This note will mainly attempt to summarize the introduction of some intuitive notions of probability used in common sense human reasoning. Most of what is said here is available here (Jaynes 2003). Three intuitive notions of probability Jaynes presents some forms of inference that are not possible in classical first order or propositional logic, yet they are frequent in human common sense reasoning. Let’s present some rules and some examples along them:...
This note extends the content Markov Processes in this specific context. Standard notions Explore-exploit dilemma 🟩 We have seen something similar also in Active Learning when we tried to model if we wanted to look elsewhere or go for the maximum value we have found. The dilemma under analysis is the explore-exploit dilemma: whether if we should just go for the best solution we have found at the moment, or look for a better one....
As we will briefly see, Kernels will have an important role in many machine learning applications. In this note we will get to know what are Kernels and why are they useful. Intuitively they measure the similarity between two input points. So if they are close the kernel should be big, else it should be small. We briefly state the requirements of a Kernel, then we will argue with a simple example why they are useful....
With variational inference we want to find a good approximation of the posterior distribution from which it is easy to sample. The objective is to approximate the posterior with a simpler one, because sometimes the prior or likelihood are difficult to compute. $$ p(\theta \mid x_{1:n}, y_{1:n}) = \frac{1}{z} p(y_{1:n} \mid \theta, x_{1:n}) p(\theta \mid x_{1:n}) \approx q(\theta \mid \lambda) $$ For Bayesian Linear Regression we had high dimensional Gaussians which made the inference closed form, in general this is not true, so we need some kinds of approximation....
Andiamo a parlare di processi Markoviani. Dobbiamo avere bene a mente il contenuto di Markov Chains prima di approcciare questo capitolo. Markov property Uno stato si può dire di godere della proprietà di Markov se, intuitivamente parlando, possiede già tutte le informazioni necessarie per predire lo stato successivo, ossia, supponiamo di avere la sequenza di stati $(S_n)_{n \in \mathbb{N}}$, allora si ha che $P(S_k | S_{k-1}) = P(S_k|S_0S_1...S_{k - 1})$, ossia lo stato attuale in $S_{k}$ dipende solamente dallo stato precedente....