fundamental statistics learning note(22)

Rcall posterior is a compromise between prior and likelihood
i.e. Bayesian analysis provides framework to update posterior based on data/evidence (e.x. college football rankings)

Bayesian Hypothesis Testing

Consider $H_O: theta in Omega_0 text{ vs } theta in Omega_1$
Since $theta$ is a r.v under Bayesian inference:

• the prior probability that $H_O$ is true is $P(thetain Omega_O)$, calculated using $pi(theta)$.
• the posteriro probability that $H_O$ is true is $P(thetain Omega_O|x_1,dots,x_n)$, calculated using $pi(theta|x_1,dots,x_n)$.

Definition: The Bayes factor in favour of $H_O$ is determined by $$underbrace{frac{P(thetainOmega_0|x_1,dots,x_n)}{P(thetainOmega_1|x_1,dots,x_n)}}_{text{posterior odds that }H_Otext{ is true}}=underbrace{frac{P(thetain Omega_0)}{P(thetainOmega_1)}}_{text{prior odds that }H_Otext{ is true}}times text{Bayes Factor}$$

Intepretation:
$$begin{matrix}
text{B.F }geq 1 & text{data support }H_O \
text{B.F }< 1 & text{data support }H_A
end{matrix}$$
Ex. $x_1,dots,x_n overset{text{iid}}sim Bern(p)$. $H_O: pleq 0.5text{ vs }H_A: p>0.5$. Suppose prior on $p$ is $psim Beta(1,2)$.
(a) prior probability $H_O$ is true?
$$begin{equation}begin{split}
P(pleq 0.5)=int_0^{0.5}pi(p)dp &=int_0^{0.5}frac{Gamma(3)}{Gamma(1)Gamma(2)}p^{1-1}(1-p)^{2-1}dp \
&=int_0^{0.5}2(1-p)dp \
&=2p-p^2|_0^{0.5} \
&=1-0.5^2 \
&=0.75
end{split}end{equation}$$

(b) Observe 5 successes out of 5, what is the posterior probability $H_O$ is true?
Given $pi(theta|x_1,dots,x_n) sim Beta(1+5,2+0)$ from last chapter.
$$begin{equation}begin{split}
P(pleq 0.5) &= int_0^{0.5}frac{Gamma(8)}{Gamma(6)Gamma(2)}p^{6-1}(1-p)^{2-1}dp \
&=int_0^{0.5}42p^5(1-p)^1dp \
&=42[frac{p^6}{6}-frac{p^7}{7}]|_0^{0.5} \
&=0.0625
end{split}end{equation}$$

(c) prior odds $H_O$ is true?
$frac{0.75}{1-0.75}=3$

(d) posterior odds $H_O$ is true?
$frac{0.0625}{1-0.0625}=0.0667$

(e) Bayes factor
$0.0667=3times B.F$
$Rightarrow B.F = 0.0222 ll 1$
Data strongly supports $H_A: p>0.5$