### 统计代写 | Statistical Learning and Decision Making代考| Inference in Bayesian Networks

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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 | Statistical Learning and Decision Making代考|Inference in Bayesian Networks

In inference problems, we want to infer a distribution over query variables given some observed evidence variables. The other nodes are referred to as hidden variables. We often refer to the distribution over the query variables given the evidence as a posterior distribution.

To illustrate the computations involved in inference, recall the Bayesian network from example $2.5$, the structure of which is reproduced in figure $3.1$. Suppose we have $B$ as a query variable and evidence $D=1$ and $C=1$. The inference task is to compute $P\left(b^{1} \mid d^{1}, c^{1}\right)$, which corresponds to computing the probability that we have a battery failure given an observed trajectory deviation and communication loss.

From the definition of conditional probability introduced in equation (2.22), we know
$$P\left(b^{1} \mid d^{1}, c^{1}\right)=\frac{P\left(b^{1}, d^{1}, c^{1}\right)}{P\left(d^{1}, c^{1}\right)}$$

To compute the numerator, we must use a process known as marginalization, where we sum out variables that are not involved, in this case $S$ and $E$ :
$$P\left(b^{1}, d^{1}, c^{1}\right)=\sum_{s} \sum_{e} P\left(b^{1}, s, e, d^{1}, c^{1}\right)$$
We know from the chain rule for Bayesian networks introduced in equation (2.31) that
$$P\left(b^{1}, s, e, d^{1}, c^{1}\right)=P\left(b^{1}\right) P(s) P\left(e \mid b^{1}, s\right) P\left(d^{1} \mid e\right) P\left(c^{1} \mid e\right)$$
All of the components on the right-hand side are specified in the conditional probability distributions associated with the nodes in the Bayesian network. We can compute the denominator in equation (3.1) using the same approach but with anditional summation over the values for $B$.

This process of using the definition of conditional probability, marginalization, and applying the chain rule can be used to perform exact inference in any Bayesian network. We can implement exact inference using factors. Recall that factors represent discrete multivariate distributions. We use the following three operations on factors to achieve this:

• We use the factor product (algorithm 3.1) to combine two factors to produce a larger factor whose scope is the combined scope of the input factors. If we have $\phi(X, Y)$ and $\psi(Y, Z)$, then $\phi \cdot \psi$ will be over $X, Y$, and $Z$ with $(\phi \cdot \psi)(x, y, z)=$ $\phi(x, y) \psi(y, z)$. The factor product is demonstrated in example $3.1$.
• We use factor marginalization (algorithm 3.2) to sum out a particular variable from the entire factor table, removing it from the resulting scope. Example $3.2$ illustrates this process.
• We use factor conditioning (algorithm 3.3) with respect to some evidence to remove any rows in the table inconsistent with that evidence. Example $3 \cdot 3$ demonstrates factor conditioning.

## 统计代写 | Statistical Learning and Decision Making代考|Inference in Naive Bayes Models

The previous section presented a general method for performing exact inference in any Bayesian network. This section discusses how this same method can be used to solve classification problems for a special kind of Bayesian network structure known as a naive Bayes model. This structure is shown in figure 3.2. An equivalent but more compact representation is shown in figure $3.3$ using a plate, shown as a rounded box. The $i=1: n$ in the bottom of the box specifies that the $i$ in the subscript of the variable name is repeated from 1 to $n$.

In the naive Bayes model, the class $C$ is the query variable, and the observed features $O_{1: n}$ are the evidence variables. The naive Bayes model is called naive because it assumes conditional independence between the evidence variables given the class. Using the notation introduced in section 2.6, we can say $\left(O_{i} \perp O_{j}\right.$ C) for all $i \neq j$. Of course, if these conditional independence assumptions do not hold, then we can add the necessary directed edges between the observed features.

We have to specify the prior $P(C)$ and the class-conditional distributions $P\left(O_{i} \mid C\right)$. As done in the previous section, we can apply the chain rule to compute the joint distribution:
$$P\left(c, o_{1: n}\right)=P(c) \prod_{i=1}^{n} P\left(o_{i} \mid c\right)$$
Our classification task involves computing the conditional probability $P\left(c \mid o_{1: n}\right)$. From the definition of conditional probability, we have
$$P\left(c \mid o_{1: n}\right)=\frac{P\left(c, o_{1: n}\right)}{P\left(o_{1: n}\right)}$$

We can compute the denominator by marginalizing the joint distribution:
$$P\left(o_{1: n}\right)=\sum_{c} P\left(c, o_{1: n}\right)$$
The denominator in equation (3.5) is not a function of $C$ and can therefore be treated as a constant. Hence, we can write
$$P\left(c \mid o_{1: n}\right)=\kappa P\left(c, o_{1: n}\right)$$
where $\kappa$ is a normalization constant such that $\sum_{c} P\left(c \mid o_{1: n}\right)=1$. We often drop $\kappa$ and write
$$P\left(c \mid o_{1: n}\right) \propto P\left(c, o_{1 \Omega}\right)$$
where the proportional to symbol $\propto$ is used to represent that the left-hand side is proportional to the right-hand side. Example $3.4$ illustrates how inference can be applied to classifying radar tracks.

We can use this method to infer a distribution over classes, but for many applications, we have to commit to a particular class. It is common to classify according to the class with the highest posterior probability, $\arg \max {c} P\left(c \mid o{1: n}\right)$. However, choosing a class is really a decision problem that often should take into account the consequences of misclassification. For example, if we are interested in using our classifier to filter out targets that are not aircraft for the purpose of air traffic control, then we can afford to occasionally let a few birds and other clutter tracks through our filter. However, we would want to avoid filtering out any real aircraft because that could lead to a collision. In this case, we would probably only want to classify a track as a bird if the posterior probability were close to $1 .$ Decision problems will be discussed in chapter $6 .$

## 统计代写 | Statistical Learning and Decision Making代考|Sum-Product Variable Elimination

A variety of methods can be used to perform efficient inference in more complicated Bayesian networks. One method is known as sum-product variable elimination, which interleaves eliminating hidden variables (summations) with applications of the chain rule (products). It is more efficient to marginalize variables out as early as possible to avoid generating large factors.

We will illustrate the variable elimination algorithm by computing the distribution $P\left(B \mid d^{1}, c^{1}\right)$ for the Bayesian network in figure $3.1$. The conditional probability distributions associated with the nodes in the network can be represented by the following factors:
$$\phi_{1}(B), \phi_{2}(S), \phi_{3}(E, B, S), \phi_{4}(D, E), \phi_{5}(C, E)$$
Because $D$ and $C$ are observed variables, the last two factors can be replaced with $\phi_{6}(E)$ and $\phi_{7}(E)$ by setting the evidence $D=1$ and $C=1$.

We then proceed by eliminating the hidden variables in sequence. Different strategies can be used for choosing an ordering, but for this example, we arbitrarily choose the ordering $E$ and then $S$. To eliminate $E$, we take the product of all the factors involving $E$ and then marginalize out $E$ to get a new factor:
$$\phi_{8}(B, S)=\sum_{e} \phi_{3}(e, B, S) \phi_{6}(e) \phi_{7}(e)$$
We can now discard $\phi_{3}, \phi_{6}$, and $\phi_{7}$ because all the information we need from them is contained in $\phi_{8}$.

Next, we eliminate $S$. Again, we gather all remaining factors that involve $S$ and marginalize out $S$ from the product of these factors:
$$\phi_{9}(B)=\sum_{s} \phi_{2}(s) \phi_{8}(B, s)$$
We discard $\phi_{2}$ and $\phi_{8}$, and are left with $\phi_{1}(B)$ and $\phi_{9}(B)$. Finally, we take the product of these two factors and normalize the result to obtain a factor representing $P\left(B \mid d^{1}, c^{1}\right) .$
The above procedure is equivalent to computing the following:
$$P\left(B \mid d^{1}, c^{1}\right) \propto \phi_{1}(B) \sum_{s}\left(\phi_{2}(s) \sum_{e}\left(\phi_{3}\left(e \mid B_{t} s\right) \phi_{4}\left(d^{1} \mid e\right) \phi_{5}\left(c^{1} \mid e\right)\right)\right)$$
This produces the same result as, but is more efficient than, the naive procedure of taking the product of all of the factors and then marginalizing:
$$P\left(B \mid d^{1}, c^{1}\right) \propto \sum_{s} \sum_{e} \phi_{1}(B) \phi_{2}(s) \phi_{3}(e \mid B, s) \phi_{4}\left(d^{1} \mid e\right) \phi_{5}\left(c^{1} \mid e\right)$$

## 统计代写 | Statistical Learning and Decision Making代考|Inference in Bayesian Networks

• 我们使用因子乘积（算法 3.1）将两个因子组合起来产生一个更大的因子，其范围是输入因子的组合范围。如果我们有φ(X,是)和ψ(是,从)， 然后φ⋅ψ将会结束X,是， 和从和(φ⋅ψ)(X,是,和)= φ(X,是)ψ(是,和). 因子积在例子中演示3.1.
• 我们使用因子边缘化（算法 3.2）从整个因子表中总结出特定变量，将其从结果范围中删除。例子3.2说明了这个过程。
• 我们针对某些证据使用因子条件（算法 3.3）来删除表中与该证据不一致的任何行。例子3⋅3演示因子调节。

## 统计代写 | Statistical Learning and Decision Making代考|Sum-Product Variable Elimination

φ1(乙),φ2(小号),φ3(和,乙,小号),φ4(D,和),φ5(C,和)

φ8(乙,小号)=∑和φ3(和,乙,小号)φ6(和)φ7(和)

φ9(乙)=∑sφ2(s)φ8(乙,s)

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