Importance sampling for Bayesian sensitivity analysis

Imprecise Monte Carlo simulation and iterative importance sampling for the estimation of lower previsions

“…Example. As a first example, we demonstrate imprecise importance sampling on the imprecise Dirichlet model, similar to the one studied in [11]. The starting point of the example is identical to the one studied in [15].…”

“…Here, we will use the theory developed in section 2 in order to estimate the lower prevision of a gamble f . O'Neill [11] studied this technique already in a Bayesian setting, although only studying the consistency of his estimatorθ(X, t) and not the bias and consistency ofθ * (X) as we do here. Given our parameterized collection of probability density functions p t , we define the lower prevision of f as…”

“…(96), with k = 2 (i.e. it is an imprecise Beta model), s = 10, and T = {t ∈ ∆ : t 1 ≥ 0.3, t 2 ≥ 0.6} (the example is presented differently in [11], but it is equivalent to this one after some iteration 1 2 n(x, τ (x)) 873.44201 1000.00000 τ 1 (x) 0.30000 0.30000 τ 2 (x) 0.70000 0.70000 time 1.61215 0.50916 Table 4. Importance sampling simulation results for each iteration.…”

“…A first approach is to use two-level Monte Carlo sampling, where first one samples distributions over the (extreme points of the) credal set, and then samples from these distributions, In the context of belief functions, one can also sample random sets, and then evaluate the resulting belief function through optimisation over these sets [8]. A third more sophisticated approach comprises of importance sampling from a reference distribution, and then solve an optimisation problem over the importance sampling weights [11,5,16].…”

“…Unlike [11], however, we do not just look at Bayesian sensitivity analysis, and admit arbitrary sets of distributions in our theoretical treatment. Also unlike for instance [11,2,5,16,4], in this paper, we use self-normalised importance sampling instead of standard importance sampling, as it is known that this drastically speeds up calculations [15]. In this paper, we will also show that it ensures coherence of the resulting estimates (regardless of bias).…”