Notes

An historical perspective The origins of motion capture One of the earliest starts of motion capturing is the famous horse in 1878 in motion “video”. This was the start of all the modern cameras. One of the earliest human body motion capture was in military for moving efficiency purposes in 1883. This website has many historical resources on the topic. The problem is still a problem in modern times. If we want to create models to mimic humans, it surely could be nice to understand how humans move and think. This is the general line of though of this line of research. ...

In questa serie di appunti proviamo a descrivere tutto quello che sappiamo al meglio riguardanti gli autoencoders Blog di riferimento Blog secondario che sembra buono Introduzione agli autoencoders L’idea degli autoencoders è rappresentare la stessa cosa attraverso uno spazio minore, in un certo senso è la compressione con loss. Per cosa intendiamo qualunque tipologia di dato, che può spaziare fra immagini, video, testi, musica e simili. Qualunque cosa che noi possiamo rappresentare in modo digitale possiamo costruirci un autoencoder. Una volta scelta una tipologia di dato, come per gli algoritmi di compressione, valutiamo come buono il modello che riesce a comprimere in modo efficiente e decomprimere in modo fedele rispetto all’originale. Abbiamo quindi un trade-off fra spazio latente, che è lo spazio in cui sono presenti gli elementi compressi, e la qualità della ricostruzione. Possiamo infatti osservare che se spazio latente = spazio originale, loss di ricostruzione = 0 perché basta imparare l’identità. In questo senso si può dire che diventa sensato solo quando lo spazio originale sia minore di qualche fattore rispetto all’originale. Quando si ha questo, abbiamo più difficoltà di ricostruzione, e c’è una leggera perdita in questo senso. ...

A motorcycle talk founded the CMMRS. I am wondering if small applications like these are enough in having actual impact. It does have some impacts on the people there mmm. The seed was eaten, as his metaphor, industry is eating out most of the talent. And academia is still important as a propulsion engine. I don’t think it is something general, you need something specific for sure. They say its not recruiting informercial and repeat it a lot. But it seems very strange. They say its more christian like opportunity to ...

This note is one of the few notes that was generated with the help of chatgpt. Bayesian Information Criterion (BIC) The Bayesian Information Criterion (BIC) is a model selection criterion that helps compare different statistical models while penalizing model complexity. It is rooted in Bayesian probability theory but is commonly used even in frequentist settings. Mathematically Precise Definition For a statistical model $M$ with $k$ parameters fitted to a dataset $\mathcal{D} = \{x_1, x_2, \dots, x_n\}$, the BIC is defined as: ...

Let’s consider first a simple model for apartments in a college. Here we are interested to predict the prices of the rooms, and how we can allocate them to students. For simplicity, we will assume that they are all equal except for the location, which could be inner or outer. Types of variables Economist will say that parameters for a model, i.e. variables that are fixed for some type of analysis exogenous variables, while the variables of interest of a model endogenous variables. In this setting, it could be the location for a certain room. ...

In this note we will first talk about briefly some of the main differences of the three main approaches regarding statistics: the bayesian, the frequentist and the statistical learning methods and then present the concept of the estimator, compare how the approaches differ from method to method, we will explain maximum likelihood estimator and the Rao-Cramer Bound. Short introduction to the statistical methods Bayesian $$ p(\theta \mid X) = \frac{1}{z}p(X \mid \theta) p(\theta) $$The quantity $P(X \mid \theta)$ could be very complicated if our model is complicated. ...

Here we analyze how demand changes when prices and income changes. Types of Goods Here we will define two main types of Goods: Normal Goods: The demand increases linearly with the income. Inferior or Ordinary Goods: The demand decreases when the income is higher, one example is low quality food. Necessary Good: The demand is increases sublinearly (e.g. constantly) with respect to the income. Luxury Good: The demand increases more than linearly with the income. Giffen Good: The decrease in price leads to a decrease in demand. Curves of Demand Income offer curve It is quite intuitive to visualize: just expand the income while keeping utility and priced fixed, you will see the optimal point translating along the curve. ...

Budget A definition for Budget Economist want simple models to start to model things. One of the things we will model here is how do you describe what you can afford about some goods. Budget Set $$ \text{Budget Constraint inequality}:p^{T}x \leq m $$ We define Budget Set to be $P_{p, m} = \left\{ x \in \mathbb{R}^{d} : p^{T}x \leq m\right\}$ Composite Goods $$ p_{1}x_{1} + x_{2} \leq m $$ Where $x_{2} = \sum_{i = 2}^{n} p_{i}x_{i}$. In this case, $x_{2}$ is called composite good, which is a sort of abstraction to simplify some calculus. ...

Indipendence of Cause and Mechanism The cause and mechanisms shall be separated. One direction is the cause, the other is the mechanism that actuates the effect of the cause. Take for example the altitude and temperature example in (Peters et al. 2017) chapter 2. The distribution of the cause p(a)p(a)p(a) and the mechanism p(t∣a)p(t|a)p(t∣a) that maps it to the effect are independent. Structural Causal Models We say that given two statistical processes $C, E$, where $C$ is the cause and E the effect, then the ...

Softmax is one of the most important functions for neural networks. It also has some interesting properties that we list here. This function is part of The Exponential Family, one can also see that the sigmoid function is a particular case of this softmax, just two variables. Sometimes this could be seen as a relaxation of the action potential inspired by neuroscience (See The Neuron for a little bit more about neurons). This is because we need differentiable, for gradient descent. The action potential is an all or nothing thing. ...