# 英国代考|  4YYD0003 Economic Analysis Of Emerging Economies新兴经济体的经济分析

Let us return to the inductive reasoning, starting from the known data, and adopting a Bayesian viewpoint. We can now use, in addition to sampling probabilities, probabilities that express our uncertainty about all possible values of the parameter. In the Bayesian inference, we consider, not the frequentist probabilities of imaginary samples but the frequentist probabilities of the observed data $\operatorname{Pr}(f=1 / 5 \mid \varphi)$ for all possible values of the parameter. This is the likelihood function that is denoted by
$$\ell(\varphi \mid \text { data }) .$$
We assume prior probabilities $\operatorname{Pr}(\varphi)$ before observations. Then, by a simple product, we get the joint probabilities of the parameter values and the data:
$$\operatorname{Pr}\left(\varphi \text { and } f=\frac{1}{5}\right)=\operatorname{Pr}\left(f=\frac{1}{5} \mid \varphi\right) \times \operatorname{Pr}(\varphi)=\ell(\varphi \mid \text { data }) \times \operatorname{Pr}(\varphi)$$

The sum of the joint probabilities gives the marginal predictive probability of the data, before observation:
$$\operatorname{Pr}\left(f=\frac{1}{5}\right)=\sum_{\varphi} \operatorname{Pr}\left(\varphi \text { and } f=\frac{1}{5}\right) .$$
The result is very intuitive since the predictive probability is a weighted average of the likelihood function, the weights being the prior probabilities.

Finally, we compute the posterior probabilities after observation, by application of the definition of conditional probabilities. The posterior distribution (given by Bayes’ theorem) is simply the normalized product of the prior and the likelihood:
$$\operatorname{Pr}\left(\varphi \mid f=\frac{1}{5}\right) \propto \ell(\varphi \mid \text { data }) \times \operatorname{Pr}(\varphi)=\frac{\operatorname{Pr}(\varphi \text { and } f=1 / 5)}{\operatorname{Pr}(f=1 / 5)}$$

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