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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Why does a quantile regression estimator underestimate the variance when using the quantile huber loss?

I have a question to quantile regression which is related to distributional Reinforcement Learning. Let the quantile loss (QL) be defined as \begin{align*} \mathcal{L}^{\tau}_{\text{QR}}(\...
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Why does importance sampling work with latent variable models?

Caveat: importance sampling doesn't actually work for variational auto-encoders, but the question makes sense regardless In "L4 Latent Variable Models (VAE) -- CS294-158-SP20 Deep Unsupervised ...
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What should be taken as random variables in the distributions of datasets?

Consider the following two paragraphs taken from the paper titles Generative Adversarial Nets by Ian J. Goodfellow et.al #1: Abstract We propose a new framework for estimating generative models via ...
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What is the distribution of autoencoder embeddings?

Is there any result on the distribution of autoencoder embeddings? For example, the following image (taken from this article) visualizes the latent space with t-SNE. As you can see, images from the ...
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How is the union bound being used?

I am trying to understand the assumption proof of Theorem 2(Page -$7$) in the paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Inequality 1 \begin{align} \mathbb{...
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What is the meaning of $p_{\text {data }}(y)$ in the CycleGAN?

In the original CycleGAN paper, on the second page, there is a sentence that I didn't quite understand In theory, this objective can induce an output distribution over $\hat{y}$ that matches the ...
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PPO: policy loss becomes nan [closed]

I'm implement PPO for a very specific problem, and it seems to be working somewhat, but after a few epochs, I always get something like this: ...
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Neural network for an output in the form of a probability distribution

I am not an expert in machine learning. Recently, I want to construct a data-driven model based the neural network. The problem is that I want from my algorithm to learn an output in the form of a ...
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Sampling form two independent normal vs sampling from uncorrelated bivariate normal

I think there is a theorem that states that "If X and Y are bivariate normal and uncorrelated, then they are independent." So, if I have two normal random variables $X \sim N(\mu_1, \sigma_1^...
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Are there any scale invariant activation functions that outputs probability distribution?

Softmax activation function is used to convert any random vector into a probability distribution. So, it is generally used as an activation function in the last layer of deep neural networks that are ...
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Why are Directed Graphical Models considered ML methods?

Consider the following problem. The probability of being born in countries [1,2,3,4] is given by [a, b, c, d] respectively. This is a categorical problem. Now, assume that the height of a person ...
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How does the distribution of the parameters change in logistic regression?

I have my own data to train a logistic regression model (for a multi-class classification task), and I want to know how the distribution of weight parameters changes after each update with gradient ...
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Proper loss function for regression with uniform target distribution

I'm doing some simulations and I would like to estimate a real number that is uniformly distributed between minValue and maxValue...
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Variational Inference: Approximate expected log likelihood via sampling

I'm working my way through a simple variational inference from scratch. For that, I assume that z denotes the probability of a coin showing ...
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Are the authors of the VAE paper writing the PDFs as a function of the random variables?

Usually, I see the conventions: discrete random variable is denoted as $X$, the pmf is written as $P(X=x)$ or $p(X=x)$ or $p_{X}(x)$ or $p(x)$, where $x$ is an instance of $X$ a continuous random ...
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What is the analytical formula for "Kaiming He" probability density function?

A probability density function is a real-valued function that roughly gives the density of probability at a particular value of a random variable. For example, the probability density function of a ...
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How does the VAE learn a joint distribution?

I found the following paragraph from An Introduction to Variational Autoencoders sounds relevant, but I am not fully understanding it. A VAE learns stochastic mappings between an observed $\mathbf{x}$...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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What does "the expectation is taken across different possible inputs, drawn from the distribution of inputs we expect the system to encounter" mean?

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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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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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 ...