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Questions tagged [calculus]

For statistical questions involving calculus. Please use also a more statistical tag. For purely mathemathical questions about the calculus, it is better to ask at math SE https://math.stackexchange.com/

4
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2answers
70 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 ...
1
vote
0answers
39 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)" ...
0
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0answers
21 views

Conditional Density Estimate Loss. Why the double integral?

I read RFCDE: Random Forests for Conditional Density Estimation. Just like it sounds, these folks trained random forests for making conditional density estimates. At inference time, density estimates ...
3
votes
1answer
52 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,...
0
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0answers
29 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 ...
2
votes
1answer
233 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^{-...
0
votes
1answer
67 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 ...
0
votes
0answers
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 ...
0
votes
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\...
1
vote
1answer
40 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(\...
0
votes
1answer
144 views

Is there local minimums in MSE function?

Here is "mean squared error" function: C = $\frac{1}{2n}$ * $\sum(length(y - a)^2)$ As I understand, this is like paraboloid in multidimensional space. So, I guess, there is only one extremum: global ...
-1
votes
1answer
504 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.!
0
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0answers
30 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 ...
2
votes
1answer
35 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 ...
1
vote
0answers
22 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 $X=Z^2$. ...
1
vote
1answer
67 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 $\...
1
vote
0answers
478 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)...
2
votes
2answers
51 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⁡ (...
1
vote
0answers
15 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 ...
6
votes
1answer
1k 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 ...
1
vote
0answers
43 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 ...
4
votes
1answer
994 views

How do you obtain the standard error for a slope at a given data point, for curvilinear regression?

A distribution looks like this: modeled by an equation $y=1.0333x^2 - .5382x + 1.6905.$ Find the rate of change (i.e. the slope at that point of the regression equation) at point 6 (the x axis value)...
2
votes
1answer
2k 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{...
2
votes
1answer
454 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 ...
3
votes
1answer
318 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, ...
3
votes
1answer
276 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 ...
5
votes
0answers
2k 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 ...
1
vote
2answers
238 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 ...
-1
votes
1answer
67 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)}{...
2
votes
2answers
2k 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 ...
0
votes
1answer
25 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 ...
0
votes
0answers
28 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{ \...
0
votes
1answer
62 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 ...
1
vote
1answer
170 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 ...
4
votes
1answer
315 views

What is the second derivative of a B-spline?

A B-spline of degree $j$ is defined at knots $\vec k$ by the Cox-de Boor recursion formula \begin{align} B_{i,1}(x) &= \left\{ \begin{matrix} 1 & \mathrm{if} \quad ...
0
votes
0answers
56 views

Simplifying equation involving logit function

I have an equation obtained from inverse logit transformations $$\left(1+e^{-(\tilde{\alpha}+x_1\tilde{\beta_1})}\right)^{-1} = (1-\phi)\left(1+e^{-(\alpha+x_1\beta_1)}\right)^{-1}+ \phi\left(1+e^{-(\...
3
votes
1answer
96 views

Finding integral of minimum function. yn(1-y)^(n-1)

This is for a homework question in which I am trying to find the $E(Y_n = min\{X_1,...,X_n\}$). So far I have found that the minimum cdf, as below. The minimum of $X_i$ is when all of $X_i > x$. ...
1
vote
0answers
108 views

Solution verification - calculation of second derivatives of multinomial probit log-likelihood function

The initial function of log-likelihood of multinomial probit model with $J$ alternatives: $ ln \ell=\sum_{i=1}^N\sum_{j=1}^{J-1} y_{ij} \cdot ln \Phi(\sum_{k=1}^Kx_{ik}\beta_{kj})+ ({n_i-\sum_{j=1}^{...
3
votes
1answer
2k views

Why is optimisation solved with gradient descent rather than with an analytical solution? [duplicate]

I'm trying to understand why, when trying to minimise an objective function, gradient descent is often used, rather than setting the gradient of the error to zero, and solving it analytically. In ...