Inverse Ising problem in continuous time: A latent variable approach

We consider the inverse Ising problem: the inference of network couplings from observed spin trajectories for a model with continuous time Glauber dynamics. By introducing two sets of auxiliary latent random variables we render the likelihood into a form which allows for simple iterative inference algorithms with analytical updates. The variables are (1) Poisson variables to linearize an exponential term which is typical for point process likelihoods and (2) Pólya-Gamma variables, which make the likelihood quadratic in the coupling parameters. Using the augmented likelihood, we derive an expectation-maximization (EM) algorithm to obtain the maximum likelihood estimate of network parameters. Using a third set of latent variables we extend the EM algorithm to sparse couplings via L1 regularization. Finally, we develop an efficient approximate Bayesian inference algorithm using a variational approach. We demonstrate the performance of our algorithms on data simulated from an Ising model. For data which are simulated from a more biologically plausible network with spiking neurons, we show that the Ising model captures well the low order statistics of the data and how the Ising couplings are related to the underlying synaptic structure of the simulated network.

Figure 1

Inference with EM algorithm on artificial data. (a) True couplings (black dots) and external fields (red triangles) vs inferred ones. (b) The log-likelihood as function of EM iterations. The parameters are set to $N=40$, $T={10}^3$, and $g=0.3$ with external fields $\mathbf{\theta }=\mathit{0}$. (c) MSE between $\mathit{J}$ and ${\mathit{J}}_{\mathrm{est}}$ as a function of scaling factor of the variance $g$ and (d) as a function of data length $T$. If not changed parameters are as in (a).

Figure 2

Inference of sparse couplings with EM and variational Bayes. Artificial data ($T=50,N=25,g=0.3$) are generated, but each coupling is set to 0 with probability $p_{\mathrm{sparse}}=1/2$. (a) Difference in likelihood (with respect to the likelihood obtained with optimal ${\lambda }^{★}$) of couplings ${\mathit{J}}_{\mathrm{est}}$ inferred by EM as a function of regularization parameter $\lambda$. Likelihood ${\mathcal{L}}_{\mathrm{test}}$ is computed on unseen test data ($T=50$). The optimal parameter is ${\lambda }^{★}=29.4$ (red diamond). The vertical line marks the variational estimation ${\lambda }_{\mathrm{Bayes}}^{★}$. (b) Difference in free energy $\mathcal{F}$ (with respect to the likelihood obtained with estimate of optimal ${\lambda }_{\mathrm{Bayes}}^{★}$) of the variational Bayes algorithm. The optimal parameter is ${\lambda }_{\mathrm{Bayes}}^{★}=34.5$ (blue diamond). (c) ROC curves for the ${\lambda }^{★}$ (EM, solid red line) and ${\lambda }_{\mathrm{Bayes}}^{★}$ (Bayes, dashed blue line), respectively. (d) The AUC for different parameters $\lambda$ for the EM result (solid black line with squares) and the variational Bayes algorithm (dashed gray line with triangles). Diamonds mark the optimal ${\lambda }^{★}$ and the estimate ${\lambda }_{\mathrm{Bayes}}^{★}$.

Figure 3

The classification of nonzero couplings depending on sparsity of couplings and data length. (a) The AUC depending on the sparsity, i.e., the probability of a true coupling being 0, and in (b) depending on length of training data $T$. Results for EM shown by the red solid and the variational Bayes algorithm by the blue dashed line. If not changed, parameters are as in Fig. 2.

Figure 4

Model fitted to data from a simulated recurrent network. (a) Second-order correlation $C_{ij}$ of the original data vs data sampled from mean couplings ${\langle \mathit{J}\rangle }_{q_1}$ obtained via variational Bayes. (b) Pearson correlation between first- and fourth-order correlations of real sampled data. (c) ROC curve for identifying synapses with posterior over couplings $\mathit{J}$ ($\mathrm{AUC}=0.65,{\lambda }_{\mathrm{Bayes}}^{★}=16.5$). (d) Mean couplings ${\langle \mathit{J}\rangle }_{q_1}$ of the variational posterior vs transpose. Couplings between neurons connected by a synapse are marked with gray triangles.