bronchitis 0.04695 0.28372</span></a>
<a class="sourceLine" id="cb10-17" data-line-number="17"><span class="co">#&gt; Either 0.10889 0.04995</span></a>
<a class="sourceLine" id="cb10-18" data-line-number="18"><span class="co">#&gt; XRay 0.00000 0.00899</span></a>
<a class="sourceLine" id="cb10-19" data-line-number="19"><span class="co">#&gt; Dyspnea 0.07093 0.00000</span></a></code></pre></div>
<a class="sourceLine" id="cb10-2" data-line-number="2"><span class="co">#&gt; Asia Smoking Tuberculosis LungCancer Bronchitis Either</span></a>
<a class="sourceLine" id="cb10-3" data-line-number="3"><span class="co">#&gt; Asia 0.0000 0.1698 0.1908 0.158 0.175 0.1229</span></a>
<a class="sourceLine" id="cb10-4" data-line-number="4"><span class="co">#&gt; Smoking 0.0829 0.0000 0.0799 0.440 0.424 0.0360</span></a>
<a class="sourceLine" id="cb10-5" data-line-number="5"><span class="co">#&gt; Tuberculosis 0.1319 0.1319 0.0000 0.277 0.123 0.1818</span></a>
<a class="sourceLine" id="cb10-6" data-line-number="6"><span class="co">#&gt; LungCancer 0.0649 0.5125 0.2198 0.000 0.146 0.1499</span></a>
<a class="sourceLine" id="cb10-7" data-line-number="7"><span class="co">#&gt; Bronchitis 0.0809 0.5764 0.1389 0.149 0.000 0.0539</span></a>
<a class="sourceLine" id="cb10-8" data-line-number="8"><span class="co">#&gt; Either 0.0170 0.0529 0.8182 0.850 0.022 0.0000</span></a>
<a class="sourceLine" id="cb10-9" data-line-number="9"><span class="co">#&gt; XRay 0.0649 0.0460 0.3047 0.271 0.032 0.7922</span></a>
<a class="sourceLine" id="cb10-10" data-line-number="10"><span class="co">#&gt; Dyspnea 0.0050 0.0000 0.0270 0.172 0.968 0.7552</span></a>
<a class="sourceLine" id="cb10-11" data-line-number="11"><span class="co">#&gt; XRay Dyspnea</span></a>
<a class="sourceLine" id="cb10-12" data-line-number="12"><span class="co">#&gt; Asia 0.1399 0.13786</span></a>
<a class="sourceLine" id="cb10-13" data-line-number="13"><span class="co">#&gt; Smoking 0.0230 0.00000</span></a>
<a class="sourceLine" id="cb10-14" data-line-number="14"><span class="co">#&gt; Tuberculosis 0.0390 0.18681</span></a>
<a class="sourceLine" id="cb10-15" data-line-number="15"><span class="co">#&gt; LungCancer 0.0440 0.03996</span></a>
<a class="sourceLine" id="cb10-16" data-line-number="16"><span class="co">#&gt; Bronchitis 0.0360 0.03197</span></a>
<a class="sourceLine" id="cb10-17" data-line-number="17"><span class="co">#&gt; Either 0.0889 0.00899</span></a>
<a class="sourceLine" id="cb10-18" data-line-number="18"><span class="co">#&gt; XRay 0.0000 0.03596</span></a>
<a class="sourceLine" id="cb10-19" data-line-number="19"><span class="co">#&gt; Dyspnea 0.0270 0.00000</span></a></code></pre></div>
<p>The <code><a href="../reference/query.html">query()</a></code> function takes formula statements. So it is possible to explicitly query: <strong>what is the probability of LungCancer node being children of the Smoking node?</strong></p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb11-1" data-line-number="1"><span class="kw"><a href="../reference/query.html">query</a></span>(<span class="dt">mcmcabn =</span> mcmc.out.asia, <span class="dt">formula =</span> <span class="op">~</span><span class="st"> </span>LungCancer<span class="op">|</span>Smoking)</a>
<a class="sourceLine" id="cb11-2" data-line-number="2"><span class="co">#&gt; [1] 0.398</span></a></code></pre></div>
<p>This means that an arrow from Smoking to LungCancer appears in 39.76% of the sampled DAGs.</p>
<a class="sourceLine" id="cb11-2" data-line-number="2"><span class="co">#&gt; [1] 0.512</span></a></code></pre></div>
<p>This means that an arrow from Smoking to LungCancer appears in 51.249% of the sampled DAGs.</p>
<p>One can also ask more complicated structural requests. If we want to know <strong>what is the probability of Smoking node being parent of both LungCancer and Bronchitis node?</strong></p>
<div class="sourceCode" id="cb12"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb12-1" data-line-number="1"><span class="kw"><a href="../reference/query.html">query</a></span>(<span class="dt">mcmcabn =</span> mcmc.out.asia, <span class="dt">formula =</span> <span class="op">~</span><span class="st"> </span>LungCancer<span class="op">|</span>Smoking <span class="op">+</span><span class="st"> </span>Bronchitis<span class="op">|</span>Smoking)</a>
<a class="sourceLine" id="cb12-2" data-line-number="2"><span class="co">#&gt; [1] 0.178</span></a></code></pre></div>
<p>It means that an arrow from Smoking to LungCancer <strong>and</strong> an arrow from Smoking to Bronchitis appears in 17.782% of the sampled DAGs. This probability cannot be read from the matrix above. Indeed, it is a structural statement.</p>
<a class="sourceLine" id="cb12-2" data-line-number="2"><span class="co">#&gt; [1] 0.271</span></a></code></pre></div>
<p>It means that an arrow from Smoking to LungCancer <strong>and</strong> an arrow from Smoking to Bronchitis appears in 27.073% of the sampled DAGs. This probability cannot be read from the matrix above. Indeed, it is a structural statement.</p>
<p>If one want to query positive and negative statements such as <strong>What is the probability of previous statement when there is no arc from Smoking to Tuberculosis and no arc from Bronchitis to XRay?</strong></p>
<div class="sourceCode" id="cb13"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb13-1" data-line-number="1"><span class="kw"><a href="../reference/query.html">query</a></span>(<span class="dt">mcmcabn =</span> mcmc.out.asia ,<span class="dt">formula =</span> <span class="op">~</span>LungCancer<span class="op">|</span>Smoking <span class="op">+</span><span class="st"> </span>Bronchitis<span class="op">|</span>Smoking <span class="op">-</span><span class="st"> </span>Tuberculosis<span class="op">|</span>Smoking <span class="op">-</span><span class="st"> </span>XRay<span class="op">|</span>Bronchitis)</a>
<a class="sourceLine" id="cb13-2" data-line-number="2"><span class="co">#&gt; [1] 0</span></a></code></pre></div>
<a class="sourceLine" id="cb13-2" data-line-number="2"><span class="co">#&gt; [1] 0.002</span></a></code></pre></div>
<p>So essentially zero!</p>
<div id="formula-statement-tutorial" class="section level3">
<h3 class="hasAnchor">
......
......@@ -2,6 +2,7 @@ pandoc: 2.3.1
pkgdown: 1.4.1
pkgdown_sha: ~
articles:
convergence: Articles/convergence.html
mcmcabn: mcmcabn.html
urls:
reference: https://www.math.uzh.ch/pages/mcmcabn//reference
......
docs/reference/plot-1.png

145 KB | W: | H:

docs/reference/plot-1.png

179 KB | W: | H:

docs/reference/plot-1.png
docs/reference/plot-1.png
docs/reference/plot-1.png
docs/reference/plot-1.png
  • 2-up
  • Swipe
  • Onion skin
docs/reference/plot-2.png

76.8 KB | W: | H:

docs/reference/plot-2.png

78.7 KB | W: | H:

docs/reference/plot-2.png
docs/reference/plot-2.png
docs/reference/plot-2.png
docs/reference/plot-2.png
  • 2-up
  • Swipe
  • Onion skin
......@@ -174,37 +174,37 @@ terms, `.` replaces all the variables in name. Additional, when one want to excl
<span class='fu'><a href='https://rdrr.io/r/utils/data.html'>data</a></span>(<span class='st'>"mcmc_run_asia"</span>)
<span class='co'>##return a named matrix with individual arc support</span>
<span class='fu'>query</span>(<span class='kw'>mcmcabn</span> <span class='kw'>=</span> <span class='no'>mcmc.out.asia</span>)</div><div class='output co'>#&gt; Asia Smoking Tuberculosis LungCancer Bronchitis
#&gt; Asia 0.00000000 0.007992008 0.01698302 0.005994006 0.007992008
#&gt; Smoking 0.01498501 0.000000000 0.03296703 0.416583417 0.434565435
#&gt; Tuberculosis 0.02497502 0.031968032 0.00000000 0.099900100 0.006993007
#&gt; LungCancer 0.01298701 0.397602398 0.15784216 0.000000000 0.057942058
#&gt; Bronchitis 0.00999001 0.484515485 0.01898102 0.102897103 0.000000000
#&gt; Either 0.02097902 0.085914086 0.75224775 0.743256743 0.018981019
#&gt; XRay 0.00999001 0.023976024 0.34065934 0.288711289 0.005994006
#&gt; Dyspnea 0.01098901 0.096903097 0.18781219 0.295704296 0.671328671
#&gt; Either XRay Dyspnea
#&gt; Asia 0.007992008 0.008991009 0.005994006
#&gt; Smoking 0.099900100 0.032967033 0.065934066
#&gt; Tuberculosis 0.138861139 0.048951049 0.205794206
#&gt; LungCancer 0.196803197 0.057942058 0.132867133
#&gt; Bronchitis 0.088911089 0.046953047 0.283716284
#&gt; Either 0.000000000 0.108891109 0.049950050
#&gt; XRay 0.631368631 0.000000000 0.008991009
#&gt; Dyspnea 0.409590410 0.070929071 0.000000000</div><div class='input'>
<span class='fu'>query</span>(<span class='kw'>mcmcabn</span> <span class='kw'>=</span> <span class='no'>mcmc.out.asia</span>)</div><div class='output co'>#&gt; Asia Smoking Tuberculosis LungCancer Bronchitis
#&gt; Asia 0.000000000 0.16983017 0.19080919 0.1578422 0.17482517
#&gt; Smoking 0.082917083 0.00000000 0.07992008 0.4395604 0.42357642
#&gt; Tuberculosis 0.131868132 0.13186813 0.00000000 0.2767233 0.12287712
#&gt; LungCancer 0.064935065 0.51248751 0.21978022 0.0000000 0.14585415
#&gt; Bronchitis 0.080919081 0.57642358 0.13886114 0.1488511 0.00000000
#&gt; Either 0.016983017 0.05294705 0.81818182 0.8501499 0.02197802
#&gt; XRay 0.064935065 0.04595405 0.30469530 0.2707293 0.03196803
#&gt; Dyspnea 0.004995005 0.00000000 0.02697303 0.1718282 0.96803197
#&gt; Either XRay Dyspnea
#&gt; Asia 0.12287712 0.13986014 0.137862138
#&gt; Smoking 0.03596404 0.02297702 0.000000000
#&gt; Tuberculosis 0.18181818 0.03896104 0.186813187
#&gt; LungCancer 0.14985015 0.04395604 0.039960040
#&gt; Bronchitis 0.05394605 0.03596404 0.031968032
#&gt; Either 0.00000000 0.08891109 0.008991009
#&gt; XRay 0.79220779 0.00000000 0.035964036
#&gt; Dyspnea 0.75524476 0.02697303 0.000000000</div><div class='input'>
<span class='co'>## what is the probability of LungCancer node being children of the Smoking node?</span>
<span class='fu'>query</span>(<span class='kw'>mcmcabn</span> <span class='kw'>=</span> <span class='no'>mcmc.out.asia</span>,<span class='kw'>formula</span> <span class='kw'>=</span> ~<span class='no'>LungCancer</span><span class='kw'>|</span><span class='no'>Smoking</span>)</div><div class='output co'>#&gt; [1] 0.3976024</div><div class='input'>
<span class='fu'>query</span>(<span class='kw'>mcmcabn</span> <span class='kw'>=</span> <span class='no'>mcmc.out.asia</span>,<span class='kw'>formula</span> <span class='kw'>=</span> ~<span class='no'>LungCancer</span><span class='kw'>|</span><span class='no'>Smoking</span>)</div><div class='output co'>#&gt; [1] 0.5124875</div><div class='input'>
<span class='co'>## what is the probability of Smoking node being parent of</span>
<span class='co'>## both LungCancer and Bronchitis node?</span>
<span class='fu'>query</span>(<span class='kw'>mcmcabn</span> <span class='kw'>=</span> <span class='no'>mcmc.out.asia</span>,
<span class='kw'>formula</span> <span class='kw'>=</span> ~ <span class='no'>LungCancer</span><span class='kw'>|</span><span class='no'>Smoking</span>+<span class='no'>Bronchitis</span><span class='kw'>|</span><span class='no'>Smoking</span>)</div><div class='output co'>#&gt; [1] 0.1778222</div><div class='input'>
<span class='kw'>formula</span> <span class='kw'>=</span> ~ <span class='no'>LungCancer</span><span class='kw'>|</span><span class='no'>Smoking</span>+<span class='no'>Bronchitis</span><span class='kw'>|</span><span class='no'>Smoking</span>)</div><div class='output co'>#&gt; [1] 0.2707293</div><div class='input'>
<span class='co'>## what is the probability of previous statement,</span>
<span class='co'>## when there is no arc from Smoking to Tuberculosis and from Bronchitis to XRay?</span>
<span class='fu'>query</span>(<span class='kw'>mcmcabn</span> <span class='kw'>=</span> <span class='no'>mcmc.out.asia</span>,
<span class='kw'>formula</span> <span class='kw'>=</span> ~<span class='no'>LungCancer</span><span class='kw'>|</span><span class='no'>Smoking</span> +
<span class='no'>Bronchitis</span><span class='kw'>|</span><span class='no'>Smoking</span> -
<span class='no'>Tuberculosis</span><span class='kw'>|</span><span class='no'>Smoking</span> -
<span class='no'>XRay</span><span class='kw'>|</span><span class='no'>Bronchitis</span>)</div><div class='output co'>#&gt; [1] 0</div></pre>
<span class='no'>XRay</span><span class='kw'>|</span><span class='no'>Bronchitis</span>)</div><div class='output co'>#&gt; [1] 0.001998002</div></pre>
</div>
<div class="col-md-3 hidden-xs hidden-sm" id="sidebar">
<h2>Contents</h2>
......
......@@ -182,27 +182,27 @@
#&gt; Thinning: 99
#&gt;
#&gt; Maximum score: -11151.13
#&gt; Empirical mean: -11322.76
#&gt; Empirical standard deviation: 337.337
#&gt; Empirical mean: -11164.74
#&gt; Empirical standard deviation: 7.407195
#&gt; Quantiles of the posterior network score:
#&gt; 0.025 0.25 0.5 0.75 0.975
#&gt; BN score -12350.22 -11278.52 -11181.1 -11156.04 -11151.13
#&gt; 0.025 0.25 0.5 0.75 0.975
#&gt; BN score -11188.58 -11168.16 -11163.84 -11159.79 -11154.2
#&gt;
#&gt;
#&gt; Global acceptance rate: 0.1638362
#&gt; Global acceptance rate: 0.2047952
#&gt; Accepted Rejected
#&gt; MBR 10 27
#&gt; MC3 144 795
#&gt; REV 10 15
#&gt; MBR 9 24
#&gt; MC3 186 752
#&gt; REV 10 20
#&gt;
#&gt;
#&gt; Sample size adjusted for autocorrelation: 325.5766
#&gt; Sample size adjusted for autocorrelation: 51.58366
#&gt;
#&gt; Autocorrelations by lag:
#&gt; 0 1 2 3 4 5 6 7
#&gt; acf 1 0.4338829 0.213675 0.1552886 0.1088207 0.0889256 0.07131714 0.01286986
#&gt; 8 9 10
#&gt; acf 0.01950408 -0.02595695 0.01850446</div></pre>
#&gt; 0 1 2 3 4 5 6 7
#&gt; acf 1 0.5684532 0.5619595 0.5225394 0.478408 0.4586218 0.4566739 0.4189232
#&gt; 8 9 10
#&gt; acf 0.4024697 0.3702477 0.3348253</div></pre>
</div>
<div class="col-md-3 hidden-xs hidden-sm" id="sidebar">
<h2>Contents</h2>
......
......@@ -30,6 +30,9 @@
<url>
<loc>https://www.math.uzh.ch/pages/mcmcabn//reference/summary.html</loc>
</url>
<url>
<loc>https://www.math.uzh.ch/pages/mcmcabn//articles/Articles/convergence.html</loc>
</url>
<url>
<loc>https://www.math.uzh.ch/pages/mcmcabn//articles/mcmcabn.html</loc>
</url>
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment