Notes

DI Law of Large Numbers e Central limit theorem ne parliamo in Central Limit Theorem and Law of Large Numbers. Usually these methods are useful when you need to calculate following something similar to Bayes rule, but don’t know how to calculate the denominator, often infeasible integral. We estimate this value without explicitly calculating that. Interested in $\mathbb{P}(x) = \frac{1}{z} \mathbb{P}^{*}(x) = \frac{1}{Z} e^{-E(x)}$ Can evaluate E(x) at any x. Problem 1 Make samples x(r) ~ 2 P Problem 2 Estimate expectations $\Phi = \sum_{x}\phi(x)\mathbb{P}(x)$) What we’re not trying to do: We’re not trying to find the most probable state. We’re not trying to visit all typical states. Law of large numbers $$ S_{n} = \sum^n_{i=1} x_{i} ,:, \bar{x}_{n} = \frac{S_{n}}{n} $$$$ \bar{x}_{n} \to \mu $$ Ossia il limite converge sul valore atteso di tutte le variabili aleatorie. ...

Sono variazioni possibili equivalenti: • Nastri addizionali • Testine addizionali • Nastri infiniti su entrambi i lati • Non-determinismo • Scelta probabilistica • Scelta quantistica Si può dire che la definizione di TM è stata robusta nella storia perché tantissimi formalismi che intuitivamente sembrano essere molto diversi rispetto alla TM alla fine possono essere dimostrate essere equivalenti. Turing con nastri addizionali Questo è presente in modo abbastanza facile sul Sipser. ...

PAC Learning is one of the most famous theories in learning theory. Learning theory concerns in answering questions like: What is learnable? Somewhat akin to La macchina di Turing for computability theory. How well can you learn something? PAC is a framework that allows to formally answer these questions. Now there is also a bayesian version of PAC in which there is a lot of research. Some definitions Empirical Risk Minimizer and Errors $$ \arg \min_{\hat{c} \in \mathcal{H}} \hat{R}_{n}(\hat{c}) $$ Where the inside is the empirical error. ...

The DP (Dirichlet Processes) is part of family of models called non-parametric models. Non parametric models concern learning models with potentially infinite number of parameters. One of the classical application is unsupervised techniques like clustering. Intuitively, clustering concerns in finding compact subsets of data, i.e. finding groups of points in the space that are particularly close by some measure. The Dirichlet Process See Beta and Dirichlet Distributions for the definition and intuition of these two distributions. One quite important thing that Dirichlet allows to do is the ability of assigning an ever growing number of clusters to data. This models are thus quite flexible to change and growth. ...

This is a quite good resource about this part of Support Vector Machines (step by step derivation). (Bishop 2006) chapter 7 is a good resource. The main idea about this supervised method is separating with a large gap. The thing is that we have a hyperplane, when this plane is projected to lower dimensional data, it can look like a non-linear separator. After we have found this separator, we can intuitively have an idea of confidence based on the distance of the separator. ...

The beta distribution The beta distribution is a powerful tool for modeling probabilities and proportions between 0 and 1. Here’s a structured intuition to grasp its essence: Core Concept The beta distribution, defined on $[0, 1]$, is parameterized by two shape parameters: α (alpha) and β (beta). These parameters dictate the distribution’s shape, allowing it to flexibly represent beliefs about probabilities, rates, or proportions. Key Intuitions a. “Pseudo-Counts” Interpretation α acts like “successes” and β like “failures” in a hypothetical experiment. Example: If you use Beta(5, 3), it’s as if you’ve observed 5 successes and 3 failures before seeing actual data. After observing x real successes and y real failures, the posterior becomes Beta(α+x, β+y). This makes beta the conjugate prior for the binomial distribution (bernoulli process). b. Shape Flexibility Uniform distribution: When α = β = 1, all values in [0, 1] are equally likely. Bell-shaped: When α, β > 1, the distribution peaks at mode = (α-1)/(α+β-2). Symmetric if α = β (e.g., Beta(5, 5) is centered at 0.5). U-shaped: When α, β < 1, density spikes at 0 and 1 (useful for modeling polarization, meaning we believe the model to only produce values at 0 or 1, not in the middle.). Skewed: If α > β, skewed toward 1; if β > α, skewed toward 0. c. Moments Mean: $α/(α+β)$ – your “expected” probability of success. Variance: $αβ / [(α+β)²(α+β+1)]$ – decreases as α and β grow (more confidence). $$ \text{Mode} = \frac{\alpha - 1}{\alpha + \beta - 2} $$The mathematical model $$ \text{Beta} (x \mid a, b) = \frac{1}{B(a, b)} \cdot x^{a -1 }(1 - x)^{b - 1} $$ Where $B(a, b) = \Gamma(a) \Gamma(b) / \Gamma( + b)$ And $\Gamma(t) = \int_{0}^{\infty}e^{-x}x^{t - 1} \, dx$ ...

Introduzione alle funzioni del markup La semantica di una parola è caratterizzata dalla mia scelta (design sul significato). Non mi dice molto, quindi proviamo a raccontare qualcosa in più. Definiamo markup ogni mezzo per rendere esplicita una particolare interpretazione di un testo. In particolare è un modo per esplicitare qualche significato. (un po’ come la punteggiatura, che da qualche altra informazione oltre le singole parole, rende più chiaro l’uso del testo). ...

A data model is an abstract view over the data that hides the way it is stored physically. The same idea from (Codd 1970) This is why we should not modify data directly, but pass though some abstraction that maintain the properties of that specific data model. Data Models Tree view We can view all JSON and XML data, as presented in Markup, as trees. This structure is usually quite evident, as it is inherent in their design. Converting from the tree structure to a memory model is known as serialization, while the reverse process is called parsing. ...

We want to know how to handle systems that have a large number of data. In previous lesson we have discovered how to quickly access and make Scalable systems with huge dimensions, see Cloud Storage. Object storage could store billions of files, we want to handle millions of petabyte files. Designing DFSs The Use Case Remember that the size of the files where heavily limited for Cloud Storage. The physical limitation was due to the limited size of a single hard disk, which was usually in the order of the Terabytes. Here, we would like to easily store petabytes of data in a single file, for example big datasets. Another feature that should be easily supported is highly concurrent access to the filesystem, last but not least being able to set up permissions in the system. ...

Machine learning cannot distinguish between causal and environment features. Shortcut learning Often we observe shortcut learning: the model learns some dataset dependent shortcuts (e.g. the machine that was used to take the X-ray) to make inference, but this is very brittle, and is not usually able to generalize. Shortcut learning happens when there are correlations in the test set between causal and non-causal features. Our object of interest should be the main focus, not the environment around, in most of the cases. For example, a camel in a grass land should still be recognized as a camel, not a cow. One solution could be engineering invariant representations which are independent of the environment. So having a kind of encoder that creates these representations. ...