Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [bayes-risk]

10 questions
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What is the link between probabilistic predictions and Bayes optimum decisions?

Frank Harrell writes in one of our community wikis about the link between "Bayes optimum decision" and the link to probabilistic predictions (and, thus, one of his favorite topics in proper scoring rules for machine learning evaluation). What is…
Dave
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Least favorable prior - Find the distribution that maximizes the Bayes risk

Suppose I've found that the Bayes risk is of the form $$r(\theta) = \int_{-a}^a \theta^2 \pi(\theta)d\theta $$ I want to show that the following distribution, $\pi(a)=\pi(-a)=0.5$, maximizes this quantity, i.e. that it's a least favorable prior…
Maverick Meerkat
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Bayes Risk Not Connected to Observed Data

It puzzles me that the Bayes risk seems not connected to the observed data. Let me illustrate this with an example. Let a coin toss follow a Bernoulli distribution with a hidden parameter $\theta$ and let the prior of $\theta$ be a uniform…
Tom Bennett
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Min and Max values of Bayesian Risk Classifier when Posteriori Probability and PDFs are Unknown

I'm struggling to come up with a well reasoned argument for this problem. Let $\tau_1$ be the posteriori probability and let $L(r^*)$ be the risk classifier. For this scenario, assume: $X\in\mathbb{X}=[0,1],Y\in\{ 0,1 \}$ $\pi_y=P(Y=y)=1/2$ for…
Major Redux
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Derivation when priori probability is Unknown

I want to derive an expression for the Bayesian classification risk $L(r^*)$ when the priori $\tau_1\in[0,1]$ is unknown. For this problem, let: $X\in\mathbb{X}=[0,1],Y\in\{ 0,1 \}$ $\pi_y=P(Y=y)=1/2$ for $y\in{0,1}$ Also, conditional distributions…
Major Redux
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Derivations of Bayesian Risk Classifier when Posteriori Probability is Unknown

I found two expressions for a Bayesian risk classifier when the posteriori probability is unknown, but I don't understand how and why the derivations were made. For this scenario, assume: $X\in\mathbb{X}=[0,1],Y\in\{ 0,1 \}$ $\pi_y=P(Y=y)=1/2$ for…
Major Redux
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How to calculate the Bayesian Risk Classifier

I'm not exactly sure how to calculate the Bayesian risk Classifier $L(r^*)$ for $Y\in\{ 0,1 \}$. For this scenario, assume: $X\in\mathbb{X}=[0,1],Y\in\{ 0,1 \}$ $\pi_y=P(Y=y)=1/2$ for $y\in{0,1}$ Conditional distributions $[X|Y=y]$ characterised…
Major Redux
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How to find the Bayesian equivalent of of $\bar{X}-\bar{Y}$

Let $X_1, \dots, X_n$ i.i.d. from $N(\mu, \sigma^2)$; $Y_1, \dots, Y_m$ i.i.d. from $N(\eta, \tau^2)$. $X_1, \dots, X_n$ are independent of $Y_1, \dots, Y_m$. And $\tau^2$ and $\sigma^2$ are considered known. I want to show that $\bar{X}-\bar{Y}$ is…
anonyx2
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The risk functions of this estimator

Let $x_1 ,\ldots ,x_n $ be a random sample taken from the Weibull distribution $$f(x;\nu ,\theta)=\frac{\nu}{\theta}x^{\nu -1}\exp(-\frac{x^\nu}{\theta}),$$ where $x,\nu,\theta >0$. I got the Bayes estimator of $\theta$ under the linear exponential…
M.Ramana
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Bayes decision rule - Basic

Good day, I'm having a problem with an exercise I have for in class. What is given for this example is the likelihood of an event happening and vice versa. Other details I have is only what the cost will be of the event occurring. However, from…
pewpew
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