The em algorithm and extensions pdf

This article is about the decision theory concept. When dealing with gains, it is referred to as “maximin”—to the em algorithm and extensions pdf the minimum gain.

For the sake of example, we consider only pure strategies. The definition is very similar to that of the maximin value—only the order of the maximum and minimum operators is inverse. B2 since the worst possible result is then no payment. So a more stable strategy is needed. Minimax is used in zero-sum games to denote minimizing the opponent’s maximum payoff.

Maximin” is a term commonly used for non-zero-sum games to describe the strategy which maximizes one’s own minimum payoff. A can, at best, draw, then player B’s best move is the one leading to a draw. Late in the game, it’s easy to see what the “best” move is. The Minimax algorithm helps find the best move, by working backwards from the end of the game. A value is associated with each position or state of the game. The player then makes the move that maximizes the minimum value of the position resulting from the opponent’s possible following moves. The above algorithm will assign a value of positive or negative infinity to any position since the value of every position will be the value of some final winning or losing position.

We can then limit the minimax algorithm to look only at a certain number of moves ahead. 12 plies, then applied a heuristic evaluation function. The number of nodes to be explored for the analysis of a game is therefore approximately the branching factor raised to the power of the number of plies. Other heuristic pruning methods can also be used, but not all of them are guaranteed to give the same result as the un-pruned search. The heuristic value is a score measuring the favorability of the node for the maximizing player.

Hence nodes resulting in a favorable outcome, such as a win, for the maximizing player have higher scores than nodes more favorable for the minimizing player. For non terminal leaf nodes at the maximum search depth, an evaluation function estimates a heuristic value for the node. The quality of this estimate and the search depth determine the quality and accuracy of the final minimax result. Suppose the game being played only has a maximum of two possible moves per player each turn. Minimax theory has been extended to decisions where there is no other player, but where the consequences of decisions depend on unknown facts. For example, deciding to prospect for minerals entails a cost which will be wasted if the minerals are not present, but will bring major rewards if they are. Minimax thus can be used on ordinal data, and can be more transparent.

Rawls defined this principle as the rule which states that social and economic inequalities should be arranged so that “they are to be of the greatest benefit to the least-advantaged members of society”. Upper Saddle River, New Jersey: Prentice Hall, pp. During the 1997 match, the software search extended the search to about 40 plies along the forcing lines, even though the nonextended search reached only about 12 plies. Can the Maximin Principle Serve as a Basis for Morality? This page was last edited on 15 November 2017, at 01:02.

Upper Saddle River, NJ: Pearson Prentice Hall, 2005. E-M algorithm such as clustering using the soft K-means algorithm, and emphasizes the variational view of the E-M algorithm. EM equations for Gaussian Mixtures and Gaussian Mixture Hidden Markov Models. Beal includes comparisons of EM to Variational Bayesian EM and derivations of several models including Variational Bayesian HMMs. Frank Dellaert, gives an easier explanation of EM algorithm in terms of lowerbound maximization. A self contained derivation of the EM Algorithm by Sean Borman. Jörg Sander and Xiaowei Xu in 1996.

DBSCAN is one of the most common clustering algorithms and also most cited in scientific literature. Consider a set of points in some space to be clustered. All points not reachable from any other point are outliers. Because they are all reachable from one another, they form a single cluster. Point N is a noise point that is neither a core point nor directly-reachable. All points within the cluster are mutually density-connected.

If a point is density-reachable from any point of the cluster, it is part of the cluster as well. It starts with an arbitrary starting point that has not been visited. This point’s ε-neighborhood is retrieved, and if it contains sufficiently many points, a cluster is started. Otherwise, the point is labeled as noise. Note that this point might later be found in a sufficiently sized ε-environment of a different point and hence be made part of a cluster. If a point is found to be a dense part of a cluster, its ε-neighborhood is also part of that cluster. Hence, all points that are found within the ε-neighborhood are added, as is their own ε-neighborhood when they are also dense.