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In this article we show the rough outline of a computer algorithm to generatelower bounds on the exponential function of (in principle) arbitrary precision.We implemented this to generate all necessary analytic terms for the Boltzmannmachine partition function thus leading to lower bounds of any order. It turnsout that the extra variational parameters can be optimized analytically. Weshow that bounds upto nineth order are still reasonably calculable in practicalsituations. The generated terms can also be used as extra correction terms(beyond TAP) in mean field expansions.

General Lower Bounds based on Computer Generated Higher Order Expansions