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An optimization system is provided utilizing a Bayesian neural network calculation of a derivative wherein an output is optimized with respect to an input utilizing a stochastical method that averages over many regression models. This is done such that constraints from first principal models are inc

An optimization system is provided utilizing a Bayesian neural network calculation of a derivative wherein an output is optimized with respect to an input utilizing a stochastical method that averages over many regression models. This is done such that constraints from first principal models are incorporated in terms of prior art distributions.

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An optimization system is provided utilizing a Bayesian neural network calculation of a derivative wherein an output is optimized with respect to an input utilizing a stochastical method that averages over many regression models. This is done such that constraints from first principal models are inc

An optimization system is provided utilizing a Bayesian neural network calculation of a derivative wherein an output is optimized with respect to an input utilizing a stochastical method that averages over many regression models. This is done such that constraints from first principal models are incorporated in terms of prior art distributions. a motorcycle, office equipment, a computer, computer equipment, a residential premises, a commercial premises, an article of personal property, and an article of commercial property.3. The apparatus of claim 1, the memory device further containing information related to at least one of one of a vehicle, a motor vehicle, a truck, construction equipment, farm equipment, a boat, a recreational vehicle, an airplane, an aircraft, a motorcycle, office equipment, a computer, computer equipment, a residential premises, a commercial premises, an article of personal property, an article of commercial property, an individual, a business entity, a repair cost, a replacement cost, a probability of damage, a probability of post-warranty repair, historical leasing information, one of locality, regional, geographical, and seasonal, information corresponding to the lease, a usage pattern, a usage habit, a manufacturer's warranty, a lease term, a lease duration, historical repair information, repair frequency information, insurance policy information, insurance premium information, insurance product information, insurance service information, an insurance premium rebate incentive program, insurance premium rebate incentive information, actuarial information, statistical information, risk information, and risk of loss information.4. The apparatus of claim 1, the third data set containing information regarding an insurance premium rebate incentive, and the processor generating the forth data set containing at least one of information, an insurance premium, an insurance policy, and an insurance product, containing at least one of a premium rebate incentive feature and an premium rebate incentive provision.5. The apparatus of claim 4, the processor determining whether the at least one of a premium rebate incentive feature and an premium rebate incentive provision is in effect, and the processor calculating an amount of an insurance premium to be refunded.6. The apparatus of claim 1, further comprising:an input device for inputting information contained in at least one of the first data set, the second data set, and the third data set,and the one of a display device and an output device facilitating a presentation of at least one of information contained in the fourth data set, the insurance premium, the insurance policy, an the insurance product, to a prospective policy holder.7. The apparatus of claim 1, further comprising:a receiver for receiving a request for information contained in at least one of the first data set, the second data set, the third data set, the fourth data set, and the fifth data set, from a remote communication device; anda transmitter for transmitting information contained in at least one of the first data set, the second data set, the third data set, the fourth data set, and the fifth data set, to the remote communication device in response to said request.8. The apparatus of claim 1, wherein the processor identifies a credit derivative for at least one of providing a hedging position, providing insurance, and providing reinsurance, for the at least one of a liability, a potential liability, and a risk of loss, associated with the at least one of an insurance policy and an insurance product.9. The apparatus of claim 1, wherein the processor determines at least one of a price, pricing information, and offering terms, for a credit derivative, the credit derivative at least one of providing a hedging position, providing insurance, and providing reinsurance, for the at least one of a liability, a potential liability, and a risk of loss, associated with the at least one of an insurance policy and an insurance product.10. The apparatus of claim 1, wherein the device for at least one of displaying and outputting information provides information concerning a credit derivative, the credit derivative at least one of providing a hedging position, providing insuran ce, and providing reinsurance, for the at least one of a liability, a potential liability, and a risk of loss, associated with the at least one of an insurance policy and an insurance product.11. An apparatus for processing lease insurance information, comprising: a memory device for storing a first data set, the first data set containing information for generating at least one of an insurance premium and an insurance policy for providing insurance for post warranty repairs for a leased entity;a processor for processing the first data set in conjunction with a second data set and a third data set, the second data set containing information regarding at least one of the entity to be leased and a term of the lease, and the third data set containing information regarding at least one of a driving history of the leasing individual, a driving history of the leasing entity, a usage history of the leasing individual, a usage history of the leasing entity, an insurance history of the leasing individual, an insurance history of the leasing entity, a past leasing history of the leasing individual, a past leasing history of the leasing entity, a desired lease insurance coverage, a lease insurance deductible, and a lease insurance policy term, the processor generating a fourth data set containing at least one of an insurance premium and an insurance policy for providing at least one of an insurance policy and an insurance product for post warranty repairs for the leased entity;a device for at least one of displaying and outputting information contained in at least one of the fourth data set, the insurance premium, the insurance policy, and the insurance product,the processor generating a fifth data set containing information regarding at least one of a liability, a potential liability, and a risk of loss, associated with the at least one of an insurance policy and an insurance product.12. The apparatus of claim 11, the entity being at least one of a vehicle, a motor vehicle, a truck, construction equipment, farm equipment, a boat, a recreational vehicle, an airplane, an aircraft, a motorcycle, office equipment, a computer, computer equipment, a residential premises, a commercial premises, an article of personal property, and an article of commercial property.13. The apparatus of claim 11, the memory device further containing information related to at least one of one of a vehicle, a motor vehicle, a truck, construction equipment, farm equipment, a boat, a recreational vehicle, an airplane, an aircraft, a motorcycle, office equipment, a computer, computer equipment, a residential premises, a commercial premises, an article of personal property an article of commercial property, an individual, a business entity, a repair cost, a replacement cost, a probability of damage, a probability of post-warranty repair, historical leasing information, one of locality, regional, geographical, and seasonal, information corresponding to the lease, a usage pattern, a usage habit, a manufacturer's warranty, a lease term, a lease duration, historical repair information, repair frequency information, insurance policy information, insurance premium information, insurance product information, insurance service information, an insurance premium rebate incentive program, insurance premium rebate incentive information, actuarial information, statistical information, risk information, and risk of loss information.14. The apparatus of claim 11, the third data set containing information regarding an insurance premium rebate incentive, and the processor generating the forth data set containing at least one of information, an insurance premium, an insurance policy, and an insurance product, containing at least one of a premium rebate incentive feature and an premium rebate incentive provision.15. The apparatus of claim 14, the processor determining whether the at least one of a premium rebate ince ntive feature and an premium rebate incentive provision is in effect, and the processor calculating an amount of an insurance premium to be refunded.16. The apparatus of claim 11, further comprising:an input device for inputting information contained in at least one of the first data set, the second data set, and the third data set,and the one of a display device and an output device facilitating a presentation of at least one of information contained in the fourth data set, the insurance premium, the insurance policy, an the insurance product, to a prospective policy holder.17. The apparatus of claim 11, further comprising:a receiver for receiving a request for information contained in at least one of the first data set, the second data set, the third data set, the fourth data set, and the fifth data set, from a remote communication device; anda transmitter for transmitting information contained in at least one of the first data set, the second data set, the third data set, the fourth data set, and the fifth data set, to the remote communication device in response to said request.18. The apparatus of claim 11, wherein the processor identifies a credit derivative for at least one of providing a hedging position, providing insurance, and providing reinsurance, for the at least one of a liability, a potential liability, and a risk of loss, associated with the at least one of an insurance policy and an insurance product.19. The apparatus of claim 11, wherein the processor determines at least one of a price, pricing information, and offering terms, for a credit derivative, the credit derivative at least one of providing a hedging position, providing insurance, and providing reinsurance, for the at least one of a liability, a potential liability, and a risk of loss, associated with the at least one of an insurance policy and an insurance product, and further wherein the device for at least one of displaying and outputting information provides information concerning the credit derivative.20. A method for processing lease insurance information, comprising: storing a first data set, the first data set containing information for generating at least one of an insurance premium and an insurance policy for providing insurance for at least one of excess wear and tear and for a leased entity and post warranty repairs for a leased entity;processing the first data set in conjunction with a second data set and a third data set, the second data set containing information regarding at least one of the entity to be leased and a term of the lease, and the third data set containing information regarding at least one of a driving history of the leasing individual, a driving history of the leasing entity, a usage history of the leasing individual, a usage history of the leasing entity, an insurance history of the leasing individual, an insurance history of the leasing entity, a past leasing history of the leasing individual, a past leasing history of the leasing entity, a desired lease insurance coverage, a lease insurance deductible, and a lease insurance policy term;generating a fourth data set containing at least one of an insurance premium and an insurance policy for providing at least one of an insurance policy and an insurance product for at least one of excess wear and tear and for a leased entity and post warranty repairs for a leased entity;at least one of displaying and outputting information contained in at least one of the fourth data set, the insurance premium, the insurance policy, and the insurance product; andgenerating a fifth data set containing information regarding at least one of a liability, a potential liability, and a risk of loss, associated with the at least one of an insurance policy and an insurance product. r transaction numbers which are sent from a transponder (14) to an interrogator (12). By incrementing the transaction counter stored in the transponder with successful transactions the toll authority can ascertain whether accounting of a transaction has been missed (i.e., a transaction number missing from the sequence), or double-counted (i.e., two transactions with the same transaction number).