**A Procedure for Simultaneous Calculation of Sensitivities
in Probabilistic Risk Assessments.* ***S. Uryasev,* *Brookhaven
National* *Laboratory, Bldg. 130, Upton, NY, 11973; and* *A.
Shlyakhter*,* Dept. of Physics, Harvard Univ., Cambridge, MA
02138 and Dept. of Environmental Health, Harvard School of Public
Health, 665 Huntington Ave., Boston MA 02115*

Sensitivity analysis of risk estimates to parameters is the
first part of uncertainty analysis. It involves calculation of
the derivatives which is often simple for risk estimates
themselves but becomes intractable (in analytical form) for the
upper percentiles of risk if the distributions other than
lognormal are involved. A new general formula for calculating the
derivatives of the integrals over sets given by inequalities was
recently developed by Uryasev [I]. Probability and quantile
functions (for example, 95 percentile) are the special cases of
this general formula. Derivative of a quantile function with
respect to parameter is presented as the ratio of two probability
functions with the same probability density and integration set.
Thus derivatives of the quantile function with respect to all
parameters are reduced to calculation of the similar integrals
over the same sets. These integrals can be calculated
simultaneously by the Monte Carlo simulation. To illustrate this
formulation, let us consider a simple risk model in the form of
the ratio of two random variables, y_{1} and y_{2}*:
*R=y_{1}*/ *y_{2}*.* Let us assume
that y_{1} follows a lognormal distribution with the
density f_{1} (y_{1}) and parameters *m* and*
s* (which may be a combination of several lognormal
distributions for exposure variables). Let us further assume that
y_{2} follows a normal distribution with the density f_{2}(y_{2})
with parameters *m* and* s *(e.g. the distribution of
bodyweights). Derivatives of the quantile value *a(m,s,m,s), *take
the form of the ratio where the numerator and denominator are the
integrals with the itegrand f_{1} (y_{1})f_{2}(y_{2})
in the numerator multiplied by the function a_{1}(y_{1},y_{2})**
**and the integrand f_{1}(y_{1}) f_{2}(y_{2})
in the denominator multiplied by the function b(y_{1},y_{2})**.
**For our example (lognormal distribution in the numerator and
normal distribution in the denominator) these functions take the
following form: a_{1}(y_{1},y_{2})=(ln y_{1}
- m)/s^{2}, a_{2}(y_{1},y_{2})=((m
- ln y_{1})^{2} - s^{2})/s^{3}, a_{3}
(y_{1},y_{2})=(y_{2} - m)/s^{2},
and a_{4} (y_{1},y_{2})=((m -** **y_{2})^{2}
- s^{2})/s^{3} for the sensitivity of *a(m,s,m,s)*
to m, s, m, and s, respectively; and b(y_{1},y_{2})=
y_{2}(m - s^{2} - ln y_{1})/y_{1}s^{2}.
In this manner, all sensitivities of the upper percentile of risk
can be obtained together with percentile *a(m,s,m,s) *in the
same Monte Carlo simulation without appreciable additional
calculations. Analytical formulas for sensitivities for the
typical models used in risk assessment must be derived only once.
After this is done, the results can be used in routine Monte
Carlo simulations.

[1]S.Uryasev "Derivatives of Probability Functions and
Integrals Over Sets Given by Inequalities," *Journal of
Computational and Applied Mathematics, *v.56 (1994), in press.

*Work performed under the auspices of the U.S. Department of Energy.