Questions tagged [probability-theory]
For questions related to probability theory in the context of artificial intelligence.
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Why don't we also need to approximate $p(x \mid z)$ in the VAE?
In the VAE, we approximate the probability distribution $p(z \mid x)$, where $z$ is the latent vector and $x$ is our data. The reason is that $p(z \mid x)$ becomes impossible to calculate for ...
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Which formula of p(x, y) to use?
The probability distribution $p(x, y)$ can be calculated in two ways :
$p(x, y) = p(y \mid x) p(x)$
$p(x, y) = p(x \mid y) p(y)$
But according to the book Deep Generative Modeling (page number 3 ...
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Derivation in paper Deep Neural Networks as Gaussian processes in ICLR 2018
I am trying to understand the derivation of the main equation in the seminal paper titled Deep Neural Networks as Gaussian processes (in ICLR 2018). I have asked this question in https://math....
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Why are today's neural networks not modeled with probability theory?
In the paper The Perceptron: A probabilistic model for information storage and organization in the brain, Rosenblatt used the probability theory to model his perceptron.
My professor told me that ...
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What do we mean by "orderly opinions" in this sentence in the context of Bayes theorem?
In this page, it's written (emphasis mine)
If probabilities are thought to describe orderly opinions, Bayes theorem describes how the opinions should be updated in the light of new information
What ...
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How would the probability of a document $P(d)$ be computed in the Naive Bayes classifier?
In naive Bayes classification, we estimate the class of a document as follows
$$\hat{c} = \arg \max_{c \in C} P(c \mid d) = \arg \max_{c \in C} \dfrac{ P(d \mid c)P(c) }{P(d)} $$
It has been said in ...
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Why is the equation $\mathbb{E} \left[ (Y - \hat{Y})^2 \right] = \left(f(X) - \hat{f}(X) \right)^2 + \operatorname{Var} (\epsilon)$ true?
In the book An Introduction to Statistical Learning, the authors claim (equation 2.3, p. 19, chapter 2)
$$\mathbb{E} \left[ (Y - \hat{Y})^2 \right] = \left(f(X) - \hat{f}(X) \right)^2 + \operatorname{...
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Bayesian Perceptron: Why to marginalize over neuron's output instead of it's weights?
I found a very interesting paper on the internet that tries to apply Bayesian inference with a gradient-free online-learning approach: Bayesian Perceptron: Towards fully Bayesian Neural Networks.
I ...
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How is the formula for the Bayes error rate with an integral derived?
My questions concern a particular formulation of the Bayes error rate from Wikipedia, summarized below.
For a multiclass classifier, the Bayes error rate may be calculated as follows: $$p = 1 - \sum_{...
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Understanding how to calculate $P(x|c_k)$ for the Bernoulli naïve Bayes classifier
I'm looking at the Bernoulli naïve Bayes classifier on Wikipedia and I understand Bayes theorem along with Gaussian naïve Bayes. However, when looking at how $P(x|c_k)$ is calculated, I don't ...
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How can the V and Q functions take the expectation over a sum where the number of summands is random?
Assume the existence of a Markov Decision Process consisting of:
State space $S$
Action space $A$
Transition model $T: S \times A \times S \to [0,1]$
Reward function $R: S \times A \times S \to \...
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Is it possible to compute $P( F \mid S )$ given $P(F \mid S,A)$ and $P(F \mid S, \lnot A)$ in Bayesian network?
I have a bayesian network, which has the following data:
$P(S) = 0.07$
$P(A) = 0.01$
$P(F \mid S,A) = 1.0$
$P(F \mid S, \lnot A) = 0.7$
$P(F \mid \lnot S, A) = 0.9$
$P(F \mid \lnot S, \lnot A) =...
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What are the main benefits of using Bayesian networks?
I have some trouble understanding the benefits of Bayesian networks.
Am I correct that the key benefit of the network is that one does not need to use the chain rule of probability in order to ...
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How do compute the table for $p(s',r|s,a)$ (exercise 3.5 in Sutton & Barto's book)?
I am trying to study the book Reinforcement Learning: An Introduction (Sutton & Barto, 2018). In chapter 3.1 the authors state the following exercise
Exercise 3.5 Give a table analogous to that ...
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Why is the equation $r(s', a, s') =\sum_{r \in \mathcal{R}} r \frac{p\left(s^{\prime}, r \mid s, a\right)}{p\left(s^{\prime} \mid s, a\right)}$true?
I am referring to eq. 3.6 (page 49) based on Sutton's online book and can be found in an image below.
I could not make sense of the final derivation of the equation $r(s, a, s')$. My question is ...
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