# 计算机代写|机器学习代写machine learning代考|COMP3670

## 计算机代写|机器学习代写machine learning代考|Beta Distribution

The beta distribution is used to describe a continuous random variable, $X$, that lakes a probability-like value $x \in \mathbb{R}$ and $0 \leq x \leq 1$. The beta distribution takes the following functional form:
$$\operatorname{Beta}(x \mid \alpha, \beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1},$$
where $\Gamma(\cdot)$ denotes the gamma function, and $\alpha$ and $\beta$ are two positive parameters of the beta distribution. Similarly, we can summarize some key properties for the beta distribution as follows:

• Parameters: $\alpha>0$ and $\beta>0$.
• Support (the domain of the continuous random variable):
$$x \in \mathbb{R} \text { and } 0 \leq x \leq 1 .$$
• $$• \int_0^1 \operatorname{Beta}(x \mid \alpha, \beta) d x=1 •$$
• We can recognize that the beta distribution shares the same functional form as the binomial distribution. They differ only in terms of swapping the roles of the parameters and random variables. Therefore, these two distributions are said to be conjugate to each other. In this sense, the beta distribution can be viewed as a distribution of the parameter $p$ in the binomial distribution. As we will learn, this viewpoint plays an important role in Bayesian learning (refer to Chapter 14).
• Depending on the choices of the two parameters $\alpha$ ard $\beta$, the beta distribution behaves quite differently. As shown in Figure 2.7, when both parameters are larger than 1 , the beta distribution is a unimodal bellshaped distribution between 0 and 1 . The mode of the distribution can be computed as $(\alpha-1) /(\alpha+\beta-2)$ in this case. It becomes a monotonic distribution when one parameter is larger than 1 and the other is smaller than 1 , particularly monotonically decaying if $0<\alpha<1<\beta$ and monotonically increasing if $0<\beta<1<\alpha$. At last, if both parameters are smaller than 1 , the beta distribution is bimodal between 0 and 1, peaking at the two ends.

## 计算机代写|机器学习代写machine learning代考|Dirichlet Distribution

The Dirichlet distribution is a multivariate generalization of the beta distribution that is used to describe multiple continuous random variables $\left{X_1, X_2, \cdots, X_m\right}$, taking values on the probabilities of observing a complete set of mutually exclusive events. As a result, the values of these random variables are always summed to 1 because these events are complete. For example, if we use some biased dice in a tossing experiment, we can define six random variables, each of which represents the probability of observing each digit when tossing a die. For each biased die, these six random variables take different probabilities, but they always sum to 1 for each die. These six random variables from all biased dice can be assumed to follow the Dirichlet distribution.

In general, the Dirichlet distribution takes the following functional form:
$$\operatorname{Dir}\left(p_1, p_2, \cdots, p_m \mid r_1, r_2, \cdots, r_m\right)$$ $$=\frac{\Gamma\left(r_1+\cdots+r_m\right)}{\Gamma\left(r_1\right) \cdots \Gamma\left(r_m\right)} p_1^{r_1-1} p_2^{r_2-1} \cdots p_m^{r_m-1},$$
where $\left{r_1, r_2, \cdots, r_m\right}$ denote $m$ positive parameters of the distribution. We can similarly summarize some key properties for the Dirichlet distribution as follows:

• Parameters: $r_i>0(\forall i=1, \cdots, m)$.
• Support: The domain of $m$ random variables is an $m$-dimensional simplex that can be represented as
$$0<p_i<1 \quad(\forall i=1, \cdots, m) \quad \text { and } \quad \sum_{i=1}^m p_i=1 .$$
For example, Figure $2.8$ shows a three-dimensional simplex for the Dirichlet distribution of three random variables $\left{p_1, p_2, p_3\right}$ when $m=3$.

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Beta Distribution

beta 分布用于描述一个连续的随机变量， $X$ ，湖有一个类似概率的值 $x \in \mathbb{R}$ 和 $0 \leq x \leq 1$. beta 分布采用 以下函数形式:
$$\operatorname{Beta}(x \mid \alpha, \beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1},$$

• 参数: $\alpha>0$ 和 $\beta>0$.
• 支持 (连续随机变量的域)：
$x \in \mathbb{R}$ and $0 \leq x \leq 1$
• \$\$
• \int_ $0^{\wedge} 1 \backslash$ \operatorname ${$ Beta $}(x \backslash \operatorname{mid} \backslash a \mid p h a$, beta) $d x=1$
• \$\$
• 我们可以认识到贝塔分布与二项分布具有相同的函数形式。它们仅在交换参数和随机变量的角色方面 有所不同。因此，这两个分布被称为彼此共轭。从这个意义上说， $\beta$ 分布可以看作是参数的分布 $p$ 在二 项分布中。正如我们将要学习的，这个观点在贝叶斯学习中扮演着重要的角色（参见第 14 章)。
• 取决于两个参数的选择 $\alpha$ 高的 $\beta$, beta 分布的行为完全不同。如图 2.7 所示，当两个参数都大于 1 时， $\beta$ 分布是一个介于 0 和 1 之间的单峰钟形分布。分布的模式可以计算为 $(\alpha-1) /(\alpha+\beta-2)$ 在这种 情况下。当一个参数大于 1 而另一个参数小于 1 时，它变成单调分布，特别是单调詉减，如果 $0<\alpha<1<\beta$ 如果单调递增 $0<\beta<1<\alpha$. 最后，如果两个参数都小于 1，则 beta 分布在 0 和 1 之间为双峰分布，在两端达到峰值。

## 计算机代写|机器学习代写machine learning代考|Dirichlet Distribution

\begin{aligned} & \operatorname{Dir}\left(p_1, p_2, \cdots, p_m \mid r_1, r_2, \cdots, r_m\right) \ =& \frac{\Gamma\left(r_1+\cdots+r_m\right)}{\Gamma\left(r_1\right) \cdots \Gamma\left(r_m\right)} p_1^{r_1-1} p_2^{r_2-1} \cdots p_m^{r_m-1} \end{aligned}

• 参数: $r_i>0(\forall i=1, \cdots, m)$.
• 支持: 领域 $m$ 随机变量是 $m$-维单纯形，可以表示为
$$0<p_i<1 \quad(\forall i=1, \cdots, m) \quad \text { and } \quad \sum_{i=1}^m p_i=1$$

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