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9

If we view minimizing cross entropy as equivalent to maximizing the log-likelihood of the same model, then I believe we can go as far back as RA Fisher. This places the date between 1912 and 1922, depending on how well-developed you wish the theory to be; see discussion in John Aldrich "R. A. Fisher and the Making of Maximum Likelihood 1912 – 1922" ...


11

The earliest I have been able to find is Good, I. J. “Rational Decisions.” Journal of the Royal Statistical Society. Series B (Methodological), vol. 14, no. 1, 1952, pp. 107–114. JSTOR, www.jstor.org/stable/2984087 Look at section 8, "Fair Fees": By itself $\log p_1$ (or $\log(1 - p_1)$) is a measure of the merit of a probability estimate I ...


1

This answer only addresses point (2), the combinatorics. Can one compute the number of admissible sequences (with N, M and δ)? Special case of $\delta=1$ In the special case of $\delta = 1$, your problem is equivalent to the $k$-composition of $N$ with $k = M+1$. If you think of the set of steps $a_i$ between elements of your sequence, you need the ...


1

The method dates back to J. Durbin (1960) “The fitting of time series models” (Intl Statis. Review, 233-244) and was for example exploited in a model selection context by Hannan and Kavaleris (1984) (Biometrika 71, 273-280).


0

To understand why we use the relative and subjective notions of under- and overfitting, we must remember that, as George Box said, "all models are wrong" (see here for an explanation of this aphorism), but some of them are useful. When confronted to data whose generative model is unknown, we can define a set of plausible and competing models to ...


0

I don't think there are papers about feature extraction, probably there are only for single components to be applied into AE in order to reduce reconstruction error. Besides that, if you want to extract features from an image I also suggest an AE using the following approach Train AE to reconstruct images Get encoder and use that to find meaningful features ...


0

I don't have enough reputations to comment, so I express my views in the form of "answer". What strikes me as strange is how your errors are modelled. Normally, measurement errors are modelled as $Z\ =\ Z^{*} \ +\ \eta ,\ \eta \sim N( \mu _{\eta } ,\ \sigma _{\eta })$, where $\ Z^{*}$ is the true measurement of the variable. This measurement error ...


5

I would recommend Seber & Lee (from which I originally learned regression.) Cover most of your topics with proofs. An alternative in the same style, but also covering glm's is Linear Models and Generalizations : Least Squares and Alternatives by Rao et al. A shorter book with a more geometric viewpoint is The Coordinate-Free Approach to Linear Models by ...


4

I think that you might be interested in Stachurski (2016) A Primer in Econometric Theory. The book is quite mathematically oriented. The book is organized into 3 sections: Background - which is all about pure mathematical foundations; vector spaces, linear algebra and matrices, foundations of probability, modeling dependence, asymptotics etc. Foundations ...


0

I am stunned at how many introductory books there are. My recommendation is William Bolstad and James Curran's Introduction to Bayesian Statistics. It has several nice elements to it. First, it covers a large percentage of the conjugate problems and covers all the standard problems in a sophomore-level test for an introduction to statistics course. It also ...


0

A really introductory book is Will Kurt's Bayesian Statistics the Fun Way. This approaches the logic of Bayesian statistics at a very high level - going through the basics of probability, and how prior information can combine with data to affect posterior probabilities. As an ecologist who didn't have a robust statistical training in gradschool, I still get ...


0

The following books on Statistical modeling using R might be helpful Title : An Introduction to Statistical Learning: with Applications in R ISBN: 9781461471370 This book presents some of the most important modeling and prediction techniques, along with relevant applications. Topics include linear regression, classification, resampling methods, support ...


6

In this context the "deviance test" is another term for "likelihood ratio test" (LRT) LRTs are well understood as a means to compare mixed effects models with different fixed effects, provided they are fitted with maximum likelihood, rather than restricted maximum likelihood. In terms of references I don't think you need anything ...


1

As already commented by @whuber, the copula of $(X,Y)$ does not contain enough information to answer this question. The copula only gives information about the dependence structure, but to compute $\DeclareMathOperator{\P}{\mathbb{P}} \P(X > Y)$ information on the marginals in also necessary! As an example, if $(X,Y)$ are independent random variables, ...


3

I need a good, intuitive explanation for why not explicitly accounting for these potential sources of variation will lead to a worse model In my experience it is fairly easy to justify the use of mixed models to a non-techincal audience, at a level that they will be able to understand. I generally try to do this with real-world examples. The main ...


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