We consider the problem of Bayesian parameter inference in the Merton structural credit risk model, where the posterior is induced by a jump-diffusion likelihood and the marginal evidence is not available in closed form. To approximate this posterior, we construct a variational family based on triangular sum-of-squares (SOS) polynomial flows, in which each component map is monotone by construction: its diagonal derivative is a positive definite quadratic form on a monomial basis, yielding a closed-form log-Jacobian and explicit gradients with respect to all flow parameters. The symmetric positive definite matrices parametrizing the flow are optimized by intrinsic Riemannian gradient ascent on the positive definite cone equipped with the affine-invariant metric, which preserves feasibility at every iterate without projection. We show that the rank-one Jacobian gradients produced by the SOS structure have unit norm in the affine-invariant metric, establishing a direct algebraic coupling between the transport family and the optimization geometry and implying a universal 1-Lipschitz bound for the log-Jacobian along geodesics. On the likelihood side, we derive exact score identities for all five structural parameters of the Merton model—drift, volatility, jump intensity, jump mean, and jump volatility—through both the Poisson log-normal mixture and the Fourier inversion representations. Strictly positive parameters are handled via exponential reparametrization, and the resulting gradients propagate end-to-end through the flow. We establish uniform truncation bounds on compact parameter sets for the infinite mixture and its associated score series, providing rigorous control over the finite approximations used in practice. The base distribution is chosen to be uniform on 0,15, whose bounded support ensures uniform control of the monomial basis and stabilizes the polynomial calculus. These ingredients are assembled into a fully explicit modified ELBO with implementable gradients, combining Euclidean updates for vector parameters and intrinsic manifold updates for matrix parameters.
Persio et al. (Sat,) studied this question.