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2

I can't follow your working after $\int_0^1\frac{\left(\frac{\partial f(x;\theta)}{\partial \theta}\right)^2}{f(x;\theta)}d x$, note that this expression is nonnegative as the function being integrated is nonnegative. Let $u = 1 + \theta \log x, \frac{du}{dx}=\frac{\theta}x$, \begin{align} &\int_0^1\frac{\left(\frac{\partial f(x;\theta)}{\partial \theta}\...


0

What you are talking about with sample 2 is multistage sampling, whereby you first select a random sample clusters, and then randomly sample from within these clusters. Exercise 5.1 in this book chapter goes over estimating variance for this type of sampling. There is some academic literature saying this is bias, but I was taught it has an undergrad, and it ...


1

One of the most popular ways to learn a (deterministic) function is Gaussian Process (GP) regression. It is commonly phrased in the Bayesian framework so that our 'prior beliefs' are $$ f(\cdot) \sim GP(m(\cdot), C(\cdot, \cdot))$$ where $m(\cdot)$ represents our prior expected value of $f(x)$ for any $x$. It's usually a 'rough and ready' approximation. E.g. ...


0

You can use ranking metrics. Examples: Precision@K - rank your prediction and calculate out of the top k what was the ratio of true positive. Average precision - averaging the precision@K for different Ks. DCG & NDCG - rank your data and the give high score for the relevant items and sum them. $\sum_{i=0}{N} \frac {rel_i} {log_2(i)}$ - relevance in ...


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You don't have to adjust residual error by volatility, since you can remove cluster volatility in the log price returns using a GARCH model before running any kind of regression. Below is a plot of volatility over time, $\sigma(t)$, as well as the SP500 log price returns with cluster volatility removed (red line). Be sure to always use the log price ...


2

You don't necessarily need more samples to make multiple inferences, but you may want to keep in mind that as you make more estimates, you're more likely to make a wrong estimate somewhere. Your sample size of 1066 will allow you to estimate proportions that will be within 3 percentage points of the true value 95% of the time (and they'll be even closer if ...


1

The "worst-case" for the width of a Wald-based CI around a proportion is when p=0.5 (the variance is p(1-p)/n; for a fixed n, p(1-p) is maximised when p=0.5 as you can prove using calculus or show by graphing), so if you want to be safe, this is a sensible choice. If you were confident that most people have a dog (at least 0.8), few people were ...


1

You can say that in your data, on average, belonging to the fifth quantile of parental income instead of the second leads to an increase in the score equal to: 7.49 - (-0.95) = 8.44. And this difference was statistically significant at an alpha level of 0.01. Regarding your second question: given your model, the difference in score between two people of the ...


4

Having read the blog post, I think the author is saying that we shouldn't use randomness in models of the real world because the real world is not random, since everything (such as a coin flip) actually has a cause. This makes probability theory the science of last resort. Only after truly exhausting your ability to investigate causal factors and processes ...


4

I think the issue with the arguments raised in the question is the naive realist philosophy of models apparently behind it. If we model an experiment in a frequentist manner, what we do is that, when using the model, we treat the experiment as if it would be infinitely repeatable, with random outcomes the relative frequency of which stabilises for a growing ...


3

Mitch Gordon confesses that he doesn't exactly understand the role of probability theory in science and you should believe him. IMHO he gives adequate evidence of his confusion in the link. If you think you may have a biased coin, a probability model for a fair coin is a hindrance to checking the coin for fairness only if you're too lazy to toss it ...


4

As to the first critique, it could be a critique of any and all branches of the sciences. There are no perfectly repeatable experiments. It isn't really possible to completely control any experiment. A meteor could strike the location of the experiment, for example. Also, the ability to repeat an experiment is irrelevant. Most Frequentist inferences are ...


1

$ f(X|mean=y,\theta)=f(X|\sum X_i =ny, \theta)=f(X| \sum X_i=ny) =f(X| mean =y) $ So by definiton, yes


0

Your reasoning seems right. I think you assume $\mathbf{X}=(X_i)_{i=1}^n$, with $X_i$ independently identically distributed from $f$. Then, you have correctly rewrititten the likelihood $\mathbb{L}(\mathbf{X},\theta)$ as a function of the parameter $\theta$ that depends (up to a proportianality constant) on the sample $\mathbf{X}$ just via the statistic $T(\...


1

Using wikipedia's intuitive definition: Roughly, given a set $\mathbf {X}$ of independent identically distributed data conditioned on an unknown parameter $\theta$ , a sufficient statistic is a function $T(\mathbf {X} )$ whose value contains all the information needed to compute any estimate of the parameter. If $T(X)$ contains all the information, $T(X)/...


1

An important reason for finding the sample standard deviation $S$ is that $S$ is an estimate of the population standard deviation $\sigma.$ For larger $n$ the estimate is more precise. Confidence intervals can help to give an idea of the precision. Suppose you are sampling from a normal distribution with standard deviation $\sigma = 5.$ I will show you the ...


0

Yes, the marginal likelihood has a closed-form for all polynomial models of the form $\mathbf{t} = X\mathbf{w} + \boldsymbol{\varepsilon}$, where, \begin{aligned} X &= \begin{bmatrix} \mathbf{1}^T & \mathbf{(x^1)}^T & (\mathbf{x}^2)^T ... (\mathbf{x}^n)^T \end{bmatrix}\\ \boldsymbol{\varepsilon} &\sim \mathcal{N}(0, \sigma_n^2I)\\ \mathbf{w} &...


5

This part of the book is wrong Your misgivings on this matter are appropriate, because this part of the book is wrong. Even when making inferences about infinite populations, random sampling does not give independent random variables --- it gives conditionally independent random variables, conditional on the empirical distribution of the underlying ...


11

This extract from the text suffers from ambiguity and incorrectness. Let's deal with the latter first. Independence of two random variables $X$ and $Y$ is not about one variable "providing no information about the first" (a remarkably ambiguous phrase in its own right!). Independence is strictly about probabilities and it means nothing more nor ...


0

I think you're overthinking it. When we conceptualize a study, and its sampling frame, we consider looking retrospectively at the outcomes, measured and unmeasured, from the study's completion. As noted in the example, the student plans to randomly sample days from the school year. So we can consider each of the year's 52*5 school days, and its associated ...


0

Your point estimate of the correlation coefficient is indeed very small, so you might do well to perform a hypothesis test to see if it is significantly different from zero. It's possible that the true correlation value is actually zero (no correlation whatsoever), and that your non-zero correlation coefficient is simply a result of statistical noise. Even ...


3

No, I don't understand same things from these explanations. For example, a constant predictor is a robust one, but it's probably not a good predictor.


8

McElreath appears to be maligning the oft repeated practice of requiring regressors (to the right of the equality sign, sometimes 'independent' variables) and regressands (to the left of the equality sign, sometimes 'dependent' variables) to be normally distributed (i.e. "Gaussian") in the context of something like OLS regression. In fact none of ...


0

You don't say anything about the actual weights you are dealing with. So I'll choose some hypothetical values for illustration. Suppose women have average weights about 230 (pounds) with standard deviation 45, and men averaging about 245 with SD 55. Then respective sample sizes 150 and 50 may not be enough to detect the 15 lbs difference in gender weights. ...


4

I answered this once on twitter, I can reproduce the answer here. Derivation (graphs licensing each step are provided below). $$ \begin{align} P(y|do(x)) &= P(y|do(x), do(z)) \qquad &\text{Rule 3: $(Y \perp\!\!\!\perp Z|X)_{G_\overline{XZ}}$}\\ &= P(y |x, do(z)) \qquad &\text{Rule 2: $(Y\perp\!\!\!\perp X)_{G_{\overline{Z}\...


2

As a preliminary note, I'm not a fan of the notation used in the excerpted material, so I'll avoid using most of it. In any case, as you can see from the formulae, the difference between the variance of the prediced value and the variance of a new value is the variance of the error term in the regression, which is estimated by the $\text{MSE}$ term. So the ...


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