.. _sampling: Sampling from the posterior distribution ---------------------------------------- When analyzing empirical networks, one should be open to the possibility that there will be more than one fit of the SBM with similar posterior probabilities. In such situations, one should instead `sample` partitions from the posterior distribution, instead of simply finding its maximum. One can then compute quantities that are averaged over the different model fits, weighted according to their posterior probabilities. Full support for model averaging is implemented in ``graph-tool`` via an efficient `Markov chain Monte Carlo (MCMC) `_ algorithm [peixoto-efficient-2014]_. It works by attempting to move nodes into different groups with specific probabilities, and `accepting or rejecting `_ such moves so that, after a sufficiently long time, the partitions will be observed with the desired posterior probability. The algorithm is designed so that its run-time (i.e. each sweep of the MCMC) is linear on the number of edges in the network, and independent on the number of groups being used in the model, and hence is suitable for use on very large networks. In order to perform such moves, one needs again to operate with :class:`~graph_tool.inference.blockmodel.BlockState` or :class:`~graph_tool.inference.nested_blockmodel.NestedBlockState` instances, and calling their :meth:`~graph_tool.inference.blockmodel.BlockState.mcmc_sweep` methods. For example, the following will perform 1000 sweeps of the algorithm with the network of characters in the novel Les Misérables, starting from a random partition into 20 groups .. testcode:: model-averaging g = gt.collection.data["lesmis"] state = gt.BlockState(g, B=20) # This automatically initializes the state # with a random partition into B=20 # nonempty groups; The user could # also pass an arbitrary initial # partition using the 'b' parameter. # Now we run 1,000 sweeps of the MCMC. Note that the number of groups # is allowed to change, so it will eventually move from the initial # value of B=20 to whatever is most appropriate for the data. dS, nattempts, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: model-averaging Change in description length: -353.848032... Number of accepted vertex moves: 37490 .. note:: Starting from a random partition is rarely the best option, since it may take a long time for it to equilibrate. It was done above simply as an illustration on how to initialize :class:`~graph_tool.inference.blockmodel.BlockState` by hand. Instead, a much better option in practice is to start from an approximation to the "ground state" obtained with :func:`~graph_tool.inference.minimize.minimize_blockmodel_dl`, e.g. .. testcode:: model-averaging state = gt.minimize_blockmodel_dl(g) state = state.copy(B=g.num_vertices()) dS, nattempts, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: model-averaging Change in description length: 31.622518... Number of accepted vertex moves: 43152 Although the above is sufficient to implement model averaging, there is a convenience function called :func:`~graph_tool.inference.mcmc.mcmc_equilibrate` that is intend to simplify the detection of equilibration, by keeping track of the maximum and minimum values of description length encountered and how many sweeps have been made without a "record breaking" event. For example, .. testcode:: model-averaging # We will accept equilibration if 10 sweeps are completed without a # record breaking event, 2 consecutive times. gt.mcmc_equilibrate(state, wait=10, nbreaks=2, mcmc_args=dict(niter=10), verbose=True) will output: .. testoutput:: model-averaging :options: +NORMALIZE_WHITESPACE niter: 1 count: 0 breaks: 0 min_S: 703.94152 max_S: 730.97213 S: 703.94152 ΔS: -27.0306 moves: 431 niter: 2 count: 1 breaks: 0 min_S: 703.94152 max_S: 730.97213 S: 708.61840 ΔS: 4.67688 moves: 413 niter: 3 count: 2 breaks: 0 min_S: 703.94152 max_S: 730.97213 S: 704.60994 ΔS: -4.00847 moves: 416 niter: 4 count: 0 breaks: 0 min_S: 700.85336 max_S: 730.97213 S: 700.85336 ΔS: -3.75658 moves: 391 niter: 5 count: 1 breaks: 0 min_S: 700.85336 max_S: 730.97213 S: 713.22553 ΔS: 12.3722 moves: 387 niter: 6 count: 2 breaks: 0 min_S: 700.85336 max_S: 730.97213 S: 703.57357 ΔS: -9.65196 moves: 434 niter: 7 count: 3 breaks: 0 min_S: 700.85336 max_S: 730.97213 S: 715.02440 ΔS: 11.4508 moves: 439 niter: 8 count: 0 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 700.68857 ΔS: -14.3358 moves: 427 niter: 9 count: 1 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 717.95725 ΔS: 17.2687 moves: 409 niter: 10 count: 2 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 720.02079 ΔS: 2.06354 moves: 435 niter: 11 count: 3 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 718.15880 ΔS: -1.86199 moves: 399 niter: 12 count: 4 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 708.06732 ΔS: -10.0915 moves: 436 niter: 13 count: 5 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 712.76007 ΔS: 4.69274 moves: 432 niter: 14 count: 6 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 705.60582 ΔS: -7.15425 moves: 409 niter: 15 count: 7 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 704.37333 ΔS: -1.23249 moves: 434 niter: 16 count: 8 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 717.54492 ΔS: 13.1716 moves: 426 niter: 17 count: 9 breaks: 0 min_S: 700.68857 max_S: 730.97213 S: 715.05767 ΔS: -2.48725 moves: 449 niter: 18 count: 0 breaks: 1 min_S: 715.77940 max_S: 715.77940 S: 715.77940 ΔS: 0.721731 moves: 448 niter: 19 count: 0 breaks: 1 min_S: 708.38072 max_S: 715.77940 S: 708.38072 ΔS: -7.39868 moves: 447 niter: 20 count: 0 breaks: 1 min_S: 705.63447 max_S: 715.77940 S: 705.63447 ΔS: -2.74625 moves: 441 niter: 21 count: 1 breaks: 1 min_S: 705.63447 max_S: 715.77940 S: 707.01766 ΔS: 1.38319 moves: 434 niter: 22 count: 2 breaks: 1 min_S: 705.63447 max_S: 715.77940 S: 708.21127 ΔS: 1.19361 moves: 447 niter: 23 count: 0 breaks: 1 min_S: 703.12325 max_S: 715.77940 S: 703.12325 ΔS: -5.08802 moves: 454 niter: 24 count: 0 breaks: 1 min_S: 703.05106 max_S: 715.77940 S: 703.05106 ΔS: -0.0721911 moves: 433 niter: 25 count: 1 breaks: 1 min_S: 703.05106 max_S: 715.77940 S: 704.77370 ΔS: 1.72264 moves: 423 niter: 26 count: 0 breaks: 1 min_S: 701.61368 max_S: 715.77940 S: 701.61368 ΔS: -3.16003 moves: 441 niter: 27 count: 0 breaks: 1 min_S: 701.61368 max_S: 721.54373 S: 721.54373 ΔS: 19.9301 moves: 434 niter: 28 count: 1 breaks: 1 min_S: 701.61368 max_S: 721.54373 S: 703.33612 ΔS: -18.2076 moves: 439 niter: 29 count: 2 breaks: 1 min_S: 701.61368 max_S: 721.54373 S: 710.79425 ΔS: 7.45813 moves: 437 niter: 30 count: 3 breaks: 1 min_S: 701.61368 max_S: 721.54373 S: 706.35044 ΔS: -4.44381 moves: 429 niter: 31 count: 4 breaks: 1 min_S: 701.61368 max_S: 721.54373 S: 713.56014 ΔS: 7.20970 moves: 463 niter: 32 count: 5 breaks: 1 min_S: 701.61368 max_S: 721.54373 S: 720.16436 ΔS: 6.60422 moves: 445 niter: 33 count: 6 breaks: 1 min_S: 701.61368 max_S: 721.54373 S: 714.76845 ΔS: -5.39591 moves: 404 niter: 34 count: 7 breaks: 1 min_S: 701.61368 max_S: 721.54373 S: 703.21572 ΔS: -11.5527 moves: 410 niter: 35 count: 0 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 701.53898 ΔS: -1.67675 moves: 434 niter: 36 count: 1 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 708.14043 ΔS: 6.60146 moves: 433 niter: 37 count: 2 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 704.07209 ΔS: -4.06835 moves: 410 niter: 38 count: 3 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 704.76811 ΔS: 0.696023 moves: 413 niter: 39 count: 4 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 703.54823 ΔS: -1.21988 moves: 398 niter: 40 count: 5 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 713.59891 ΔS: 10.0507 moves: 388 niter: 41 count: 6 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 704.40168 ΔS: -9.19724 moves: 403 niter: 42 count: 7 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 707.57723 ΔS: 3.17556 moves: 400 niter: 43 count: 8 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 704.09679 ΔS: -3.48044 moves: 423 niter: 44 count: 9 breaks: 1 min_S: 701.53898 max_S: 721.54373 S: 704.64514 ΔS: 0.548354 moves: 419 niter: 45 count: 10 breaks: 2 min_S: 701.53898 max_S: 721.54373 S: 715.92329 ΔS: 11.2781 moves: 411 Note that the value of ``wait`` above was made purposefully low so that the output would not be overly long. The most appropriate value requires experimentation, but a typically good value is ``wait=1000``. The function :func:`~graph_tool.inference.mcmc.mcmc_equilibrate` accepts a ``callback`` argument that takes an optional function to be invoked after each call to :meth:`~graph_tool.inference.blockmodel.BlockState.mcmc_sweep`. This function should accept a single parameter which will contain the actual :class:`~graph_tool.inference.blockmodel.BlockState` instance. We will use this in the example below to collect the posterior vertex marginals (via :class:`~graph_tool.inference.blockmodel.BlockState.collect_vertex_marginals`), i.e. the posterior probability that a node belongs to a given group: .. testcode:: model-averaging # We will first equilibrate the Markov chain gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10)) pv = None def collect_marginals(s): global pv pv = s.collect_vertex_marginals(pv) # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps: gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_marginals) # Now the node marginals are stored in property map pv. We can # visualize them as pie charts on the nodes: state.draw(pos=g.vp.pos, vertex_shape="pie", vertex_pie_fractions=pv, edge_gradient=None, output="lesmis-sbm-marginals.svg") .. figure:: lesmis-sbm-marginals.* :align: center :width: 450px Marginal probabilities of group memberships of the network of characters in the novel Les Misérables, according to the degree-corrected SBM. The `pie fractions `_ on the nodes correspond to the probability of being in group associated with the respective color. We can also obtain a marginal probability on the number of groups itself, as follows. .. testcode:: model-averaging h = np.zeros(g.num_vertices() + 1) def collect_num_groups(s): B = s.get_nonempty_B() h[B] += 1 # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps: gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_num_groups) .. testcode:: model-averaging :hide: figure() Bs = np.arange(len(h)) idx = h > 0 bar(Bs[idx], h[idx] / h.sum(), width=1, color="#ccb974") gca().set_xticks([6,7,8,9]) xlabel("\$B\$") ylabel(r"\$P(B|\boldsymbol A)\$") savefig("lesmis-B-posterior.svg") .. figure:: lesmis-B-posterior.* :align: center Marginal posterior probability of the number of nonempty groups for the network of characters in the novel Les Misérables, according to the degree-corrected SBM. Hierarchical partitions +++++++++++++++++++++++ We can also perform model averaging using the nested SBM, which will give us a distribution over hierarchies. The whole procedure is fairly analogous, but now we make use of :class:`~graph_tool.inference.nested_blockmodel.NestedBlockState` instances. .. note:: When using :class:`~graph_tool.inference.nested_blockmodel.NestedBlockState` instances to perform model averaging, they need to be constructed with the option ``sampling=True``. Here we perform the sampling of hierarchical partitions using the same network as above. .. testcode:: nested-model-averaging g = gt.collection.data["lesmis"] state = gt.minimize_nested_blockmodel_dl(g) # Initialize he Markov # chain from the "ground # state" # Before doing model averaging, the need to create a NestedBlockState # by passing sampling = True. # We also want to increase the maximum hierarchy depth to L = 10 # We can do both of the above by copying. bs = state.get_bs() # Get hierarchical partition. bs += [np.zeros(1)] * (10 - len(bs)) # Augment it to L = 10 with # single-group levels. state = state.copy(bs=bs, sampling=True) # Now we run 1000 sweeps of the MCMC dS, nattempts, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: nested-model-averaging Change in description length: 15.483135... Number of accepted vertex moves: 57684 Similarly to the the non-nested case, we can use :func:`~graph_tool.inference.mcmc.mcmc_equilibrate` to do most of the boring work, and we can now obtain vertex marginals on all hierarchical levels: .. testcode:: nested-model-averaging # We will first equilibrate the Markov chain gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10)) pv = [None] * len(state.get_levels()) def collect_marginals(s): global pv pv = [sl.collect_vertex_marginals(pv[l]) for l, sl in enumerate(s.get_levels())] # Now we collect the marginals for exactly 100,000 sweeps gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_marginals) # Now the node marginals for all levels are stored in property map # list pv. We can visualize the first level as pie charts on the nodes: state_0 = state.get_levels()[0] state_0.draw(pos=g.vp.pos, vertex_shape="pie", vertex_pie_fractions=pv[0], edge_gradient=None, output="lesmis-nested-sbm-marginals.svg") .. figure:: lesmis-nested-sbm-marginals.* :align: center :width: 450px Marginal probabilities of group memberships of the network of characters in the novel Les Misérables, according to the nested degree-corrected SBM. The pie fractions on the nodes correspond to the probability of being in group associated with the respective color. We can also obtain a marginal probability of the number of groups itself, as follows. .. testcode:: nested-model-averaging h = [np.zeros(g.num_vertices() + 1) for s in state.get_levels()] def collect_num_groups(s): for l, sl in enumerate(s.get_levels()): B = sl.get_nonempty_B() h[l][B] += 1 # Now we collect the marginal distribution for exactly 100,000 sweeps gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_num_groups) .. testcode:: nested-model-averaging :hide: figure() f, ax = plt.subplots(1, 5, figsize=(10, 3)) for i, h_ in enumerate(h[:5]): Bs = np.arange(len(h_)) idx = h_ > 0 ax[i].bar(Bs[idx], h_[idx] / h_.sum(), width=1, color="#ccb974") ax[i].set_xticks(Bs[idx]) ax[i].set_xlabel("\$B_{%d}\$" % i) ax[i].set_ylabel(r"\$P(B_{%d}|\boldsymbol A)\$" % i) locator = MaxNLocator(prune='both', nbins=5) ax[i].yaxis.set_major_locator(locator) tight_layout() savefig("lesmis-nested-B-posterior.svg") .. figure:: lesmis-nested-B-posterior.* :align: center Marginal posterior probability of the number of nonempty groups :math:`B_l` at each hierarchy level :math:`l` for the network of characters in the novel Les Misérables, according to the nested degree-corrected SBM. Below we obtain some hierarchical partitions sampled from the posterior distribution. .. testcode:: nested-model-averaging for i in range(10): state.mcmc_sweep(niter=1000) state.draw(output="lesmis-partition-sample-%i.svg" % i, empty_branches=False) .. image:: lesmis-partition-sample-0.svg :width: 200px .. image:: lesmis-partition-sample-1.svg :width: 200px .. image:: lesmis-partition-sample-2.svg :width: 200px .. image:: lesmis-partition-sample-3.svg :width: 200px .. image:: lesmis-partition-sample-4.svg :width: 200px .. image:: lesmis-partition-sample-5.svg :width: 200px .. image:: lesmis-partition-sample-6.svg :width: 200px .. image:: lesmis-partition-sample-7.svg :width: 200px .. image:: lesmis-partition-sample-8.svg :width: 200px .. image:: lesmis-partition-sample-9.svg :width: 200px