# 统计代写|贝叶斯分析代写Bayesian Analysis代考|Prediction

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## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Prediction

To make inferences about an unknown observable, often called predictive inferences, we follow a similar logic. Before the data $y$ are considered, the distribution of the unknown but observable $y$ is
$$p(y)=\int p(y, \theta) d \theta=\int p(\theta) p(y \mid \theta) d \theta$$
This is often called the marginal distribution of $y$, but a more informative name is the prior predictive distribution: prior because it is not conditional on a previous observation of the process, and predictive because it is the distribution for a quantity that is observable.
After the data $y$ have been observed, we can predict an unknown observable, $\tilde{y}$, from the same process. For example, $y=\left(y_1, \ldots, y_n\right)$ may be the vector of recorded weights of an object weighed $n$ times on a scale, $\theta=\left(\mu, \sigma^2\right)$ may be the unknown true weight of the object and the measurement variance of the scale, and $\tilde{y}$ may be the yet to be recorded weight of the object in a planned new weighing. The distribution of $\tilde{y}$ is called the posterior predictive distribution, posterior because it is conditional on the observed $y$ and predictive because it is a prediction for an observable $\tilde{y}$ :
\begin{aligned} p(\tilde{y} \mid y) & =\int p(\tilde{y}, \theta \mid y) d \theta \ & =\int p(\tilde{y} \mid \theta, y) p(\theta \mid y) d \theta \ & =\int p(\tilde{y} \mid \theta) p(\theta \mid y) d \theta . \end{aligned}
The second and third lines display the posterior predictive distribution as an average of conditional predictions over the posterior distribution of $\theta$. The last step follows from the assumed conditional independence of $y$ and $\tilde{y}$ given $\theta$.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Likelihood

Using Bayes’ rule with a chosen probability model means that the data $y$ affect the posterior inference (1.2) only through $p(y \mid \theta)$, which, when regarded as a function of $\theta$, for fixed $y$, is called the likelihood function. In this way Bayesian inference obeys what is sometimes called the likelihood principle, which states that for a given sample of data, any two probability models $p(y \mid \theta)$ that have the same likelihood function yield the same inference for $\theta$.

The likelihood principle is reasonable, but only within the framework of the model or family of models adopted for a particular analysis. In practice, one can rarely be confident that the chosen model is correct. We shall see in Chapter 6 that sampling distributions (imagining repeated realizations of our data) can play an important role in checking model assumptions. In fact, our view of an applied Bayesian statistician is one who is willing to apply Bayes’ rule under a variety of possible models.
Likelihood and odds ratios
The ratio of the posterior density $p(\theta \mid y)$ evaluated at the points $\theta_1$ and $\theta_2$ under a given model is called the posterior odds for $\theta_1$ compared to $\theta_2$. The most familiar application of this concept is with discrete parameters, with $\theta_2$ taken to be the complement of $\theta_1$. Odds provide an alternative representation of probabilities and have the attractive property that Bayes’ rule takes a particularly simple form when expressed in terms of them:
$$\frac{p\left(\theta_1 \mid y\right)}{p\left(\theta_2 \mid y\right)}=\frac{p\left(\theta_1\right) p\left(y \mid \theta_1\right) / p(y)}{p\left(\theta_2\right) p\left(y \mid \theta_2\right) / p(y)}=\frac{p\left(\theta_1\right)}{p\left(\theta_2\right)} \frac{p\left(y \mid \theta_1\right)}{p\left(y \mid \theta_2\right)}$$
In words, the posterior odds are equal to the prior odds multiplied by the likelihood ratio, $p\left(y \mid \theta_1\right) / p\left(y \mid \theta_2\right)$.

# 贝叶斯分析代考

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Prediction

$$p(y)=\int p(y, \theta) d \theta=\int p(\theta) p(y \mid \theta) d \theta$$

\begin{aligned} p(\tilde{y} \mid y) & =\int p(\tilde{y}, \theta \mid y) d \theta \ & =\int p(\tilde{y} \mid \theta, y) p(\theta \mid y) d \theta \ & =\int p(\tilde{y} \mid \theta) p(\theta \mid y) d \theta . \end{aligned}

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Likelihood

$$\frac{p\left(\theta_1 \mid y\right)}{p\left(\theta_2 \mid y\right)}=\frac{p\left(\theta_1\right) p\left(y \mid \theta_1\right) / p(y)}{p\left(\theta_2\right) p\left(y \mid \theta_2\right) / p(y)}=\frac{p\left(\theta_1\right)}{p\left(\theta_2\right)} \frac{p\left(y \mid \theta_1\right)}{p\left(y \mid \theta_2\right)}$$

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