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EVIDENCE AND CHANGING BELIEFS

Bayesian statisticians believe that Bayesian inference uses aspects of the Scientific Method , which involves collecting Evidence that is meant to be consistent or inconsistent with a given Hypothesis . As evidence accumulates, the degree of belief in a hypothesis changes. With enough evidence, it will often become very high or very low. Bayesian statisticians also believe that Bayesian inference is a suitable logical basis to discriminate between conflicting hypotheses. Hypotheses with a very high degree of belief should be accepted as true; those with a very low degree of belief should be rejected as false.

:An example of Bayesian inference is
::''For billions of years, the sun has risen after it has set. The sun has set tonight. With very high probability (or I strongly believe that or it is true that) the sun will rise tomorrow. With very low probability (or I do not at all believe that or it is false that) the sun will not rise tomorrow.''
Bayesian inference uses a numerical estimate of the degree of belief in a hypothesis before evidence has been observed and calculates a numerical estimate of the degree of belief in the hypothesis after evidence has been observed. Bayesian inference usually relies on degrees of belief, or subjective probabilities, in the induction process and does not necessarily claim to provide an objective method of induction. Nonetheless, some Bayesian statisticians believe probabilities can have an objective value and therefore Bayesian inference can provide an objective method of induction. See Scientific Method .

Bayes' theorem adjusts probabilities given new evidence in the following way:
  <math>P(EH 0)</math> Is Called The '' "http://wwwinformationdelightinfo/encyclopedia/entry/conditional_probability" class="copylinks">Conditional Probability '' of seeing the evidence <math>E</math> given that the hypothesis <math>H_0</math> is true It is also called the '' Likelihood Function '' when it is expressed as a function of <math>H_0</math> given <math>E</math>
  <math>P(E)</math> Is Called The '' "http://wwwinformationdelightinfo/encyclopedia/entry/marginal_probability" class="copylinks">Marginal Probability '' of <math>E</math>: the probability of witnessing the new evidence <math>E</math> under all mutually exclusive hypotheses It can be calculated as the sum of the product of all probabilities of mutually exclusive hypotheses and corresponding conditional probabilities: <math>\sum P(EH_i)P(H_i)</math>
  <math>P(H 0E)</math> Is Called The '' "http://wwwinformationdelightinfo/encyclopedia/entry/posterior_probability" class="copylinks">Posterior Probability '' of <math>H_0</math> given <math>E</math>
  The Probability Of <math>E</math> Given <math>H 0</math>, <math>P(EH 0)</math>, Can Be Represented As A Function Of Its Second Argument With Its First Argument Held At A Given Value Such A Function Is Called A "http://wwwinformationdelightinfo/encyclopedia/entry/likelihood_function" class="copylinks">Likelihood Function it is a function of <math>H_0</math> given <math>E</math> A ratio of two likelihood functions is called a likelihood ratio, <math>\Lambda </math> For example,
  :<math>\Lambda rac{L(H_0E)}{L(\mbox{not } H_0E)} = rac{P(EH_0)}{P(E\mbox{not } H_0)} </math>
  :<math>P(H 0E) rac{P(EH_0)P(H_0)}{P(EH_0)P(H_0)+ P(E\mbox{not }H_0)P(\mbox{not }H_0)} = rac{\Lambda P(H_0)}{\Lambda P(H_0) + P(\mbox{not } H_0)}</math>
  :<math>P(E 1, E 2 H 0) P(E_1 H_0) imes P(E_2 H_0)</math>
  :<math>P(E 1,E 2\mbox{not }H 0) P(E_1\mbox{not }H_0) imes P(E_2\mbox{not }H_0)</math>
  :<math>P(H 0E 1, E 2) rac{P(E_1H_0) imes P(E_2H_0)\P(H_0)}{P(E_1) imes P(E_2)}</math>
  :<math>P(H 0E 1, E 2) rac{\Lambda_1 \Lambda_2 P(H_0)}{\Lambda_1 \Lambda_2 P(H_0) + P(\mbox{not } H_0)} </math>,
  The Datum ''D'' Is The Observation Of A Plain Cookie From The Contents Of The Bowls, We Know That P(''D'' ''H''<sub>1</sub>) 30/40 = 075 and P(''D'' ''H''<sub>2</sub>) = 20/40 = 05 Bayes' formula then yields
  \begin{matrix} P(H 1 D) & & rac{P(H_1) \cdot P(D H_1)}{P(H_1) \cdot P(D H_1) + P(H_2) \cdot P(D H_2)} \ \ \ & =& rac{05 imes 075}{05 imes 075 + 05 imes 05} \ \ \ & =& 06 \end{matrix}
  :<math>\begin{matrix} P(A B) & & rac{P(B A) P(A)}{P(B A)P(A) + P(B \mbox{not } A)P(\mbox{not }A)} \ \
  P(AB) & & rac{099 imes 0001}{099 imes 0001 + 005 imes 0999}\, ,\ ~\ &\approx &0019\, \end{matrix}</math>
  :<math>P(AB) rac{099 imes 0001}{099 imes 0001 + 0001 imes 0999} \approx 05 </math>,
  :<math>\begin{matrix} P(A \mbox{not } B) & & rac{P(\mbox{not }B A) P(A)}{P(\mbox{not }B A)P(A) + P(\mbox{not }B \mbox{not } A)P(\mbox{not }A)} \ \
  P(A\mbox{not }B) & & rac{001 imes 0001}{001 imes 0001 + 095 imes 0999}\, ,\ ~\ &\approx &00000105\, \end{matrix}</math>
  :<math>\begin{matrix} P(A \mbox{not } B) & & rac{P(\mbox{not }B A) P(A)}{P(\mbox{not }B A)P(A) + P(\mbox{not }B \mbox{not } A)P(\mbox{not }A)} \ \
  P(A\mbox{not }B) & & rac{001 imes 06}{001 imes 06 + 095 imes 04}\, ,\ ~\ &\approx &00155\, \end{matrix}</math>
  :p(G E) p(G) p(E G) / p(E)
  :p(G E) (03 &times 10) /(03 &times 10 + 07 &times 10<sup>-6</sup>) = 099999766667
  : <math> P(m,na) \begin{pmatrix} n+m \ m \end{pmatrix} a^m (1-a)^n </math>
  : <math> P(am,n) rac{p(m,na)\,p(a)}{\int_0^1 p(m,na)\,p(a)\,da}