Questions tagged [mean-square-error]
This tag is for questions about mean-square-error. In statistics, the mean squared error (MSE) of an estimator measures the average of the squares of the errors or deviations, that is, the difference between the estimator and what is estimated.
278 questions
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Explanation of why MSE and log-likelihood of a group of bernoulli distributions are not optimised at the true probability
I have 2 random variables X and Y:
$$X_i \sim Unif(0,1)$$
$$Y_i \sim Bernoulli(X_i^{exp(\delta)})$$
I wanted to be able to test how well a set of success probabilities fit to observed set of ...
- 131
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122
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Matrix calculus chain rule for MSE gradient: Why the transpose?
I'm working through the derivation of the gradient for the Mean Squared Error (MSE) loss function in a standard linear regression model and have a question about the application of the chain rule in ...
- 31
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Formalizing the Link: Bias-Variance, Generalization Error, and Robustness (LFP)
I'm looking for input on a proposed formal connection between three key concepts in machine learning. I'm hoping to get feedback on the validity of this framework and pointers to any existing ...
- 101
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Role of $\sigma(X)$ in the theorem that the conditional expectation $\mathbb{E}[Y|X]$ minimizes mean sqaured error
So it is very well known that the conditional expectation minimizes the mean squared error. That is, given a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and random variables $X:(\Omega, \...
- 766
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1
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61
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MSE between sample mean and true mean for vectors
From a population of $N$ vectors I select a sample of $n$ vectors. What will be an error in a sample mean compared with population mean, i.e. how "close" the vectors will be?
Does it matter ...
- 2,413
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36
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Example of Rao-Blackwell $\tilde{\theta}$ estimator case
Question: I am aiming to understand part of my notes on Rao-Blackwell theorem. Specifically I am looking for examples which clarify what is meant by $(Condition \hspace{1mm} 1.)$. I already have two ...
- 81
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1
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101
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What is the formula of the bias contained in the Mean Squared Error?
I have a question regarding the formula of the Bias contained in the Mean Squared Error, more specifically about the denominator.
I know that the MSE can be written as:
MSE=(1/N)(sum(observed value i-...
- 11
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Matrix form PCA
Background
Given
Data points matrix $X$ is given.
We look for $W$, an orthonormal column matrix
$$
W = \begin{bmatrix}
| & | & & | \\
w_1 & w_2 & \cdots & w_k \\
| & | &...
- 431
1
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1
answer
87
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Isolate matrix in the middle by rearranging terms
I wish to isolate the terms $\vec{x}\vec{x}^\intercal$ from the below expression. I wish to do this in order to find some sort of closed form for combining $A$ and $B$.
$$A\vec{x}\vec{x}^\intercal A +...
- 21
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0
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78
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Nesterov Accelerated Gradient method for convex non-smooth objective functions
I need to solve an optimization problem involving an Extreme Learning Machine $z=W_2\sigma(W_1 x)$, where the weight matrix for the hidden layer $W_1$ is a fixed random matrix, $\sigma()$ is the ...
- 11
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1
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330
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Relationship between Variance and Consistency?
This is something I have been struggling to understand. Our prof told us that a consistent estimator has $0$ variance as $n$ becomes large. Even though this intuitively makes sense, I don't understand ...
- 4,183
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1
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73
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Computing MSE of a linear combination of order statistics
Let $X_1 , \dots , X_n$ be IID Uniform$(\theta, 2\theta)$ with $\theta>0$. Consider estimators for $\theta$ of the form:
\begin{equation}
Z_a = aX_{(1)} + \frac{1}{2}(1-a)X_{(n)}
\end{equation}
...
- 467
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40
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MAE and RMSE by groups
Consider five real numbers $A_1$, $A_2$, $A_3$, $B_1$, $B_2$. They are errors, the five differences between real values and estimated values. The MAE(mean absolute error) is
$$\text{MAE} = \frac{|A_1|+...
- 860
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1
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81
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Mean squared error of MLE of $\theta$ where $f(x) = 3x^2 \theta e^{-\theta x^3} 1_{(0,\infty)}(x)$
Given $f(x) = 3x^2 \theta e^{-\theta x^3} 1_{(0,\infty)}(x)$ I want to find the MSE of the MLE estimator for $\theta$. I've found that $\hat{\theta} = \frac{n}{\sum_{i=1}^n X_i^3} = \frac{1}{\bar{X^3}}...
- 119
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How to measure the error between modeled and observed data?
Consider a scenario where observed data is represented in grey and modelled data in red, as below
Here, the x-axis is a position, and the y-axis is an expected time, so that the slope defines, in a ...
- 3,585