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. ...

Metodi di registrazione informazione Ci stiamo chiedendo in che modo possiamo registrare attività del cervello e quindi cercare di fare decoding delle informazioni presenti Prima parliamo di alcune tecniche non invasive che ci permettono di vedere alcune attività presenti nel cervello. Metodi macroscopici Functional Magnetic Resonance Imaging Un metodo è fMRI. (ci sono cose ) TODO capire come funziona Electro-Encephalo-Gram EEG che prende direttamente dai segnali Ma il drawback di entrambi è che non registrano attività del singolo array. ...

The Potassium Exchange values We use the measurement by Cole and Curthis 40mS/cm squared was their measure of Potassium ions leaving the membrane $$ \Delta Q = Idt = GA \Delta E dt $$The potassium concentration is 0.155 moles per litre. Where $G$ is the conductance per unit area, $A$ the membrane surface, $E$ voltage deflection Remember that the conductance is the reciprocal of the resistance, and $V = IR \implies I = \frac{V}{R} = GV$ ...

What is a neural system? A neural system is an intricately organized network of specialized cells—primarily neurons, along with a variety of supportive glial cells—that processes and transmits information via electrical and chemical signals. In biological organisms, such systems underpin the entire nervous system, coordinating functions that range from basic reflexes to the complex interplay of perception, thought, and behavior. Early studies in neurobiology revealed that even simple neural circuits can generate coordinated responses, while modern neuroscience has shown that vast, hierarchically structured networks (such as the central and peripheral nervous systems) are responsible for the rich tapestry of animal behavior and cognition ...

Synapses are the connections that exist between one neuron and another, so we can think of them as the communication channel between neurons. Gap Junctions Electrical based 🟩 These are also called Gap Junctions These are more direct connections between neurons, allowing excitation ions to pass through quite directly (this is the difference compared to chemically based ones). It’s a circuit more similar to an electronic one because it’s faster. Another characteristic of these kinds of synapses is that they are two-way channels. ...

Some history: Reticular Theory vs Neuron Doctrine The late 19th century witnessed a debate in neuroscience between Camillo Golgi and Santiago Ramón y Cajal, two pioneers whose opposing views shaped our understanding of the nervous system. This debate centered on the structural and functional organization of neurons, culminating in their joint reception of the 1906 Nobel Prize in Physiology or Medicine. Golgi’s Reticular Theory Golgi proposed the Reticular Theory based on his staining techniques (see #Staining methods), which held that: ...

il livello isa è il livello delle istruzioni 8.1 Struttura Potremmo definire l’architettura di un elaboratore come tutte le parti del processore che una persona abbia bisogno di sapere per scrivere codice assembly. Istruzioni possibili Registri Solitamente le istruzioni sono divise in due parti: 8.1.1 Opcode e indirizzamento Opcode Questo opcode indica la tipologia di istruzione. Per esempio per l’architettura HACK è il primo bit, che indica se è una istruzione C oppure una istruzione A. ...

This is a new framework that is faster than MapReduce (See Massive Parallel Processing). It is written in Scala and has a more functional approach to programming. Spark extends the previous MapReduce framework to a generic distributed dataflow, properly modeled as a DAG. There are other benefits of using Spark instead of the Map reduce Framework: Spark processes data in memory, avoiding the disk I/O overhead of MapReduce, making it significantly faster. Spark uses a DAG to optimize the entire workflow, reducing data shuffling and stage count. But MapReduce sometimes has its advantages: ...

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: $$ \text{BIC} = -2 \cdot \ln \hat{L} + k \cdot \ln(n) $$where: ...

We have a prior $p(\text{model})$, we have a posterior $p(\text{model} \mid \text{data})$, a likelihood $p(\text{data} \mid \text{model})$ and $p(\text{data})$ is called the evidence. Classical Linear regression $$ y = w^{T}x + \varepsilon $$ Where $\varepsilon \sim \mathcal{N}(0, \sigma_{n}^{2}I)$ and it’s the irreducible noise, an error that cannot be eliminated by any model in the model class, this is also called aleatoric uncertainty. One could write this as follows: $y \sim \mathcal{N}(w^{T}x, \sigma^{2}_{n}I)$ and it’s the exact same thing as the previous, so if we look for the MLE estimate now we get ...