- Dirichlet Process has been proposed to overcome these issues. The basic idea is to fix <math>s > 0</math> but do not choose any precise base measure <math>G_0</math>.
- measures. In other words, the IDP is the set of all Dirichlet Processes (with a fixed <math>s > 0</math>) obtained
- Since <math>F(x)=E[\mathbb{I}_{(\infty,x]}</math>, where <math>\mathbb{I}_{(\infty,x]}</math> is the [[indicator function]], we can use
- The lower and upper posterior mean of <math>F(x)</math are
- \underline{\mathcal{E}}\big[F(x)|X_1,\dots,X_n\big]=\underline{\mathcal{E}}\big[E(\mathbb{I}_{(\infty,x]})|X_1,\dots,X_n\big]&=& \frac{n}{s+n} \frac{\sum\limits_{i=1}^n \mathbb{I}_{(\infty,x]}(X_i)}{n}&=&\frac{n}{s+n}\hat{F}(x),\\
- \overline{\mathcal{E}}\big[F(x)|X_1,\dots,X_n\big]=\overline{\mathcal{E}}\big[E(\mathbb{I}_{(\infty,x]})|X_1,\dots,X_n\big]&=&\frac{s}{s+n}+ \frac{n}{s+n} \frac{\sum\limits_{i=1}^n \mathbb{I}_{(\infty,x]}(X_i)}{n}&=&\frac{s}{s+n}+ \frac{n}{s+n} \hat{F}(x).
- function]]. Here, to obtain the lower we have exploited the fact that <math>\inf \mathbb{I}_{(\infty,x]}=0</math> and for the upper that <math>\sup \mathbb{I}_{(\infty,x]}=1</math>.
- IDP can also be used for hypothesis testing, for instance to test the hypothesis <math>F(0)<0.5</math> , i.e., the median of <math>F</math> is greater than zero.
- By considering the partition <math>(-\infty,0],(0,\infty)</math> and the property of the Dirichlet process, it can be shown that
- F(0) \sim Beta(\alpha_0+n_{<0},\beta_0+n-n_{<0})
- where <math>n_{<0}</math> is the number of observations that are less than zero, <math>\alpha_0=s\int_{-\infty}^0
- \underline{\mathcal{P}}\big[F(0)<0.5|X_1,\dots,X_n\big]=\int\limits_{0}^{0.5} Beta(\theta;s+n_{<0},n-n_{<0})d\theta=I,\\
- \overline{\mathcal{P}}\big[F(0)<0.5|X_1,\dots,X_n\big]=\int\limits_{0}^{0.5} Beta(\theta;n_{<0},s+n-n_{<0})d\theta.\\
- # if both the inequalities are satisfied we can declare that <math>F(0)<0.5</math> with probability larger than <math>1-\gamma </math>;
- # if both are not satisfied, we can declare that the probability that <math>F(0)<0.5</math> is lower than the desired probability of <math>1-\gamma </math>.
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