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stat0011 Decision and Risk 决策与风险 伦敦大学学院

STAT0011 is specified as a formal option for third and fourth year undergraduates from the Department of Mathematics. Any such student who has taken one of the standard prerequisites should simply register for STAT0011 on Portico and await a decision. Mathematics students who do not satisfy the standard prerequisites must additionally consult a member of staff in the Department of Statistical Science.

Content: Organisational decisions are often made in situations where there is uncertainty as to whether the decision-maker’s choice will bring benefit or disaster. Risk is the probability that a decision will result in a loss or undesirable outcome.

stat0011 Decision and Risk 决策与风险 伦敦大学学院
问题 1.

We now derive the limits of the multivariate uniform prior for $k$ regression coefficients. The basic idea used here is a generalization of the very simple idea that was used to derive the limits for the univariate case. This generalization will involve a transition from univariate line pieces to multivariate ellipsoids.

Say we have $k$ independent predictors $\mathbf{x}_1, \ldots, \mathbf{x}_k$, that is, $\mathbf{x}_i^T \mathbf{x}_j=0$ for $i \neq j$. Then we have that
$$
\beta_i=\frac{\mathbf{x}_i^T \mathbf{y}}{\mathbf{x}_i^T \mathbf{x}_i}=\frac{|\mathbf{y}|}{\left|\mathbf{x}_i\right|} \cos \theta_i, \quad-\frac{\pi}{2} \leq \theta_i \leq \frac{\pi}{2}
$$

Because of the independence of the $k$ independent variables we have that if one of the angles $\theta_i=0$, then $\theta_j=\pi / 2$ for $j \neq i$. It follows that all the possible values of $\beta_i$ must lie in an $\mathrm{k}$-variate ellipsoid centered at the origin and with respective axes of
$$
r_i=\frac{|\mathbf{y}|_{\max }}{\left|\mathbf{x}i\right|{\min }} .
$$
If we substitute (15) in the identity for the volume of an k-variate ellipsoid
$$
V=\pi\left(\frac{4}{3}\right)^{k-2} \prod_{i=1}^k r_i .
$$
We find that
$$
V=\pi\left(\frac{4}{3}\right)^{k-2} \frac{|\mathbf{y}|_{\max }^k}{\prod_{i=1}^k\left|\mathbf{x}i\right|{\min }}
$$

问题 2.

Suppose we wish the compute the evidence of a specific model $M$, with
$$
M: \mathbf{y}=X \beta+\mathbf{e},
$$

where $\mathbf{e}=\left(e_1, \ldots, e_N\right)$ and $e_i \tilde{N}(0, \sigma)$ for $i=1, \ldots, N$, and for some known value of $\sigma$, then the corresponding likelihood ( $7 \mathrm{~b})$ is
$$
p(\mathbf{y} \mid X, \beta, \sigma, M)=\frac{1}{\left(2 \pi \sigma^2\right)^{N / 2}} \exp \left[-\frac{(\mathbf{y}-X \beta)^T(\mathbf{y}-X \beta)}{2 \sigma^2}\right] .
$$
Combining the likelihood ( $7 \mathrm{~b}$ ) with the derived proper non-informative prior (21), we get the posterior
$$
p(\boldsymbol{\beta} \mid I)=\frac{1}{\pi}\left(\frac{3}{4}\right)^{k-2} \frac{\left|X^T X\right|{\min }^{1 / 2}}{|\mathbf{y}|{\max }^k}, \quad \boldsymbol{\beta} \in \text { Ellipsoid. }
$$
For the regression coefficients $\boldsymbol{\beta}$, we get the following multivariate distribution
$$
\begin{aligned}
& p(\mathbf{y}, \boldsymbol{\beta} \mid X, \sigma, M)= \
& \quad\left(\frac{3}{4}\right)^{k-2} \frac{\left|X^T X\right|{\min }^{1 / 2}}{\pi|\mathbf{y}|{\max }^k} \frac{1}{\left(2 \pi \sigma^2\right)^{N / 2}} \exp \left[-\frac{(\mathbf{y}-X \boldsymbol{\beta})^T(\mathbf{y}-X \boldsymbol{\beta})}{2 \sigma^2}\right] .
\end{aligned}
$$

学习资料推荐

针对 Decision and Risk 推荐三本教材:

✅ Comparative risk assessment and environmental decision making

✅ The Credit Scoring Toolkit: Theory and Practice for Retail Credit Risk Management and Decision Automation

✅ Risk, Uncertainty and Decision-Making in Property

商业和经济预测介绍:

  • 在研讨会上,来自美国国家环境健康科学研究所的克里斯托弗-波蒂尔强调,人类与环境之间复杂的相互作用要求采用系统方法来理解环境健康和实施环境健康决策。基本需求、住所因素和内源因素等环境因素相互作用,决定了一个人的健康状况。他认为,关于风险评估的一般假设–很容易找出问题的主要原因并制定解决方案–是一种过时的方法。此外,他还指出,关于人体系统以及科学如何解决其暴露于危害的问题,目前正在进行大量的研究和测试,从人口和临床层面到分子层面。虽然所有这些科学都有助于了解环境对健康的影响,但大多数风险评估都是以毒理学和流行病学证据为基础,而不是以遗传学和毒物基因组学等新兴科学为基础。因此,科学家和决策者需要关注新兴科学领域,找到将这些研究纳入环境健康决策过程的方法。