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Lempel Ziv coding (recurrence time coding). Let Rn (X n ) be the last time in the past that we have seen a block of n symbols X n . Then 1 n log Rn H (X), and encoding by describing the recurrence time is asymptotically optimal. Lempel Ziv coding (sequence parsing). If a sequence is parsed into the shortest phrases not seen before (e.g., 011011101 is parsed to 0,1,10,11,101,...) and l(x n ) is the description length of the parsed sequence, then 1 lim sup l(X n ) H (X) n with probability 1 (13.137)

program running time and program length Kolmogorov complexity focuses on minimizing along the second axis, and computational complexity focuses on minimizing along the rst axis Little work has been done on the simultaneous minimization of the two Physics (Thermodynamics) Statistical mechanics is the birthplace of entropy and the second law of thermodynamics Entropy always increases Among other things, the second law allows one to dismiss any claims to perpetual motion machines We discuss the second law brie y in 4 Mathematics (Probability Theory and Statistics) The fundamental quantities of information theory entropy, relative entropy, and mutual information are de ned as functionals of probability distributions In turn, they characterize the behavior of long sequences of random variables and allow us to estimate the probabilities of rare events (large deviation theory) and to nd the best error exponent in hypothesis tests Philosophy of Science (Occam s Razor).

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and also P(bi = +1) = P(bi = 1) = e L(bi ) 1 = L(bi ) 1+e 1 + e L(bi ) (15) (16)

Of course, unlike a normal Linux system, the disk space situation is very different:

William of Occam said Causes shall not be multiplied beyond necessity, or to paraphrase it, The simplest explanation is best Solomonoff and Chaitin argued persuasively that one gets a universally good prediction procedure if one takes a weighted combination of all programs that explain the data and observes what they print next Moreover, this inference will work in many problems not handled by statistics For example, this procedure will eventually predict the subsequent digits of When this procedure is applied to coin ips that come up heads with probability 07, this too will be inferred When applied to the stock market, the procedure should essentially nd all the laws of the stock market and extrapolate them optimally In principle, such a procedure would have found Newton s laws of physics.

Goodness-of-Fit Issues As stated previously, standard missing data procedures are available for LCA and LTA that successfully adjust parameter estimates for MAR missingness. However, the G2 statistic used to assess the t of the resulting models is potentially misleading. This is because the G2 re ects two components: deviations between the expected response pattern frequencies and the observed frequencies, and deviations from MCAR missingness. To the extent that missing data are not MCAR, the G2 can be in ated, making model t appear poorer than it really is. This happens even when MAR missingness has been successfully adjusted for in the parameter estimates. By separating the two contributions to the traditional G2 statistic, lack of t and departures from MCAR, we can clearly assess the t of latent class models for incomplete data. The contribution due to departures from MCAR can be identi ed by tting the saturated model to the variables in the model. By subtracting this value from the traditional G2 value we can determine the t of the latent class model after taking missing data into account (Little & Rubin, 1987). Note that when the effect of missing data on model t is ignored because the G2 value is in ated, in general researchers are more likely to reject models that actually provide adequate t to the data. Both WinLTA (Collins & Wugalter, 1992; Hyatt & Collins, 2000) and Mplus (Muth n & Muth n, 2001) incorporate this adjustment to model t for latent class models with incomplete data.

Of course, such inference is highly impractical, because weeding out all computer programs that fail to generate existing data will take impossibly long We would predict what happens tomorrow a hundred years from now Economics (Investment) Repeated investment in a stationary stock market results in an exponential growth of wealth The growth rate of the wealth is a dual of the entropy rate of the stock market The parallels between the theory of optimal investment in the stock market and information theory are striking We develop the theory of investment to explore this duality Computation vs Communication As we build larger computers out of smaller components, we encounter both a computation limit and a communication limit Computation is communication limited and communication is computation limited These become intertwined, and thus.

Select an instance of the symbol on the Stage and right-click (Windows) or Control-click (Macintosh), and choose Edit in New Window from the context menu. Edit the symbol as needed. Click the Close box in the upper right corner (Windows) or upper left corner (Macintosh) to close the new window, and click in the main movie window to return to editing the main movie.

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