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Backpropagation computes the gradient of a loss function with respect to the weights of the network for a single input–output example, and does so efficiently, computing the gradient one layer at a time, iterating backward from the last layer to avoid redundant calculations of intermediate terms in the chain rule; this can be derived through ...
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms.
The above system is known as the inverse Wiener-Hopf factor. The backward recursion is the adjoint of the above forward system. The result of the backward pass may be calculated by operating the forward equations on the time-reversed and time reversing the result. In the case of output estimation, the smoothed estimate is given by
The backward algorithm complements the forward algorithm by taking into account the future history if one wanted to improve the estimate for past times. This is referred to as smoothing and the forward/backward algorithm computes (|:) for < <. Thus, the full forward/backward algorithm takes into account all evidence.
The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, [1] and the LaTeX symbol.
Here two sets of prediction equations are combined into a single estimation scheme and a single set of normal equations. One set is the set of forward-prediction equations and the other is a corresponding set of backward prediction equations, relating to the backward representation of the AR model:
An example of backward chaining. If X croaks and X eats flies – Then X is a frog; If X chirps and X sings – Then X is a canary; If X is a frog – Then X is green; If X is a canary – Then X is yellow; With backward reasoning, an inference engine can determine whether Fritz is green in four steps.
The ultimatum game does have several other Nash Equilibria which are not subgame perfect and therefore do not arise via backward induction. The ultimatum game is a theoretical illustration of the usefulness of backward induction when considering infinite games, but the ultimatum games theoretically predicted results do not match empirical ...