Probability smoothing

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On Probability Estimation by Exponential Smoothing

Probability estimation is essential for every statistical data compression algorithm. In practice probability estimation should be adaptive, recent observations should receive a higher weight than older observations. We present a probability estimation method based on exponential smoothing that satisfies this requirement and runs in constant time per letter. Our main contribution is a theoretical analysis in case of a binary alphabet for various smoothing rate sequences: We show that the redundancy w.r.t. a piecewise stationary model with $s$ segments is $O\left(s\sqrt n\right)$ for any bit sequence of length $n$, an improvement over redundancy $O\left(s\sqrt{n\log n}\right)$ of previous approaches with similar time complexity

Pointwise Convergence in Probability of General Smoothing Splines

Establishing the convergence of splines can be cast as a variational problem which is amenable to a $\Gamma$-convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, $n$, as $\lambda_n=n^{-p}$. Using standard theorems from the $\Gamma$-convergence literature, we prove that the general spline model is consistent in that estimators converge in a sense slightly weaker than weak convergence in probability for $p\leq \frac{1}{2}$. Without further assumptions we show this rate is sharp. This differs from rates for strong convergence using Hilbert scales where one can often choose $p>\frac{1}{2}$

Smoothing in Probability Estimation Trees

Classification learning is a type of supervised machine learning technique that uses a classification model (e.g. decision tree) to predict unknown class labels for previously unseen instances. In many applications it can be very useful to additionally obtain class probabilities for the different class labels. Decision trees that yield these probabilities are also called probability estimation trees (PETs). Smoothing is a technique used to improve the probability estimates. There are several existing smoothing methods, such as the Laplace correction, M-Estimate smoothing and M-Branch smoothing. Smoothing does not just apply to PETs. In the field of text compression, PPM in particular, smoothing methods play a important role. This thesis migrates smoothing methods from text compression to PETs. The newly migrated methods in PETs are compared with the best of the existing smoothing methods considered in this thesis under different experiment setups. Unpruned, pruned and bagged trees are considered in the experiments. The main finding is that the PPM-based methods yield the best probability estimate when used with bagged trees, but not when used with individual (pruned or unpruned) trees