of magnitude in frequency [@Lasky:2015lej; @Akrami:2018odb].

While we have targeted an isotropic, stationary, and Gaussian background, other search techniques can probe backgrounds that violate one or more of these assumptions. Upper limits on an anisotropic gravitational-wave background from O1 were presented in [@stoch_dir_O1]. Furthermore, non-Gaussian searches targeting the compact binary  are currently being developed [@popcorn1; @popcorn2; @popcorn3; @tbs_methods]. A successful detection of the  by any of these approaches would offer a new probe of the gravitational-wave sky.

*Acknowledgments* —

This article has been assigned the document number LIGO-P1800258.

[**Supplement To: A search for the isotropic stochastic background in Advanced LIGO’s second observing run**]{}\
(The LIGO Scientific Collaboration and Virgo Collaboration)

Stochastic Upper Limit Construction
===================================

Here we describe the parameter estimation procedure used to set upper limits on the amplitude of the stochastic background.

We assume a Gaussian likelihood for the measured cross-correlations $C(f)$ (see Eq. ), such that $$\mathcal{L}(C | \Theta, \mathcal{H})
    \propto \exp\left(
    -\frac{1}{2}\sum_k \frac{\left[C(f_k)-\langle C(\Theta; f_k)\rangle_\mathcal{H}\right]^2}{\sigma^2(f_k)}
       \right).$$ Here, $\langle C(\Theta; f_k)\rangle_\mathcal{H}$ is the expected cross-correlation spectrum given hypothesis $\mathcal{H}$ and model parameters $\Theta$. The variance $\sigma^2(f)$ is defined in Eq. . To perform parameter estimation using the combined results from Advanced LIGO’s O1 and O2 observing runs, 