Questions tagged [probability-theory]

For questions related to probability theory in the context of artificial intelligence.

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58 views

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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1answer
23 views

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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46 views

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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152 views

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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1answer
79 views

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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1answer
181 views

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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1answer
68 views

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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0answers
59 views

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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3answers
95 views

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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1answer
118 views

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 ...