Questions tagged [probability-distribution]

For questions related to AI theory that relies on the knowledge of a distribution of probabilities across one or more dimensions affecting probability. Such a distribution may be in discrete buckets, such as quartile, octile, or percentile conventions or continuous functions based on some closed form (algebraic formula). Distributions of probability are key in planning, natural language handling, and other AI objectives.

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58 views

Is knowing underlying probability distribution mandatory for deciding iid property of random variables?

Consider the following information regarding iid random variables The acronym IID stands for "Independent and Identically Distributed". A sequence of random variables (or random vectors) is ...
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Which of the following probability distribution is generating an iid dataset?

Let $X_1, X_2$ be two discrete random variables. Each random variable takes two values: $1, 2$ The probability distribution $p_1$ over $X_1, X_2$ is given by $$p_1(X_1=1, X_2 = 1) = \dfrac{1}{4}$$ $$...
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How can we "draw i.i.d" from any probability distribution?

Consider the following paragraph from 2 Learning in High Dimensions in from of the paper titled Geometric Deep Learning Grids, Groups, Graphs, Geodesics, and Gauges ...
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74 views

Would it be any more difficult for AI/ML to beat more poker players?

Would the number of players at a poker table alter a capable AI/ML's chances of winning? I don't see how it would, so long as every player was playing fairly. I would assume that it would not matter ...
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What are mathematically the factors of variation in deep learning?

The following paragraph from an answer tells us about factors of variation Factors of variation are some factors which determine varieties in observed data. If that factors change, the behaviour of ...
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How to calculate policy probability ratio in multiple action space

I try to solve a navigation problem with PPO; my actions space have three-part: robot linear velocity that is in [-3,3] range (getting from a tanh activation func) robot linear angular that is in [-...
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Where can I read about the multinoulli distribution?

I encountered the term multinoulli distribution in the following sentence from Chapter 4: Numerical Computation of the deep learning book. The softmax function is often used to predict the ...
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Are there any other metrics available for calculating the distance between two probability distributions other than those mentioned?

The divergence between two probability distributions is used in calculating the difference between the true distribution and generated distribution. These divergence metrics are used in loss functions....
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1answer
144 views

Is it abuse of notation to use tilde operator in this context?

The following is a way to use tilde (∼) in context of random variables or random vectors. In statistics, the tilde is frequently used to mean "has the distribution (of)," for instance, $X∼N(...
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Which probability distribution a generator in Generative Adversarial Network (GAN) is capturing: dataset or ground truth?

Consider the following statement from the abstract of the paper titled Generative Adversarial Nets We propose a new framework for estimating generative models via an adversarial process, in which we ...
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How to define a continuous action distribution with a specific range for Reinforcement Learning?

Specifically for continuous control PPO, let's say my action space range is between $X$ (low) and $Y$ (high) and they are all sampled from a Gaussian Action Distribution with mean $\mu$ and standard ...
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Rotationally independent distributions

Maxwell's theorem states that multivariate normal distribution $\mathcal{N}(\mathbf{0}, \sigma^2\mathbf{I})$ is the only distribution of a random vector that is invariant and have independent ...
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Is it possible to use Neural Networks with Contextual Bandit to learn the probability distributions instead of providing them?

I want to ask you if it's possible by using neural networks jointly with the Contextual Bandit algorithm to learn the probability distributions by which the rewards are computed as a function of the ...
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Difference between a distribution model and a sampling environment in Reinforcement Learning

The book from Sutton and Barto define a model in Reinforcement Learning as "something that mimics the behavior of the environment, or more generally, that allows inferences to be made about how ...
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32 views

Predicting the probability of a periodically happening event occurring at a given time

I have encountered this problem on how to predict the probability of a periodically happening event occurring at a given time. For example, we have an event called being_an_undergrad. There are many ...
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58 views

Why do we sample vectors from a standard normal distribution for the generator?

I am new to GANs. I noticed that everybody generates a random vector (usually 100 dimensional) from a standard normal distribution $N(0, 1)$. My question is: why? Why don't they sample these vectors ...
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Is there an effective way of obtaining the topic distribution for a given document from a VAE-LDA?

Is there an effective way of obtaining the topic distribution for a given document from a Variational AutoEncoder Latent Dirichlet Allocation (VAE-LDA)? Most existing public VAE-LDA codebases seem to ...
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1answer
32 views

Why are CNN binary classifier output probability distributions often skewed?

I've been working on a lot of simple resnet18 binary classifiers lately and I've started to notice that the probability distributions are often skewed one way or the other. This figure shows one such ...
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How can a probability density value be used for the likelihood calculation?

Consider our parametric model $p_\theta$ for an underlying probabilistic distribution $p_{data}$. Now, the likelihood of an observation $x$ is generally defined as $L(\theta|x) = p_{\theta}(x)$. The ...
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What is emperical distribution in MLE?

I was reading the book Deep Learning by Ian Goodfellow. I had a doubt in the Maximum likelihood estimation section (Pg 131). I understand till the Eq 5.58 which describes what is being maximized in ...
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Why is KL divergence used so often in Machine Learning?

The KL Divergence is quite easy to compute in closed form for simple distributions -such as Gaussians- but has some not-very-nice properties. For example, it is not symmetrical (thus it is not a ...
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Is this referring to the true underlying distribution, or the distribution of our sample?

I am currently studying the paper Learning and Evaluating Classifiers under Sample Selection Bias by Bianca Zadrozny. In the introduction, the author says the following: One of the most common ...
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129 views

When should one prefer using Total Variational Divergence over KL divergence in RL

In RL, both the KL divergence (DKL) and Total variational divergence (DTV) are used to measure the distance between two policies. I'm most familiar with using DKL as an early stopping metric during ...
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Estimating $\sigma_i$ according to maximum likelihood method

Let be a Bayesian multivariate normal distribution classifier with distinct covariance matrices for each class and isotropic, i.e. with equal values over the entire diagonal and zero otherwise, $\...
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81 views

How to calculate v min and v max for C51 DQN

Background: In C51 DQNs you must specify a v-min/max to be used during training. The way this is generally done is you take the max score possible for the game and set that to v-max, then v-min is ...
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Why is the mean used to compute the expectation in the GAN loss?

From Goodfellow et al. (2014), we have the adversarial loss: $$ \min_G \, \max_D V (D, G) = \mathbb{E}_{x∼p_{data}(x)} \, [\log \, D(x)] \\ \quad\quad\quad\quad\quad\quad\quad + \, \mathbb{E}_{z∼p_z(...
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Are there any deep learning tools so that can be used to estimate long-tail distributions?

Are there any deep learning tools so that can be used to estimate long-tail distributions? update: my context is there are is there is a long-tail distribution (probability distribution) in my data of ...
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In continuous action spaces, how is the standard deviation, associated with Gaussian distribution from which actions are sampled, represented?

I have a question about implementing policy gradient methods for problems with continuous action spaces. Assume that actions are sampled from a diagonal Gaussian distribution with mean vector $\mu$ ...
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46 views

Intuitively, why can the training of a neural network be formulated as a probability estimation problem?

Neural network training problems are oftentimes formulated as probability estimation problems (such as autoregressive models). How does one intuitively understand this idea?
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Is the generator distribution in GAN's continuous or discrete?

I have some trouble with the probability densities described in the original paper. My question is based on Goodfellow's paper and tutorial, respectively: Generative Adversarial Networks and NIPS ...
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1answer
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How does $\mathbb{E}$ suddenly change to $\mathbb{E}_{\pi'}$ in this equation?

In Sutton-Barto's book on page 63 (81 of the pdf): $$\mathbb{E}[R_{t+1} + \gamma v_\pi(S_{t+1}) \mid S_t=s,A_t=\pi'(s)] = \mathbb{E}_{\pi'}[R_{t+1} + \gamma v_\pi(S_{t+1}) \mid S_{t} = s]$$ How does $...
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What is the most efficient data type to store probabilities?

In ML we often have to store a huge amount of values ranging from 0 to 1, mostly being probabilities. The most common data structure to do so seems to be a floating point? Indeed, the range of ...
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1answer
133 views

Can a variational auto-encoder learn images composed of random noise at each pixel (each drawn from the same distribution)?

Can a variational auto-encoder (VAE) learn images whose pixels have been generated from a Gaussian distribution (e.g. $N(0, 1)$), i.e. each pixel is a sample from $N(0, 1)$? My gut feeling says no, ...
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What is the difference between model and data distributions?

Is there any difference between the model distribution and data distribution, or are they the same?
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What does a joint probability density function have to do with Stochastic Optimal Control and Reinforcement Learning?

I stumbled upon a job offer from a company that was looking for someone who was good with Reinforcement Learning (applied to finance) and something in their offer caught my eye. It goes something like ...
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1answer
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How can I use the success and failure data to estimate parameters of a Dirichlet distribution?

I have used Beta function to estimate the performance of the agent. I have failure and success data of the task that runs on the agent. The parameter $\alpha$ is a number of successful tasks, while $\...
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1answer
425 views

Why do we regularize the variational autoencoder with a normal distribution?

When we define the loss function of a variational autoencoder (VAE), we add the Kullback-Leibler divergence between the sample taken according to a normal distribution of parameters: $$ N(\mu,\sigma)...
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125 views

Generalization error: Inputs drawn from distributions

I am currently studying Deep Learning by Goodfellow, Bengio, and Courville. In chapter 5.2 Capacity, Overfitting and Underfitting, the authors say the following: Typically, when training a machine ...
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1answer
72 views

Solving the supervised learning problem of learning $p(y \vert \mathbf{x})$ by using traditional unsupervised technologies to learn $p(\mathbf{x}, y)$

I am currently studying Deep Learning by Goodfellow, Bengio, and Courville. In chapter 5.1.2 The Performance Measure, $P$, the authors say the following: Unsupervised learning and supervised learning ...
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1answer
121 views

Many of the best probabilistic models represent probability distributions only implicitly

I am currently studying Deep Learning by Goodfellow, Bengio, and Courville. In chapter 5.1.2 The Performance Measure, P, the authors say the following: The choice of performance measure may seem ...
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116 views

Why does the machine learning algorithm need to learn a set of functions in the case of missing data?

I am currently studying the textbook Deep Learning by Goodfellow, Bengio, and Courville. Chapter 5.1 Learning Algorithms says the following: Classification with missing inputs: Classification ...
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Formulation of a Markov Decision Process Problem

Given a list of $N$ questions. If question $i$ is answered correctly (given probability $p_i$), we receive reward $R_i$; if not the quiz terminates. Find the optimal order of questions to maximize ...
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Why we multiply probabilities with support to obtain Q-values in Distributional C51 algorithm?

In 'Deep Reinforcement Learning Hands-On' book and chapter about Distributional C51 algorithm I'm reading, that to obtain Q-values from the distribution I need to calculate the weighted sum of the ...
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Why am I getting the logarithm of the probability bigger than zero when using Neural Spline Flows?

I am using a normalizing flow (Neural Spline Flows) to approximate a probability. After some training, the average loss is around 0.5 (so the logarithm of the probability = -0.5). However, when I am ...
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Deciding std. deviation for policy network output?

When I try to fit a Normal Distribution to the output of a policy network, for a continuous action space problem, what should be its standard deviation? mean for the distribution will directly be the ...
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722 views

What is a probability distribution in machine learning?

If we were learning or working in the machine learning field, then we frequently come across the term "probability distribution". I know what probability, conditional probability, and ...
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1answer
74 views

How does maximum approximation of the posterior choose a distribution?

I was learning about the maximum a posteriori probability (MAP) estimation for machine learning and I found a nice short video that essentially explained it as finding a distribution and tweaking the ...
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398 views

Why is Jensen-Shannon divergence preferred over Kullback-Leibler divergence in measuring the performance of a generative network?

I have read articles on how Jensen-Shannon divergence is preferred over Kullback-Leibler in measuring how good a distribution mapping is learned in a generative network because of the fact that JS-...
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1answer
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In deep learning, do we learn a continuous distribution based on the training dataset?

At least at some level, maybe not end-to-end always, but deep learning always learns a function, essentially a mapping from a domain to a range. The domain and range, at least in most cases, would be ...
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Probabilistic classification - normalize results

I have a probabilistic classifier that produces a distribution over my 3 classes - C1, C2, C3. I want to compare some new points I'm classifying to each other, to see which one is the best fit for a ...