### 统计代写|贝叶斯网络代写Bayesian network代考|PHYS4016

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|贝叶斯网络代写Bayesian network代考|A simple example: Implicit estimation in binomial distribution case

To illustrate how the Implicit method proceeds let us consider a simple example. Let $X=$ $\left(N_1, N_2\right)$ be a random variable following a binomial distribution with unknown parameters $N=N_1+N_2$ and $\theta=\left(\theta_1, \theta_2\right)$. We first estimate $N$ by the Implicit method after that we use the estimate $\widehat{N}$ to estimate $\theta$. After some calculations, we obtain
$$P(N / X)=\frac{P(X / N)}{C(X)}=C_N^{N_1} \theta_1^{N-N_1}\left(1-\theta_1\right)^{N_1+1},$$
where $\stackrel{\vee}{N}1=N-N_1=\sum{i=2}^r N_i$
So, the Implicit distribution of $N$ given $X=\left(N_1, \ldots, N_r\right)$ is a Pascal distribution with parameters $1-\theta_1$ and $N_1+1$. Suppose that $\theta_1$ is known, the Implicit estimator $\hat{N}$ of $N$ is the mean of the Pascal distribution:
$$\widehat{N}=E(N / X)=\sum_{N \geq 0} N C_N^{N_1} \theta_1^{N-N_1}\left(1-\theta_1\right)^{\stackrel{v}{N_1}+1} .$$
Let $N_{o b}$ be the number of observations and take
$$\theta_{k_0}=\max \left{\frac{N_k}{N_{o b}} ; \frac{N_k}{N_{o b}} \leq \frac{1}{r-1} \text { and } 1 \leq k \leq r\right} .$$
After some calculations, we have
$$\widehat{N}=\frac{\left(\stackrel{\vee}{N_{k_0}}+1\right)}{1-\theta_{k_0}}=N_{o b}+\frac{N_{k_0}}{N_{k_0}},$$
where $N_{k_0}=N_{o b}-N_{k_0}$
Consequently, the probability of the next observation to be in state $x^k$ given a dataset $D$ is obtained by
$$\hat{\theta}k=P\left(X{N_{a b}+1}=x^k / D\right)=\frac{N_k+1}{\hat{N}+r}, 1 \leq k \leq r \text { and } k \neq k_0$$
and $\hat{\theta}{k_0}=1-\sum{i \neq k_0} \hat{\theta}_i$
other examples and selected applications of Implicit distributions can be found in the original paper (Hassairi et al., 2005).

## 统计代写|贝叶斯网络代写Bayesian network代考|Implicit inference with Bayesian Networks

Formally, a Bayesian network is defined as a set of variables $X=\left{X_1, \ldots, X_n\right}$ with :(1) a network structure $S$ that encodes a set of conditional dependencies between variables in $X$, and (2) a set $P$ of local probability distributions associated with each variable. Together, these components define the joint probability distribution of $X$.
The network structure $S$ is a directed acyclic graph (DAG). The nodes in $S$ correspond to the variables in $X_i$. Each $X_i$ denotes both the variable and its corresponding node, and $\mathrm{Pa}\left(X_i\right)$ the parents of node $X_i$ in $S$ as well as the variables corresponding to those parents. The lack of possible arcs in $S$ encode conditional independencies. In particular, given structure $S$, the joint probability distribution for $X$ is given by the product of all specified conditional probabilities:
$$P\left(X_1, \ldots, X_n\right)=\prod_{i=1}^n P\left(X_i / P a\left(X_i\right)\right)$$
a factorization that is known as the local Markov property and states that each node is independent of its non descendant given the parent nodes. For a given $B N$ the probabilities will thus depend only on the structure of the parameters set.

## 统计代写|贝叶斯网络代写Bayesian network代考|A simple example: Implicit estimation in binomial distribution case

$$P(N / X)=\frac{P(X / N)}{C(X)}=C_N^{N_1} \theta_1^{N-N_1}\left(1-\theta_1\right)^{N_1+1},$$

$$\widehat{N}=E(N / X)=\sum_{N \geq 0} N C_N^{N_1} \theta_1^{N-N_1}\left(1-\theta_1\right)^{\stackrel{n}{N_1+1}} .$$

$$\widehat{N}=\frac{\left(N_{k_0}^{\vee}+1\right)}{1-\theta_{k_0}}=N_{o b}+\frac{N_{k_0}}{N_{k_0}},$$

$$\hat{\theta} k=P\left(X N_{a b}+1=x^k / D\right)=\frac{N_k+1}{\hat{N}+r}, 1 \leq k \leq r \text { and } k \neq k_0$$

## 统计代写|贝叶斯网络代写Bayesian network代考|Implicit inference with Bayesian Networks

$$P\left(X_1, \ldots, X_n\right)=\prod_{i=1}^n P\left(X_i / P a\left(X_i\right)\right)$$

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## MATLAB代写

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