Questions tagged [calculus]

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45 views

Maths for deeply understanding backpropgation

I have been trying to develop a deeper understanding of Neural Networks so I can understand the libraries such as tensorflow and others. I have had good success with pereceptron models, and have a ...
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17 views

Bonusball lottery question to have 1/150M probability [closed]

I follow guidance on wikipedia to calculate new lottery game i make up. I want chances to be 1:150,000,000 for jeck pot. I find two option so far: 72 numbers and play pick 6. that 1/156,238,908 two ...
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26 views

This is some variational calculus used in Bishop's Pattern Recognition and Machine Learning Book (section 1.5.5) on “Loss function for regression”

The expected regression loss is given as:$$E[L]=\int\int \{y(\mathbf x)-t\}^2 p(\mathbf x,t)d\mathbf xdt$$ To minimise the expected loss,Euler Lagrange equation is used which goes like this in the ...
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14 views

Expected Predicted Error (EPE) with L1 loss

In Element of Statistical learning it is saying on page 20, equation 2.18. That using the L1 norm instead of the usual L2 norm leads to an $f(X)$ optimising the EPE being the median instead of the ...
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28 views

Why does the dimension of gradient and Hessian matrix not conform for this function?

The function is $f(\mathbf{x}) = e^{-\frac{1}{2}\mathbf{x^TAx}}$, where $\mathbf{A}$ is a square symmetric matrix, and $\mathbf{x}$ is an n-vector. What I found were: $$ \begin{align*} \nabla f ...
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25 views

How Hessian matrix is helping in taking big step towards minimization and how is it better than usual Gradient Descent?

I know what Hessian is and $θ:=θ−H^{-1}f′(θ)$ this relation too from Newton Raphson but what i dont understand is how Hessian is really helping with big step and also how is this efficient in ...
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9 views

Does Fisher scoring always outperform Newton optimization?

My understanding is that Fisher scoring has several advantages over Newton raphson optimization such as Computational efficiency: if certain conditions are met (example:During MLE estimation, if link ...
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27 views

What is the first order derivative of linear regression's cost function using matrix calculus?

For linear regression's cost function J(b), where X is a n*m matrix, b is a m*1 vector and y is n*1 vector: First order derivative with respect to vector b (coefficients) is shown to be Using the ...
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138 views

On the convergence of infinite sum of a hypergeometric function resulted from a nested sum

I am interested in finding the CDF of the sum $U=\sum_{i=0}^N U_i$ where: $$F_{U_i}(x)=\sum_{n=0}^\infty \frac {2(-1)^nR^{k\alpha+n\alpha}x^{k+n}}{\Gamma(k)n!\theta^{k+n}(k+n)(k\alpha+n\alpha+2)}$$ ...
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31 views

Will the transformations and operations on a convergent series result in a convergent series?

This question is in the light of the question and answer of: Probability distribution function expressed in terms of a divergent series. We saw that the CDF represented in this form of infinite ...
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How to approximate expectation and variance of an integral from a discrete Time series financial dataset?

I have discrete time series financial data, with time($u$), price($S$) and someVariable($q$) which looks something like this. ...
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33 views

Calculus with quadratic B-spline

After fitting a quadratic B-spline in R with the cobs package : Rbs <- cobs(x,y, constraint= "decrease", pointwise = con) I would like to do some calculus on ...
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42 views

what is the relation between change of variables in probability and calculus?

I want to know if the change of variables formula in probability is a special case of the change of variables in calculus, or something different. Trying to think for myself, first write the CoV for ...
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1answer
34 views

Symmetry between integrals including absolute value

So I came across below symmetry in my probability course that I can't understand. I understand how the lower bound changes when removing the absolute value operator, but how does the 2 disappear?
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A maximization problem involving random variables: a special case

Consider random variables $X$ and $Y$ that are jointly normally distributed, $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \color{blue}0 \\ \mu_Y \end{pmatrix} , \...
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28 views

A maximization problem involving random variables

Consider random variables $X$ and $Y$ that are jointly normally distributed, $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \mu_X \\ \mu_Y \end{pmatrix} , \begin{...
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13 views

How to calculate logarithmized average monthly returns

I am currently reading the following paper https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2750064 . On page 57 the researchers state that they calculate the logarithmized average monthly returns ...
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1answer
95 views

Find marginal distribution

The random vector $(X,Y)$ is uniformly distributed over $$D=\{(x,y): 0 \leq x \leq 2 , 0 \leq y \leq 2-x\}.$$ Find the marginal distribution of the random variables $X$ and $Y$. For the radom vector $...
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1answer
39 views

Derivative of expectation where the variable appears in the integration limit and in the integrand?

I want to calculate the derivative of $$\varphi(\mu) = \int_{-\infty}^{\mu} r(x-\mu) f(x)dx,$$ wrt to $\mu$, where $r$ is a function and $f$ is a density function. How can I account for the presence ...
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140 views

Why does Judea Pearl call his causal graphs Markovian?

In his texts on causality, Judea Pearl always refers to the simplest graphs he uses, i.e. the acyclic graphs with independent confounders, as Markovian. I don't see why these graphs contain anything ...
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191 views

Solving integrals in R

I would like to write an R function for solving the following equation: Essentially I would like to be able to set or vary the parameters values of "m" and "s" and those parameters in "p(t)" ...
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720 views

How to manually calculate odds ratio for continuous variables?

In school, long before learning about logistic models, I've been taught how to calculate odds ratios by hand. Formula was based on a contingency table, just like this: This is very easy to ...
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1answer
137 views

Properties of Kernel Density Estimators

Given Let $X \in \mathbb{R}$ be a real-valued random variable with theoretical probability density function (pdf) $f(x)$ and corresponding cumulative distribution function (cdf) $F(x)$. Let $X_1, X_2,...
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9 views

Threshold optimization with two parameters where first parameter need to be minimum and second value to be maximum

We have threshold values ranging from 1 to 10 where attributes p1, p2 increase with increase in threshold. Our intention is to find threshold with minimum p1 and maximum p2. It would be of great help ...
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1answer
281 views

Law of Iterated Expectations Example

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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1answer
25 views

Expectation / Summation inequality with multiple indices

Part of a solution to an exercise in the book Stochastic Processes From Application to Theory (exercise 87) is the following summations: $$\sum_{i\ge1}P\left(I_i \ge i \right) = \sum_{j \ge i\ge1}P\...
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1answer
495 views

Relationship between first and second order condition of convexity

Suppose we have a function $f(\boldsymbol{x})$ and its hessian, i.e $\nabla_{\boldsymbol{x}}^2f(\boldsymbol{x})$, equals $\mathbf{0}$. We know that for convexity $\nabla_{\boldsymbol{x}}^2f(\...
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166 views

What does gradient with respect to a function mean [duplicate]

I am trying to understand this paper better Greedy function approximation: A gradient boosting machine, but I start having difficulty at around Equation (6) and (7). What does a gradient w.s.t. to an ...
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1answer
40 views

Simple Appplication of Law of Iterated Expectation

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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0answers
25 views

Verifying sub-exponential property when a random variable is not sub-gaussian pro

I am referring to the Example 2.4 (page 16) in this book chapter https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf Suppose $Z \sim N(0,1)$ and random variable $X=Z^2$. ...
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1answer
191 views

Expected Loss Calculation

How do you solve the integral $$E(L(\theta_A,\theta_B)) = \int_0^1\int_{\theta_B}^1(\theta_A - \theta_B)f(\theta_A)f(\theta_B)d\theta_Ad\theta_B$$ where $\theta_A \sim Beta(\alpha_1, \beta_1)$ and $\...
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2answers
56 views

Simple Log-likelihood question

I've got a simple question about deriving log-likelihoods. I am stumped by the following--> If the log-likelihood is: 𝑙(𝜆1,𝜆2) = 𝑦1 log(𝜆1𝐹1)−𝜆1𝐹1 −log((𝑦1)!)+𝑦2 log(𝜆2𝐹2) −𝜆2𝐹2 −log⁡ (...
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17 views

Maximize profits by setting price

I am working on a model to set a price that maximizes profits. The equation for profits is: Profits=price x (# sold) - (fixed cost) x (# sold) I have models ...
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1answer
3k views

Finding the slope at different points in a sigmoid curve

This is my data. x <- c(0.5,3.0,22.2,46.0,77.3,97.0,98.9,100.0) plot(x, pch = 19) I want to fit a curve through these points and then calculate the slope at ...
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49 views

Can Regularization by achieved using Relative Sensitivity?

In a Mathematical Model we measure the sensitivity of the output with respect to the parameters and it is desirable that a small change in a parameter doesn't lead to wild fluctuations in the output ...
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1answer
4k views

Under the 0-1 loss function, the Bayesian estimator is the mode of the posterior distribution

My notes are rather light when it comes to this topic. I understand that the bayesian estimator, defined as (for sample space $\hat{x}$): $E[\Theta | \hat{x}] = \int_{ \forall \Theta}yf_{\theta|\hat{...
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1answer
734 views

gradient descent momentum vs step size

In the gradient descent method, the learning rate (which is multiplied by the results of the gradient on each weight) identifies the size of the step (steep down) that the algorithm takes in each ...
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1answer
397 views

Help with derivation of Mean Field Variational Inference

I am studying Variational Inference using Bishop's book: Pattern Recognition and Machine Learning. At the moment, I am struggling to understand the Lower Bound derivation for the Mean-Field ...
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1answer
383 views

Conditional expectation student's t distribution

I was looking at the student's t distribution and was interested in the following conditional expectation: $$ E[X|X\geq t_v^{-1}(\alpha)] = \frac{g_v(t_v^{-1}(\alpha)}{1-\alpha}\left( \frac{v + (t_v^{-...
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5k views

Deriving linear regression gradient with MSE

So I've been tinkering around with the backpropagation algorithm and to try to get a better understanding of how it works and my calculus is quite rusty. I've derived the gradient for linear ...
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2answers
381 views

deriving z score for 95% confidence interval [duplicate]

95% of the area under the standard normal distribution lies within 1.96 standard deviations away from the mean (0). This 1.96 number is used to construct 95% confidence intervals. I was just ...
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1answer
479 views

Calculating t-SNE gradient (a mistake in the original t-SNE paper)

This is specific to the way the gradient of the KL divergence Loss function was derived in the original paper Visualizing Data using tSNE. In the Appendix A (Page 21), where they derive the gradient, ...
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1answer
846 views

How to differentiate the distribution function of lognormal distribution with respect to its parameters?

How to differentiate the distribution function of lognormal distribution with respect to its parameters? What solution will we get? I know if differentiate wrt variable, we will get density function.!
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85 views

Transformation of random variables in Generative Adversarial Nets

Let $x$ and $z$ be real-valued random vectors, where $x = g(z)$, and $g$ is an invertible, continuous and differentiable transformation of $z$. Then $p_z(z) = p_x(g(z)) \lvert det(\frac{\partial g(z)}{...
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2answers
3k views

gradient versus partial derivatives

how exactly is partial derivative different from gradient of a function? In both the case, we are computing the rate of change of a function with respect to some independent variable. While I was ...
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1answer
32 views

What is the right method to estimate rate of change in daily values over a period of time?

I would like to ask you what is the best (or the right way) to calculate the rate of change in air temperature over a period of time and then see how this rate changes over time. Daily air temperature ...
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30 views

Re-express the lognormal density in terms of its mean value

I wonder if it is possible to rewrite the lognormal density function, $$f(x)=\frac 1 {x\sigma\sqrt{2\pi}}\exp\bigg(-\frac{(\log x-\mu)^2}{2\sigma^2}\bigg)$$ so that the mean value, $\exp(\mu+\frac{ \...
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1answer
70 views

Seeing chi square minimization problem as calculus of variation problem

My question will be less informative because right now I myself don't know where I am heading. Basically I want to know that whether it is possible to think a chi-square minimization problem as a ...
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0answers
581 views

Meaning of Jacobian of the transformation for pdf of function of random vectors

I am studying multivariate statistics and I don't understand the meaning of Jacobian of the transformation for pdf of function of random vectors. If I have a random vector, let's say bivariate, (X,Y)...
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1answer
197 views

What is the intuition behind neural networks?

Everywhere in the theory of neural networks, authors saying that idea came about by observing the work of the human brain. But I can not believe in that. I guess, everything is much simpler and neural ...