P (m ∣ d_{1}, \dots, d_{S}) \propto P (m) \prod_{s = 1}^{S} P (d_{s} ∣ m), S = N_{sat} .

P (d_{s} ∣ m) \propto P (m ∣ d_{s}) / P (m)

. Therefore

P (m ∣ d) \propto P (m)^{1 - S} \prod_{s = 1}^{S} P (m ∣ d_{s}) .

p_{0} (m)

is a kernel density estimator for the simulated marginal masses and

p_{s} (m)

estimates the posterior based on satellite

s

, then

\overset{m}{ˉ} = \frac{\int m p _{0} ( m ) ^{1 - S} \prod _{s = 1}^{S} p _{s} ( m ) d m}{\int p _{0} ( m ) ^{1 - S} \prod _{s = 1}^{S} p _{s} ( m ) d m} .

The one-dimensional integrals can be evaluated by numerical integration on a common mass grid.

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