应用贝叶斯方法|  Applied Bayesian Methods代写STAT0031代考

这是一份UCL伦敦大学学院STAT0031作业代写的成功案

应用贝叶斯方法|  Applied Bayesian Methods代写STAT0031代考
问题 1.


A random variable $X \in \mathbb{R}$ has a normal $\left(\theta, \sigma^{2}\right)$ distribution if $\sigma^{2}>0$ and
$$
p\left(x \mid \theta, \sigma^{2}\right)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{1}{2}(x-\theta)^{2} / \sigma^{2}} \quad \text { for }-\infty<x<\infty
$$


证明 .

For this distribution,
$$
\begin{aligned}
\mathrm{E}\left[X \mid \theta, \sigma^{2}\right] &=\theta, \
\operatorname{Var}\left[X \mid \theta, \sigma^{2}\right] &=\sigma^{2} \
\operatorname{mode}\left[X \mid \theta, \sigma^{2}\right] &=\theta \
p\left(x \mid \theta, \sigma^{2}\right) &=\operatorname{dnorm}(x, \text { theta,sigma })
\end{aligned}
$$

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PHAS00030 COURSE NOTES :

This condition implies that our density for $\theta$ must be the uniform density:
$$
p(\theta)=1 \text { for all } \theta \in[0,1]
$$
For this prior distribution and the above sampling model, Bayes’ rule gives
$$
\begin{aligned}
p\left(\theta \mid y_{1}, \ldots, y_{129}\right) &=\frac{p\left(y_{1}, \ldots, y_{129} \mid \theta\right) p(\theta)}{p\left(y_{1}, \ldots, y_{129}\right)} \
&=p\left(y_{1}, \ldots, y_{129} \mid \theta\right) \times \frac{1}{p\left(y_{1}, \ldots, y_{129}\right)} \
& \propto p\left(y_{1}, \ldots, y_{129} \mid \theta\right)
\end{aligned}
$$




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