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In dynamic programming, a method of mathematical optimization, backward induction is used for solving the Bellman equation. [ 3 ] [ 4 ] In the related fields of automated planning and scheduling and automated theorem proving , the method is called backward search or backward chaining .
Dynamic programming is both a mathematical optimization ... by working backwards, ... any previous time can be calculated by backward induction using the ...
Bellman showed that a dynamic optimization problem in discrete time can be stated in a recursive, step-by-step form known as backward induction by writing down the relationship between the value function in one period and the value function in the next period. The relationship between these two value functions is called the "Bellman equation".
Stochastic dynamic programming deals with problems in which the current period reward and/or the next period state are random, i.e. with multi-stage stochastic systems. The decision maker's goal is to maximise expected (discounted) reward over a given planning horizon.
The algorithm makes use of the principle of dynamic programming to efficiently compute the values that are required to obtain the posterior marginal distributions in two passes. The first pass goes forward in time while the second goes backward in time; hence the name forward–backward algorithm.
Forward induction is so called because just as backward induction assumes future play will be rational, forward induction assumes past play was rational. Where a player does not know what type another player is (i.e. there is imperfect and asymmetric information), that player may form a belief of what type that player is by observing that ...
Backward chaining is implemented in logic programming by SLD resolution. Both rules are based on the modus ponens inference rule. It is one of the two most commonly used methods of reasoning with inference rules and logical implications – the other is forward chaining. Backward chaining systems usually employ a depth-first search strategy, e ...
In value iteration (Bellman 1957), which is also called backward induction, the function is not used; instead, the value of () is calculated within () whenever it is needed. Substituting the calculation of π ( s ) {\displaystyle \pi (s)} into the calculation of V ( s ) {\displaystyle V(s)} gives the combined step [ further explanation needed ] :