Bayesian Field Theory of the Rate Estimation

Abstract

We address the statistical inference of a time-dependent rate of events in the framework of Bayesian field theory. This maps the problem to a Langevin equation which, beyond the local linear regime taken as reference, involves nonlinearities and an explicit dependence on the local shape of the maximum likelihood curve. We study the corresponding impacts in a perturbative expansion, formulating a scaling hypothesis for the order of shape corrections. We find that the pure nonlinearities dominate the mean and skewness. Crucially, we uncover that the leading correction to the variance is driven by noise propagation from the signal's effective curvature. We test the derived expansion with numerical simulations and illustrate its applicability on real neural spike data.

Figures (3)

• Figure 1: The time-dependent rate $r_t$ (gray line) generates a spike train $D$ (not shown). The posterior probability $p(\boldsymbol{s} \vert D)$ of the log-process $s_t\equiv \ln (r_t)$ admits a maximum likelihood curve $\exp (s^*_t)$ (black line) found by numerically solving Eq. \ref{['Classical']}. The volatility parameter is $\sigma = 0.02$.
• Figure 2: The deviation $\delta p$ from the local linear regime is evaluated numerically (dots) from the direct simulation of Eq. \ref{['SPDE']} with an implicit Euler method, and compared to the Gram-Charlier expansion corresponding to the perturbative expansion moments of Eqs. \ref{['Variance impact']}-\ref{['path impact on expectation']} (solid line). The impact of path-dependent terms is highlighted by comparison to the homogeneous case $f_t=0$ (dashed line). The parameters are $\sigma = 0.02$, while $\alpha \approx r_0 = 1$ and $f_t$ correspond to the ML curve of Fig. (\ref{['fig:ML']}). More examples are in the SM.
• Figure 3: The ML log-rate $s^*_t$ and expected log-rate $s^*_t+\langle x\rangle$ for the most active neuron in the dataset siegle2021survey is shown on the left axis, with the volatility hyperparameter fixed to its maximum likelihood value $\sigma^* \approx 0.08$. The posterior standard deviation $Std[x]\equiv \sqrt{\langle x^2\rangle -\langle x \rangle^2}$, shown on the right axis, expands significantly compared to the local linear solution $\sqrt{\nu}$ at peaks of the ML curve where $J_t f^t > 0$, demonstrating the noise propagation driven by the signal's effective curvature. The time snippet has been selected randomly, more examples in the SM.