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The double-layer Heisenberg antiferromagnet with intra-and inter-layer couplings J and J ⊥ exhibits a zero temperature quantum phase transition between a quantum disordered dimer phase for g > g c and a Neel phase with long range antiferromagnetic order for g < g c , where g = J ⊥ /J and g c ≈ 2.5. We consider the behavior of the system at finite temperature for g ≥ g c using two different and complementary approaches; an analytical Brueckner approximation and numerically exact quantum Monte Carlo simulations. We calculate the temperature dependent spin excitation spectrum (including the triplet gap), dynamic and static structure factors, the specific heat, and the uniform magnetic susceptibility. The agreement between the analytical and numerical approaches is excellent. For T → 0 and g → g c , our analytical results for the specific heat and the magnetic susceptibility coincide with those previously obtained within the nonlinear σ model approach for N → ∞. Our quantum Monte Carlo simulations extend to significantly lower temperatures than previously, allowing us to obtain accurate results for the asymptotic quantum critical behavior. We also obtain an improved estimate for the critical coupling: g c = 2.525 ± 0.002.

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In this paper we examine the claims reserving problem using Tweedie's compound Poisson model. We develop the maximum likelihood and Bayesian Markov chain Monte Carlo simulation approaches to fit the model and then compare the estimated models under different scenarios. The key point we demonstrate relates to the comparison of reserving quantities with and without model uncertainty incorporated into the prediction. We consider both the model selection problem and the model averaging solutions for the predicted reserves. As a part of this process we also consider the sub problem of variable selection to obtain a parsimonious representation of the model being fitted.