Both the frequency school and the Bayesian school solve the problem of statistical inference.
The frequency school is also called the classic school. This school sees a fixed value for the probability of an event.
The Bayesian School sees the probability of an event as a random variable.
For example: In the description of coal storage in a coal mine, the classic school describes coal storage A = 10Kg. Bayes described coal storage A at about 10KG, and then based on historical data or other information, it was estimated that 2 <A <8, the probability is 80%, and the probability in other cases is 20%.
There are three main points of statistical inference: 1. Sampling analysis 2. Parameter estimation 3. Hypothesis testing. Three important sources of information: 1. Overall information 2. Sample information 3. Prior information
Both general information and sample information are used by both schools, and the Bayesian school adds the use of this information source of “prior information” .
Bayesian statistical case 1: Laplace studied in 1786 whether the birth rate of baby boys in Paris was greater than 0.5.
Step 1: Likelihood function
Suppose the event A of a baby is born, there are only two possibilities. Born and unborn, so it conforms to the binomial distribution X ~ b (n, p).
Therefore, its birth rate \ ({P \ left (X = x \ left | \ theta \ left) = C \ mathop {{}} \ nolimits _ {{n}} ^ {{x}} \ theta \ mathop {{}} \ nolimits ^ {{x}} \ left (1- \ theta \ left) \ mathop {{}} \ nolimits ^ {{nx}}, x = 0,1, ..., n \ right. \ right. \ right. \ right. \ right.} \)
This is the likelihood function, the probability of the occurrence of the A event.
If it is a strict likelihood function, it should be converted into a vector distribution.
\ ({P \ left (\ overline {X} \ left | \ theta \ left) = {\ mathop {\ prod} \ limits _ {{i = 1}} ^ {{n}} {P \ left (x = x \ mathop {{}} \ nolimits _ {{i}} \ left | \ theta \ right) \ right.}} \ right. \ right. \ right.} \)
Second step: prior conditions
if determined prior Conditions, there is a principle that can be used here. If you do n’t know anything about an event, you can first assume that the event is evenly distributed within a certain interval (that is, the Bayesian hypothesis principle: the principle of equal ignorance, the cognition is the same). According to this assumption, the prior distribution function of θ is obtained.
\({ \pi \left( \theta \left) { \left\{ \begin{array}{*{20}{l}} {1,0 \le \theta \le 1;}\\ {0,other;} \end{array}\right. }\right. \right. }\)
Step 3: Multiply the likelihood function and the prior distribution function to obtain the joint density function.
\ ({h \ left (x, \ theta \ left) = C \ mathop {{}} \ nolimits _ {{n}} ^ {{x}} \ theta \ mathop {{}} \ nolimits ^ {{x} } \ left (1- \ theta \
left) \ mathop {{}} \ nolimits ^ {{nx}}, 0 \ le \ theta \ le 1 \ right. \ right. \ right. \ right.} \) See It is the same as the likelihood function of each step, but its meaning is different.
Fourth point: find the m (x) marginal distribution
\ ({m \ left (x \ left) = {\ mathop {\ int} \ limits _ {{\ theta}} ^ {{}} {h \ left (x , \ theta \ left) d \ theta \ right. \ right.}} \ right. \ right.} \)
Fifth point: get the posterior distribution (Bayesian distribution)
\ ({\ pi \ left (\ theta \ left | x \ left) = \ frac {{h \ left (x, \ theta \ right)}} { {m \ left (x \ right)}} = \ frac {{P \ left (x \ left | \ theta \ left) \ pi \ left (\ theta \ right) \ right. \ right. \ right.}} {{{\ mathop {\ int} \ limits _ {{\ theta}} ^ {{}} {P \ left ({x \ left | \ theta \ right.} \ right)}} {\ pi \ left (\ theta \ right.} \ text {）} d \ theta}} \ right. \ right. \ right.} \)
Sixth point: According to the limited sample, it is substituted into the posterior distribution formula,
such as collecting data and discovering from 1745 to 1770 Born, baby boy 251527, baby girl 241945 baby girl.
\ ({P \ left (\ theta \ le 0.5 \ left | x \ left) = {\ mathop {\ int} \ limits _ {{0}} ^ {{0.5}} \ begin {array} {* {20} {l}} {\ pi \ left (dx \ left) d \ theta = 1.15 \ times 10 \ mathop {{}} \ nolimits ^ {{-42}} \ right. \ right.} \\ {} \ end {array}} \ right. \ right. \ right.} \)
The probability that θ is less than 0.5 is very small, so
\ ({\ theta> 0.5 \ text {YES} \ text {this} \ text {结} \ text {果} \ text {被} \ text {证} \ text {明}} \)