The so-called “log sum of exponentials” is a functional form commonly encountered in dynamic discrete choice models in economics. It arises when the choice-specific error terms have a type 1 extreme value distribution–a very common assumption. In these problems there is typically a vector of length which represents the non-stochastic payoff associated with each of choices. In addition to , the payoff associated with choice has a random component . If these components are independent and identically distributed across and follow the type 1 extreme value distribution, then the expected payoff from choosing optimally is We need to numerically evaluate the expression on the right hand side for many different vectors .
Special care is required to calculate this expression in compiled languages such as Fortran or C to avoid numerical problems. The function needs to work for with very large and very small components. A large can cause overflow due to the exponentiation. Similarly, when the are large (in absolute value) and negative, the exponential terms vanish. Taking the logarithm of a very small number can result in underflow.
A simple transformation can avoid both of these problems. Consider the case where . Note that we can write for any . Furthermore, we have We can choose in a way that reduces the possibility of overflow. Underflow is also possible when taking the logarithm of a number close to zero, since as . Thus, we also need to account for large negative elements of .
The following code is a Fortran implementation of a function which
makes the adjustments described above. In order to prevent numerical
overflow, the function shifts the vector
v by a constant
log, and then adjusts the result to account for
function log_sum_exp(v) result(e) real, dimension(:), intent(in) :: v ! Input vector real :: e ! Result is log(sum(exp(v))) real :: c ! Shift constant ! Choose c to be the element of v that is largest in absolute value. if ( maxval(abs(v)) > maxval(v) ) then c = minval(v) else c = maxval(v) end if e = log(sum(exp(v-c))) + c end function log_sum_exp
Note that this function is still not completely robust to very poorly
v and so over- or underflow is still possible (just
much less likely).