### 统计代写|贝叶斯统计代写Bayesian statistics代考|Bayesian inference methods

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|贝叶斯统计代写beyesian statistics代考|The Bayes theorem for probability

The Bayes theorem allows us to calculate probabilities of events when additional information for some other events is available. For example, a person may have a certain disease whether or not they show any symptoms of it. Suppose a randomly selected person is found to have the symptom. Given this additional information, what is the probability that they have the disease? Note that having the symptom does not fully guarantee that the person has the disease.

To formally state the Bayes theorem, let $B_{1}, B_{2}, \ldots, B_{k}$ be a set of mutually exclusive and exhaustive events and let $A$ be another event with positive probability (see illustration in Figure 4.1). The Bayes theorem states that for any $i, i=1, \ldots, k$,
$$P\left(B_{i} \mid A\right)=\frac{P\left(B_{i} \cap A\right)}{P(A)}=\frac{P\left(A \mid B_{i}\right) P\left(B_{i}\right)}{\sum_{j=1}^{k} P\left(A \mid B_{j}\right) P\left(B_{j}\right)}$$
Example 4.1. We can understand the theorem using a simple example. Consider a rare disease that is thought to occur in $0.1 \%$ of the population. Using a particular blood test a physician observes that out of the patients with disease $99 \%$ possess a particular symptom. Also assume that $1 \%$ of the population without the disease have the same symptom. A randomly chosen person from the population is blood tested and is shown to have the symptom. What is the conditional probability that the person has the disease?

Here $k=2$ and let $B_{1}$ be the event that a randomly chosen person has the disease and $B_{2}$ is the complement of $B_{1}$. Let $A$ be the event that a randomly chosen person has the symptom. The problem is to determine $P\left(B_{1} \mid A\right)$.
We have $P\left(B_{1}\right)=0.001$ since $0.1 \%$ of the population has the disease, and $P\left(B_{2}\right)=0.999$. Also, $P\left(A \mid B_{1}\right)=0.99$ and $P\left(A \mid B_{2}\right)=0.01$. Now
\begin{aligned} P(\text { disease } \mid \text { symptom })=P\left(B_{1} \mid A\right) &=\frac{P\left(A \mid B_{1}\right) P\left(B_{1}\right)}{P\left(A \mid B_{1}\right) P\left(B_{1}\right)+P\left(A \mid B_{2}\right) P\left(B_{2}\right)} \ &=\frac{0.99 \times 0.001}{0.99 \times 0.001+0.999 \times 0.01} \ &=\frac{99}{99+999}=0.09 . \end{aligned}
The probability of disease given symptom here is very low, only $9 \%$, since the

disease is a very rare disease and there will be a large percentage of individuals in the population who have the symptom but not the disease, highlighted by the figure 999 as the second last term in the denominator above.

It is interesting to see what happens if the same person is found to have the same symptom in another independent blood test. In this case, the prior probability of $0.1 \%$ would get revised to $0.09$ and the revised posterior probability is given by:
$$P(\text { disease } \mid \text { twice positive })=\frac{0.99 \times 0.09}{0.99 \times 0.09+0.91 \times 0.01}=0.908 .$$
As expected, this probability is much higher since it combines the evidence from two independent tests. This illustrates an aspect of the Bayesian world view: the prior probability gets continually updated in the light of new evidence.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Bayes theorem for random variables

The Bayes theorem stated above is generalized for two random variables instead of two events $A$ and $B_{i}$ ‘s as noted above. In the generalization, $B_{i}$ ‘s will be replaced by the generic parameter $\theta$ which we want to estimate and $A$ will be replaced by the observation random variable denoted by $Y$. Also, the probabilities of events will be replaced by the probability (mass or density) function of the argument random variable. Thus, $P\left(A \mid B_{i}\right)$ will be substituted by $f(y \mid \theta)$ where $f(\cdot$ ) denotes the probability (mass or density) function of the random variable $X$ given a particular value of $\theta$. The replacement for $P\left(B_{i}\right)$ is $\pi(\theta)$, which is the prior distribution of the unknown parameter $\theta$. If $\theta$ is

a discrete parameter taking only finite many, $k$ say, values, then the summation in the denominator of the above Bayes theorem will stay as it is since $\sum_{j=1}^{k} \pi\left(\theta_{j}\right)$ must be equal to 1 as the total probability. If, however, $\theta$ is a continuous parameter then the summation in the denominator of the Bayes theorem must be replaced by an integral over the range of $\theta$, which is generally taken as the whole of the real line.

The Bayes theorem for random variables is now stated as follows. Suppose that two random variables $Y$ and $\theta$ are given with probability density functions (pdfs) $f(y \mid \theta)$ and $\pi(\theta)$, then
$$\pi(\theta \mid y)=\frac{f(y \mid \theta) \pi(\theta)}{\int_{-\infty}^{\infty} f(y \mid \theta) \pi(\theta) d \theta},-\infty<\theta<\infty .$$
The probability distribution given by $\pi(\theta)$ captures the prior beliefs about the unknown parameter $\theta$ and is the prior distribution in the Bayes theorem. The posterior distribution of $\theta$ is given by $\pi(\theta \mid y)$ after observing the value $y$ of the random variable $Y$. We illustrate the theorem with the following example.
Example 4.2. Binomial Suppose $Y \sim \operatorname{binomial}(n, \theta)$ where $n$ is known and we assume $\operatorname{Beta}(\alpha, \beta)$ prior distribution for $\theta$. Here the likelihood function is
$$f(y \mid \theta)=\left(\begin{array}{l} n \ y \end{array}\right) \theta^{y}(1-\theta)^{n-y}$$
for $0<\theta<1$. The function $f(y \mid \theta)$ is to be viewed as a function of $\theta$ for a given value of $y$, although its argument is written as $y \mid \theta$ instead of $\theta \mid y$. This is because we use the probability density function of $Y$, which is more widely known, and we avoid introducing further notation for the likelihood function e.g. $L(\theta ; y)$.

Suppose that the prior distribution is the beta distribution (A.22) having density
$$\pi(\theta)=\frac{1}{B(\alpha, \beta)} \theta^{\alpha-1}(1-\theta)^{\beta-1}, \quad 0<\theta<1 .$$

## 统计代写|贝叶斯统计代写beyesian statistics代考|Sequential updating of the posterior distribution

Consider the denominator in the posterior distribution. The denominator, given by, $\int_{-\infty}^{\infty} f(y \mid \theta) \pi(\theta) d \theta$ or $\int_{-\infty}^{\infty} f\left(y_{1}, \ldots, y_{n} \mid \theta\right) \pi(\theta) d \theta$ is free of the unknown parameter $\theta$ since $\theta$ is only a dummy in the integral, and it has been integrated out in the expression. The posterior distribution $\pi\left(\theta \mid y_{1}, \ldots, y_{n}\right)$ is to be viewed as a function of $\theta$ and the denominator is merely a constant. That is why, we often ignore the constant denominator and write the posterior distribution $\pi\left(\theta \mid y_{1}, \ldots, y_{n}\right)$ as
$$\pi\left(\theta \mid y_{1}, \ldots, y_{n}\right) \propto f\left(y_{1}, \ldots, y_{n} \mid \theta\right) \times \pi(\theta)$$
By noting that $f\left(y_{1}, \ldots, y_{n} \mid \theta\right)$ provides the likelihood function of $\theta$ and $\pi(\theta)$ is the prior distribution for $\theta$, we write:
Posterior $\propto$ Likelihood $\times$ Prior.
Hence we always know the posterior distribution up-to a normalizing constant. Often we are able to identify the posterior distribution of $\theta$ just by looking at the numerator as in the two preceding examples.

The structure of the Bayes theorem allows sequential updating of the posterior distribution. By Bayes theorem we “update” the prior belief $\pi(\theta)$ to $\pi(\theta \mid \mathbf{y})$. Note that $\pi\left(\theta \mid y_{1}\right) \propto f\left(y_{1} \mid \theta\right) \pi(\theta)$ and if $Y_{2}$ is independent of $Y_{2}$ given the parameter $\theta$, then:
\begin{aligned} \pi\left(\theta \mid y_{1}, y_{2}\right) & \propto & f\left(y_{2} \mid \theta\right) f\left(y_{1} \mid \theta\right) \pi(\theta) \ & \propto f\left(y_{2} \mid \theta\right) \pi\left(\theta \mid y_{1}\right) \end{aligned}
Thus, at the second stage of data collection, the first stage posterior distribution, $\pi\left(\theta \mid y_{1}\right)$ acts as the prior distribution to update our belief about $\theta$ after. Thus, the Bayes theorem shows how the knowledge about the state of nature represented by $\theta$ is continually modified as new data becomes available. There is another strong point that jumps out of this sequential updating. It is possible to start with a very weak prior distribution $\pi(\theta)$ and upon observing data sequentially the prior distribution gets revised to a stronger one, e.g. $\pi\left(\theta \mid y_{1}\right)$ when just one observation has been recorded – of course, assuming that data are informative about the unknown parameter $\theta$.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Bayes theorem for random variables

F(是∣θ)=(n 是)θ是(1−θ)n−是

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