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Deriving predicted probabilities from gologit2 (proportional odds models) output

The Williams approach uses the Brant test for proportional odds. That this test is invalid has been pointed out since 1991 (Peterson & Harrell J Royal Stat Soc C). For related information see ...
Frank Harrell's user avatar
0 votes

How to show that many functions (a hundred, a thousand) have the same shape an distribution of values over an interval?

Based on your description 0.4 +ve derivative from zero to 0.4 and around zero or slightly negative derivative up to 1. You could fit $y = a_0 + a_1 x + a_2 (x-0.4)^+$, which is a piecewise linear ...
seanv507's user avatar
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0 votes
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How to show that many functions (a hundred, a thousand) have the same shape an distribution of values over an interval?

A Q-Q plot approach would not be appropriate for your objective. Not only that, it would give you misleading results. This is because a Q-Q plot changes the order of the data, arranging it in ...
jginestet's user avatar
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0 votes

Equivalence of first/second choice with naive probability - I don't buy it

Let $b_1, \dots, b_r$ be the $r$ red balls and let $b_{r+1}, \dots, b_{r+g}$ be the $g$ green balls. Let $\mathcal{B_1}, \mathcal{B_2}$ denote the first and the second drawn ball, respectively. If we ...
CrabMan's user avatar
  • 172
4 votes

How to show that many functions (a hundred, a thousand) have the same shape an distribution of values over an interval?

To show that several functions are more or less the same you could just superimpose them graphically. I don't think quantile plots of any flavour are directly relevant or likely to be helpful. The ...
Nick Cox's user avatar
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1 vote

Calculating the joint pdf of linearly dependent random variables $X$ and $Y=X$

As already explained in comments, there is no joint density in the plane, because all the probability mass of $(X, X)$ is concentrated on the diagonal $y=x$. There is a density on that diagonal, but ...
kjetil b halvorsen's user avatar
0 votes

Are all random variables estimators?

FWIW you define yourself random variable and estimator as different functions, performing mapping between at least three different spaces: domain of possible outcomes a sample space to a measurable ...
Roger V.'s user avatar
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4 votes

Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?

The misinterpretation of confidence intervals is related to what Blitzstein and Hwang (in their probability textbook) call "sympathetic magic". Sympathetic magic is an anthropology term for ...
Abhishek Divekar's user avatar
3 votes

Equivalence of first/second choice with naive probability - I don't buy it

Imagine there are only two balls, one of each color. If the first ball is red, the probability of the second one being green is $100\%$. If the first ball is green, the probability of the second one ...
David's user avatar
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3 votes

Equivalence of first/second choice with naive probability - I don't buy it

The abstract mathematical machinery is intuitive for many people provided it is clearly set up in a way that parallels the original probability problem. Before reading any more of this post, though, ...
whuber's user avatar
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4 votes

Equivalence of first/second choice with naive probability - I don't buy it

What's "intuitive" to one may be less so to another. I find @Flounderer's answer to be quite intuitive, but here is an alternative if that didn't do it for you: An equivalent process is to ...
Max's user avatar
  • 281
0 votes

Comparability of classifier probabiliy estimates

Maybe not the answer you are looking for, but I agree comparing the outputs of the models without having a lot of testing to be confident in their probability calculations is problematic. You should ...
noNameTed's user avatar
  • 135
12 votes

Equivalence of first/second choice with naive probability - I don't buy it

The solutions for exercises marked with the circled S symbol are available from this page: https://projects.iq.harvard.edu/stat110/strategic-practice-problems The intended solution is: This is true ...
Flounderer's user avatar
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21 votes

Equivalence of first/second choice with naive probability - I don't buy it

Let $\mathcal{G}_1, \mathcal{G}_2, \mathcal{G}_3, ...$ be the events that each consecutive drawn ball is green. You are correct to point out that: $$\begin{align} \mathbb{P}(\mathcal{G}_1) &= \...
Ben's user avatar
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0 votes

Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances?

If the $X_i$ are positively correlated, then the variance of the empirical mean is going to be bigger than in the independent case. If the $X_i$ are negatively correlated, then the variance of the ...
Guillaume Dehaene's user avatar
1 vote

Bayes' Theorem applied in real study

This looks like a homework problem (correct me if I am wrong) which have their own guidelines. Here are some hints. You write: This ratio is an estimated relative risk, not an absolute probability. ...
Peter Flom's user avatar
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5 votes

Equivalence of inverse transformations under distributional equivalence

Since the unit standard multivariate normal $\mathcal{N}(0, I)$ is rotationally symmetric, let $R$ be any orthogonal matrix, that is, $R^TR = R R^T =I$, then also $RY \sim \mathcal{N}(0, I)$. Since $g$...
kjetil b halvorsen's user avatar
0 votes

Find expected value using CDF

Alternative derivation of $EX = \int_0^\infty \left(1-F_X(x)\right)\mathrm d x$ (for a positive r.v. $X$): For any $x \ge 0$ one has that $x=\int_0^x 1 \mathrm dt = \int_0^\infty \mathbf I_{t\le x}\...
Flo's user avatar
  • 103
2 votes

If the variance converges to zero, when do we have almost sure convergence

To show $\{X_n\}$ converges to $c$ almost surely, one can add the extra condition that $$\sum_{n=1}^\infty \mathrm{Var}(X_n)<\infty \tag{*}\label 1$$ which can be satisfied by, for example, if $\...
Mingzhou Liu's user avatar
2 votes

Why do Denoising Diffusion Probabilistic Models (DDPM) add noise according to $\sigma_t$ during sampling?

This is very late, but I had the exact same confusion and managed to resolve it! Your comment earlier about the expression making more sense if it were $\sigma_{t-1}$ is the correct intuition, and is ...
Alex Nguyen-Le's user avatar
2 votes
Accepted

Let $X(t)$ be a Gaussian process. Does $\mathbb{E}[X(t)^2 X(s)^2] = \mathbb{E}[X(t)^2 ] \mathbb{E}[X(s)^2 ] + 2 (\mathbb{E}[X(t) X(s)])^2 $?

Yes. $(X(t),X(t),X(s),X(s))$ is a 4-variate Gaussian distribution, so Isserlis' theorem applies to it.
Thomas Lumley's user avatar
3 votes

Evaluating the Validity of Election Results

Is this a parliamentary election, for a parliament with many seats? If so, a 1/243 likelihood is not very surprising. Are any details of the counting method available? For example if these are ...
chrishmorris's user avatar
  • 1,855
4 votes
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If the variance converges to zero, when do we have almost sure convergence

No. Consider $X_n \in \{0,1\}$, therefore bounded, and the sequence of probabilities $P_n(X_n = 1) = 1/n$ (and therefore $P_n(X_n = 0) = 1 - 1/n$.) Clearly $X_n \to 0$ in probability, but Borel-...
jbowman's user avatar
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2 votes

What does "Aleatoric and Epistemic uncertainties" mean?

Aleatoric Uncertainty: This is the uncertainty of the process which you are trying to model. Say, you want to train a model with some sensor output where the sensor is itself producing some random ...
hafiz031's user avatar
  • 235
5 votes
Accepted

Applying Bayesian probability to a generalized Monty Hall problem

I agree with you, but there's a big "however" following this opening section: it doesn't matter why he picks the door, as long as the door has the goat behind it. The presence of the goat ...
jbowman's user avatar
  • 40.5k
1 vote

How to evaluate whether the results of multiple games conform to the probability distribution of their respective states

Using surprisal as a test statistic, we can use Monte Carlo simulation to test if the observed results are consistent with the hypothesized probabilities ($p_i$). Demonstrating in R: ...
jblood94's user avatar
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9 votes
Accepted

Is $F_X(t) > F_Y(t)$ a sufficient and necessary condition of $\mathbb{P}(X < Y) > 0.5$

A trivial counterexample for "$P(X < Y) > 0.5$" does not necessarily imply $F_X(t) > F_Y(t)$ for all $t$ is letting $X \sim N(0, 1)$ and $Y \equiv 1$. Then $P(X < Y) = P(X < 1)...
Zhanxiong's user avatar
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4 votes

Is $F_X(t) > F_Y(t)$ a sufficient and necessary condition of $\mathbb{P}(X < Y) > 0.5$

A simple counterexample for necessity follows. Consider $Y \sim U(0,4)$ and $X \sim U(1,2)$. Since $P(Y > 2) = 0.5$, clearly $P(X < Y) > 0.5$, but $F_Y(t) < F_X(t)$ for $t \in (4/3,4)$ ...
jbowman's user avatar
  • 40.5k
4 votes

Is $F_X(t) > F_Y(t)$ a sufficient and necessary condition of $\mathbb{P}(X < Y) > 0.5$

Just contributing some details to the other answers and comments. Regarding sufficiency, suppose that we have two independent random variables $X\sim F$ and $Y\sim G$, and that $X$ is absolutely ...
Zen's user avatar
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1 vote
Accepted

Formal definition of sufficient statistic

I am failing to see any problem. For a collection of probability measures $\mathfrak M:=\{P_\theta,\theta\in\Theta\},$ the statistic $T$ is sufficient if there exists a version of $\mathbf E_\theta [1\...
User1865345's user avatar
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0 votes

Neural networks output probability estimates?

I am not a data scientist, so my answer may not be very useful; however, I am facing the same question, and my idea is to actually calculate the probability on the validation subset of data once the ...
fede72bari's user avatar
2 votes

How to work out the expected rate of success when there is a guaranteed success on the nth attempt?

A simple question, stated in an elementary fashion, deserves a short elementary solution. It also deserves some consideration of what it is actually asking for. This post offers both. To establish ...
whuber's user avatar
  • 328k
2 votes

How to work out the expected rate of success when there is a guaranteed success on the nth attempt?

You could simulate it as follows: Perform $n$ Bernoulli trials with probability of success $p$; If any trials were successful, note the first success; Otherwise, note the maximum, $n$; Compute the ...
Frans Rodenburg's user avatar
0 votes

Can MAE be interpreted as the average standard deviation around the true value of a prediction?

the rule of thumb is that 95% of data should fall between two standard deviations of the mean value. The estimate of the standard deviation is different for MAE compared to RMSE. The unbiased ...
Sextus Empiricus's user avatar
0 votes

How to work out the expected rate of success when there is a guaranteed success on the nth attempt?

I wonder whether this could be modelled via Hypergeometric distribution. Lets say you have $M=\left(N-1\right)+A$ boxes. And out of them $A$ boxes have a prize. So choosing the boxes randomly you will ...
Cryo's user avatar
  • 783
1 vote
Accepted

Bayesian updating with affine transformation of random variable

The posterior distribution is invariant to (deterministic) reparametrizations of the likelihood when the prior distribution is maintained, and the transformation does not depend on the target ...
Johan de Aguas's user avatar
2 votes

(THEORY) Do Tree models output probabilities?

I think this can be made simpler. Single independent variable, and classification. In this case, I would argue, tree-based model is not that dissimilar from learning the distribution of your binary ...
Cryo's user avatar
  • 783
1 vote

Looking for a mathematical book on probability and statistics

I recommend the Information Processing series by David Blower. It is a long read. But every page sharpens your critical thinking. VOLUME I: Boolean Algebra, Classical Logic, Cellular Automata, and ...
Romke Bontekoe's user avatar
9 votes

(THEORY) Do Tree models output probabilities?

Your assessment of the situation is excellent. I would just add that in my practice random forests suffer some of the worst miscalibration that I’ve ever witnessed as a statistician. Even a single ...
Frank Harrell's user avatar
0 votes

expected value of a fishing strategy

The approach works the same as the dice problem, but now you work with the lightest fish instead of a single fish. You try to improve the expected value of the lightest fish and by doing that you ...
Sextus Empiricus's user avatar
3 votes

Do tail bounds on probability translate into bounds on expectations?

To compliment Adrian's answer, for your specific problem I think the linked notes imagined the following approach. As Whuber highlighted in the comments, for any non-negative random variable $X$, we ...
Joseph Basford's user avatar
0 votes

expected value of a fishing strategy

Day5 (last) Let $A,B$ be respective the min and max fish in hand. Let $t_5$ be our threshold for rerolling the minimum fish on day5. Let $E_{5}[{A,B}]$ be the value of having the option to reroll when ...
enryuxbt's user avatar
3 votes

Do tail bounds on probability translate into bounds on expectations?

Step 1. Taking the cue from whuber, we begin with a non-negative RV $X$ (we say non-negative because eventually $X=\left\|\hat{f}-f\right\|_n^2\ge 0$), and assume that $P(X\ge t)\le e^{-t},$ for all $...
Adrian Keister's user avatar
-1 votes

Posterior expectation of normal distribution with "truncated" observation

\begin{gather*} E[Y\mid X, Y\in A]=\frac{E[YI_A(Y)\mid X]}{E[I_A(Y)\mid X]}=\frac{\int_A yf_{Y|X}(y|x)dy}{\int_A f_{Y|X}(y|x)dy}=\frac{\int_A\int_R yf(_{\theta}, x,y)d_{\theta}dy}{\int_A\int_R f(_{\...
Speltzu's user avatar
  • 336
5 votes
Accepted

Posterior expectation of normal distribution with "truncated" observation

Since $Y\sim\mathcal N(\theta,\tau_Y^{-1})$, the indicator variable $Z=\mathbb I_A(Y)$ has a binomial distribution $$Z\sim\mathcal B(\mathbb P_\theta(Y\in A))$$ Since $X$ and $Y$ are independent, the ...
Xi'an's user avatar
  • 107k
1 vote

Posterior expectation of normal distribution with "truncated" observation

$$ \begin{aligned} &\mathbb{E}[\theta\mid X=x,Y\in A] \\ &=\int_{y\in A}\mathbb{E}[\theta\mid X=x,Y=y]\, p_Y(y\mid X=x, Y\in A)\, dy\\ &= \int_{y\in A}[\gamma_X x + (1-\gamma_X)y]\frac{\...
Johan de Aguas's user avatar
2 votes

Approximation function for MLP and LSTM

Based on your answer to Sycorax I infer that you want to look at the distribution of residuals and compare the two empirical distributions of the residuals (MLP vs LSTM). For comparison you could look ...
Ggjj11's user avatar
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