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?@????̎???́A??͓I?Ɏ??㕪?z?̃p?????[?^?[?????߂邱?Ƃ??ł???̂ŁA?ȒP?ł??B?܂??A???̃v???O?????????Ă݂܂??傤?B?ڂ????v?Z???@?͏ȗ????܂????A???̗?̂悤?ɁA???O???z???x?[?^???z?ŁA?ޓx???????s??n?A??????k?̓??z?̌`?ɂȂ??Ă???ꍇ?́A???㕪?z?????ɓ?????p?????[?^?[???A???{k?ƂȂ?A???㕪?z?????ɓ?????p?????[?^?[???A???{n?|k?ƂȂ??̂ŁA
- ???O???z?F Beta(??, ??)
- ???㕪?z?F Beta(???{k, ???{n?|k) ?i2?j
?Ƃ????ɂ߂ĊȒP?Ȍ`?ɂȂ?܂??iBeta?̓x?[?^???z??\???܂??j?B?v????ɁA?i1?j???̉E?ӂ??v?Z????ƁA?i2?j???ŕ\????鍶?ӂ̎??㕪?zBeta(???{k, ???{n?|k)????????A?Ƃ??????Ƃł??B???Ȃ݂ɁA???O???z???i?ޓx???Ƒg?ݍ??킹???Ƃ??Ɂj???㕪?z?Ɠ????`?ɂȂ???̂????????O???z?i???傤?₭??????Ԃ?Ձj?ƌĂт܂??i???̏ꍇ?A??͓I?Ɏ??㕪?z?????߂??܂??j?B
?@?Ƃ???ŁA???z?̘b?Ȃ̂ɁA?Ȃ??x?[?^???z???o?ꂷ??̂??A?Ɖ??b?i??????j?Ɏv?????????????̂ł͂Ȃ????Ǝv???܂??B???̕ӂ???x?C?Y???v?̃n?[?h???ɂȂ??Ă???悤?ł????A?ȒP?Șb?ł??B???z?́A??????k???????Ɏ?????悤?Ȋm?????z?ł???ˁB???????A?????Œm?肽???͕̂ꐔ?ł??鐬???m?????̕??z?ł??B?܂?A?????????Ɏ?????悤?Ȋm?????z?????߂????킯?ł??B?????m?????́A?????ʂ?u?m???v?Ȃ̂ŁA0?`1?̒l?????͂??ł??B?????????ϐ??ƂȂ??Ă??āA0?`1?͈̔͂̂??ꂼ??̒l???u?ǂ̒??x?ނ??炵?????v??\?????z?????߂????Ƃ??????Ƃł??B???łɏq?ׂ??悤?ɁA?x?[?^???z?͑?i?ϐ??̎?蓾??l?͈̔́j??0?`1?ŁA?p?????[?^?[??,???̎w??ɂ??A???܂??܂Ȍ`?̕??z???\????̂ŁA???̂悤?ȁu?m?????ǂ????z???邩?v??\???̂ɓK???Ă???Ƃ????킯?ł??B???[?ށA?悭???????Ƃ??????́A?Ƃ肠?????A???z?̕ꐔ???𐄒肷??ɂ̓x?[?^???z???g???Ɠs?????????A?Ɗo???Ă????Ă?????Ă??????????ł??B
?@?x?[?^???z?͘A???^?̊??Ȃ̂ŁA?Ⴆ?A???????傤??0.5?ł???m???????߂???킯?ł͂???܂???B?m?????x???̒l?́A???̒l????????m?????̂??̂ł͂Ȃ??A???̒l?̋߂??̒l???ǂ̒??x?????₷??????\?????̂ł????ˁB?Ȃ??A?ݐϕ??z???̒l?????߂?A?Ⴆ???? ≤ 0.5?Ȃǂ́A????͈͂ɓ???m???͓????܂??B
?@?ȉ??̃R?[?h?́A???O???z??????1,????1?̃x?[?^???z?i??l???z?Ɠ????`?ɂȂ?j?Ƃ??A30?l??23?l???????b?N?X?ł????Ƃ????ꍇ?i???s??n??30?A??????k??23?j?̗?ł??B???̏ꍇ?A?i2?j???ɓ??Ă͂߂?ƁA???㕪?z?̃p?????[?^?[??????1?{23, ????1?{30?|23?A?܂?A????24, ????8?ƌv?Z????܂??B?????̒l????Ɏ??O???z?Ǝ??㕪?z?????????A???ς̓_????l??95???M?p????????߂Ă??܂??B
?@?T???v???v???O?????????????ɂ???܂??B?????N???J????Python?̃v???O???????\???????̂ŁA?ŏ??̃R?[?h?Z?????N???b?N???A?mShift?n?{?mEnter?n?L?[???????ăv???O?????????s????ƁA???ʂ??\??????܂??B?R?[?h?̏ڍׂɂ??ẮA???X?g???̃R?????g?≺?̃L???v?V???????Q?Ƃ??Ă????????B
import numpy as np
from scipy.stats import beta
import matplotlib.pyplot as plt
# ?ϑ????ꂽ?f?[?^
n = 30 # ???s??
k = 23 # ??????
# ???O???z?iprior?j?̃p?????[?^?[
alpha_prior = 1
beta_prior = 1
# ???㕪?z?ipost?j?̃p?????[?^?[
alpha_post = alpha_prior + k
beta_post = beta_prior + n - k
# ???O???z?Ǝ??㕪?z?̊m?????x???????߂?
theta = np.linspace(0, 1, 500)
pdf_prior = beta.pdf(theta, alpha_prior, beta_prior) # ???O???z
pdf_post = beta.pdf(theta, alpha_post, beta_post) # ???㕪?z
# ???O???z?Ǝ??㕪?z??????????
plt.figure(figsize=(10, 6))
plt.plot(theta, pdf_prior, label=f"prior:a={alpha_prior},b={beta_prior}")
plt.plot(theta, pdf_post, label=f"posterior:a={alpha_post},b={beta_post}")
plt.xlabel("theta")
plt.ylabel("Density")
plt.legend()
plt.show()
???̎???ł́A???㕪?z?̃p?????[?^?[alpha_post??beta_post???ȒP?ɋ??߂???̂ŁA???????ɃO???t??`???悢?Bscipy.stats???W???[???Ɋ܂܂??beta.pdf?̓x?[?^???̊m?????x???̒l?????߂???Bnumpy???W???[????linspace?????g????theta?̒l??0?`1?͈̔͂?500???ɍ??݁A?m?????x???̒l?????ꂼ??v???b?g????B
?@???s???ʂ͐}2?̂悤?ɂȂ?܂??B
?}2?@???O???z?Ǝ??㕪?z?̃O???t???X?g1?????s???????ʁB???O???z??prior?Ə????ꂽ?u???[?̃O???t?i??l???z?j?A???㕪?z??posterior?Ə????ꂽ?I?????W?̃O???t?B0.767?ӂ?ɎR?̂??镪?z?ƂȂ??Ă???B?f?[?^???ϑ????邱?Ƃɂ??A???O???z?i????1,????1?̃x?[?^???z????l???z?j???A???㕪?z?i????24,????8?̃x?[?^???z?j?ɍX?V???ꂽ?B
?@????ɁA???ρi???Ғl?j??_????Ƌ?Ԑ??肵?Ă݂܂??傤?B?x?[?^???z?̕??ρi???Ғl?j???? / (?? ?{ ??)?Ƃ????????ŋ??߂??܂??B?M?p??Ԃɂ??ẮA?x?[?^???z?̗ݐϕ??z???̋t?????g???ċ??߂܂??B?R?[?h?͈ȉ??̒ʂ?ł??B
# ???ς̓_????Ƌ?Ԑ???i??̃R?[?h?????s??????Ŏ??s???邱?Ɓj
# ?_????i???㕪?z?̕??ρj?F???? alpha / (alpha + beta)?ɂ??
post_mean = alpha_post / (alpha_post + beta_post)
# ??Ԑ???i95???M?p??ԁj
ci_low, ci_high = beta.ppf([0.025, 0.975], alpha_post, beta_post)
# ???ʂ̕\??
print(f"????:{post_mean:.3f}")
print(f"95%?M?p???: [{ci_low:.3f}, {ci_high:.3f}]")
# theta <= 0.5?ł???m???????߂???
prob_le05 = beta.cdf(0.5, alpha_post, beta_post)
print(f"theta <= 0.5?ł??鎖??m??: {prob_le05:.4f}")
# ?o?͗?F
# ????:0.7500
# 95%?M?p???: [0.5890, 0.8814]
# theta <= 0.5?ł???m??: 0.0017
?x?[?^???z?̕??ρi???Ғl?j?͌????ŋ??߂???̂ŊȒP?B95???M?p??Ԃ͗ݐϕ??z???̒l??0.025?ƂȂ???????0.975?ƂȂ????܂ł͈̔́Bscipy.stats???W???[???Ɋ܂܂??beta.ppf?????g???ăx?[?^???̗ݐϕ??z???̋t???̒l?????߂?悢?B????ɁA?ݐϕ??z???̒l?????߂?beta.cdf?????g????????0.5?ȉ??ł???m???????߂Ă݂??B
?@?O??̂????炢?ɂȂ?܂????A???̌??ʂŏd?v?Ȃ̂́A?M?p????͕ꕽ?ς?95???̊m???ł??͈̔͂ɓ????Ă??邱?Ƃ?\???Ă???A?Ƃ??????Ƃł??B
?@?ÓT?I?Ȏ?@?ł̐M????Ԃ́A?????ł͂???܂???ł????B?u?M????Ԃ????߂邱?Ƃ???????J??Ԃ??ƁA95???̉?ŕꕽ?ς????̐M????ԂɊ܂܂??v?Ƃ??????Ƃł????ˁB?Ȃ?Ƃ??????Ƀ??m?̂͂??܂????悤?ȕ\?????????Ȃ??Ƃ????Ȃ????????Ǝv???܂??B?ÓT?I?Ȏ?@?ł́A?ꐔ?͒萔?ŁA?M????Ԃ????????̂ƍl?????邩??ł??B
?@????A?x?C?Y???v?ł́A?ꐔ???m???ϐ??Ƃ??ă??f???????Ă???A?f?[?^???ϑ????ꂽ??ł͐M?p??Ԃ͓????Ȃ??̂ŁA?f???Ɂu95???̊m???ŐM?p??Ԃɓ????Ă???v?ƌ??????Ƃ??ł??܂??B?]???āA???X?g2?̍Ō?̃R?[?h?̂悤?ɁA?? ≤ 0.5?ł???m???Ȃǂ????߂???킯?ł??B???????A0.5?Ɣ?r???邱?ƂɈӖ??????邩?ǂ????́A?^?X?N?ɂ???ĈقȂ?܂??B?Ⴆ?A????܂ł̌?????o??????A70???i??0.7?j?ȏ?̐l???u?????b?N?X?ł????v?Ɠ????Ȃ???Ό??ʂ?????Ƃ݂Ȃ??Ȃ??A?Ƃ???Ȃ?A?? ≥ 0.7?ł???m???????߂?ׂ??ł??B
?^?b?`???O?????₵?ɖ𗧂??ǂ??????x?C?Y????Œ??ׂ悤
?@?????ɂ͌???Ƃ͈قȂ?l?????ł????A?ÓT?I?Ȍ???ɑ???ȒP?ȕ??@?Ƃ??Ď????I?ɂ悭?g???Ă?????̂ɁAROPE?iRegion Of Practical Equivalence?j?ƌĂ????@??????܂??B?Ⴆ?A????0.5?ƍ??????邩?ǂ????ׂ?ꍇ?A??????x?????????????͈́A?Ⴆ?A0.45 ≤ ?? ≤ 0.55?????S?ɐM?p??ԂɊ܂܂?Ă??邩?i???Ȃ????j?ׂ܂??B???̗?ł???A[0.45, 0.55]?́A95%?M?p??Ԃ?[0.5890, 0.8814]?̊O?ɂ???̂ŁA????????Ƃ݂Ȃ??܂??B
?@?܂??A?ÓT?I?Ȍ???ɑΉ???????̂Ƃ??ẮA?x?C?Y???q?i?x?C?Y?t?@?N?^?[?j?ɂ?郂?f????r??????܂??B?x?C?Y???q??2?̉????i???f???j?̎??Ӗޓx?̔?ŋ??߂???̂ł????A?O???G?ꂽSavage-Dickey?i?T?x?[?W?E?f?B?b?L?[?j?@?ł́A????_?i?Ⴆ?A0.5?j?ɂ????鎖?O???z?Ǝ??㕪?z?̊m?????x???̒l?̔?ɂ???ĊȒP?ɋ??߂邱?Ƃ??ł??܂??B?????ł́A?ߋ??̌o??????A?^?b?`???O???Ȃ??ꍇ?ɂ̓????b?N?X?ł???m????1/2??0.5?ł??邱?Ƃ????????Ă??????̂Ƃ??܂??B???̏ꍇ?A?^?b?`???O?????ꍇ?̌??ʂ??ᖡ???邽?߂̋A???????ƑΗ??????͈ȉ??̂悤?ɂȂ?܂??B
- ?A??????H0?F ?? ?? 0.5
- ?Η?????H1?F ?? ?? 0.5
?@???ۂ̎????v??ł́A?????Q?Ƃ??āu?^?b?`???O?????Ȃ??Q?v??݂??āA?ǂ???̌Q?????₳?ꂽ?i?????b?N?X?ł????j?䗦???傫???????????r????̂???ʓI?ł??B???????A?????ł̎???͂????܂Ńx?C?Y????̍l???????Љ?邽?߂̂??́A?Ƃ??????Ƃł??̂܂ܐi?߂܂??傤?B???̗?ł́A???O???z?????㕪?z???x?[?^???z?Ȃ̂ŁA?m?????x???̒l?͊ȒP?ɋ??߂??܂??B?ȉ??̂悤?ȃv???O?????ɂȂ?܂??B
# ?x?C?Y???qBF10?????߂?i?ŏ??̃R?[?h?????s??????Ŏ??s???邱?Ɓj
prior_density = beta.pdf(0.5, alpha_prior, beta_prior)
posterior_density = beta.pdf(0.5, alpha_post, beta_post)
# Savage-Dickey?@?ɂ??x?C?Y???q?F??=0.5?ɑ???m?????x???̔?????߂?i?M?p??Ԃ̃R?[?h?ɂ??̃R?[?h??lj??????j
BF10 = prior_density / posterior_density
print(f"?x?C?Y???qBF10: {BF10:.3f}")
# ?o?͗?F
# ?x?C?Y???qBF10: 17.014
???O???z?Ǝ??㕪?z?̂??ꂼ??ɂ??āAscipy.stats???W???[???Ɋ܂܂??beta.pdf?????g???ăx?[?^???̊m?????x???̒l?????߁A???????g???āu??????O?v?̒l?????߂?Ƃ悢?B???ʂ?BF10??17.014?ƂȂ?B???̂??Ƃ́A?Η?????H1???A?A??????H0????17?{?x??????邱?Ƃ??Ӗ?????B
?@?x?C?Y???qBF10?́A?Η?????H1???A?A??????H0?̉??{?x??????邩?Ƃ????w?W?ł??B?t?ɁA?x?C?Y???qBF01?́A?A??????H0???A?Η?????H1?̉??{?x??????邩?Ƃ????w?W?ɂȂ?܂??BBF10?́A????_?ɂ????鎖?O???z?̊m?????x???̒l?????㕪?z?̊m?????x???̒l?ŋ??߂??ABF01?͂??̋t???ƂȂ??Ă??܂??B?x?C?Y???q?̒l???ǂ̂悤?ɕ]?????邩?ɂ??ẮA?ȉ??̂悤??Jeffreys?i1961?j?̕]???ړx???ڈ??Ƃ??Ă悭?g???܂??B
| ?x?C?Y???q | ?]?? |
|---|---|
| 1?`3.2 | ?قƂ?nj??y???鉿?l?͂Ȃ? |
| 3.2?`10 | ?????I?ȏ؋??ƂȂ??Ă??? |
| 10?`32 | ?????؋??ƂȂ??Ă??? |
| 32?`100 | ???ɋ????؋??ƂȂ??Ă??? |
| ?? 100 | ????I?ȏ؋??ƂȂ??Ă??? |
Jeffreys?i1961?j?ɂ??x?C?Y???q?̕]???ړx?B??̗?ł???ABF10??17.014?Ȃ̂ŁA?Η?????H1???x??????u?????؋??v?ƂȂ??Ă???B???????A???̕]???ړx?͂????܂Ŗڈ??ł??邱?Ƃɒ??ӁB?????ɂ???ẮA3.2??32?̑???ɃL???̂???3??30?Ƃ??Čf?ڂ??Ă???ꍇ??????B
?@???̗?ł́ABF10?̒l??17.014?ƂȂ?AJeffreys?̕]???ړx?ɂ??A?Η?????H1???????x??????邱?Ƃ???????܂????B?܂?A?^?b?`???O?ɂ????₵???ʂ???????m????0.5?ł͂Ȃ??Ƃ????܂??i???̗?́A?????闼??????ɑΉ????錋?ʂł????A?????I?Ɂu0.5?????傫???v?ƌ????Ă????ł??傤?j?B
?@???Ȃ݂ɁAH0???Ɓ?0.7?Ƃ??Čv?Z???Ă݂?ƁABF10??0.265?ƂȂ?A???̋t???????߂??BF01??3.777?ƂȂ?܂??B?܂?AH0?ɑ???u?????I?ȏ؋??ƂȂ??Ă???v?Ƃ????킯?ł??B???̂??Ƃ́A?Ɓ?0.7?i???₵???ʂ?????m????0.7?ł???j?Ƃ??????Ƃ??????I?Ɏx?????Ă???A?Ƃ??????ƂɂȂ?܂??B???̂悤?ɁA?A?????????x??????؋??̋?????]???ł???̂??A?x?C?Y?I?Ȏ?@?̗??_?ł??B
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?@?Ƃ???ŁA????̗?Ȃ?Excel?ł??ł??邶??Ȃ????Ǝv????????????Ǝv???܂??B??ł??G?ꂽ?悤?ɁA???㕪?z?̃p?????[?^?[????͓I?Ɍv?Z?ł???ꍇ?́A?????Ă??̏ꍇExcel?ł??v?Z?ł??܂??B???????A???Ӗޓx?̌v?Z??????ꍇ?ɂ́A?????????킯?ɂ͂????܂???B???̏ꍇ?́A?V?~?????[?V???????s???Ď??㕪?z?????߂܂??B??̗?ł́A?킴?킴?V?~?????[?V???????s???K?v?͂Ȃ??̂ŁA????͏Љ?܂??A?????Ă???Ă݂??ꍇ?̃v???O????????T???v???t?@?C????4?Ԗڂ̃R?[?h?Z???Ɋ܂߂Ă????܂????B?????A?????_?ŃR?[?h?????Ă??A?킯????????Ȃ????????Ǝv????̂ŁA???ɉ???͂??Ȃ????Ƃɂ??܂??B?T???v???t?@?C?????Q?Ƃ??Ă????????u????ȍ~?A?A?ڂ?ǂݐi?߂?A???????????R?[?h???????ł??A?????ŏ?????悤?ɂȂ?v?Ǝv???Ă?????????悢???Ǝv???܂??B
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?@????́A?ꕽ?ς̃x?C?Y????^?x?C?Y????ł??B???????ꂽ???َq?i??̓I?ɂ̓V???[?N???[???j?̕ꕽ?ς𐄒肵?A?ꕽ?ς?100?O?????ł??邩?ǂ????????肷??Ƃ????????p?I?Ȃ??b?????܂??B?ǂ????A???y???݂ɁI
????g??????Ȋ?
?@????????Ȃ??̂??????f?ڂ??܂??B?????Ɂu=?l?v?Ə????ꂽ???̂͊???l?i??F??True?j??\???܂??B
linspace?F ??????????쐬????
- ???W???[???F numpy
- ?`???F linspace(start, stop, num=50, endpoint=True)
- ?????F
- start?F ?????̒l?B???X?g??z????w?肷??ƕ????̓??????쐬?ł???
- stop?F ?????̒l?B???X?g??z????w?肷??ƕ????̓??????쐬?ł???
- num?F start????stop?܂ł???ɋ?邩???w?肷??
- endpoint?F ???????܂ޏꍇ??True?A?܂܂Ȃ??ꍇ??False
beta.pdf?F ?x?[?^???z?̊m?????x???̒l?????߂?
- ???W???[???F scipy.stats
- ?`???F beta.pdf(x, a, b)
- ?????F
- x?F ?m???ϐ??̒l?B???X?g??z????w?肷??ƕ????̒l?????߂???
- a?F ???z?̌`???\???p?????[?^?[?i???z?̉E???s1?ɋ߂????t?ɏd?݂??????j
- b?F ???z?̌`???\???p?????[?^?[?i???z?̍????s0?ɋ߂????t?ɏd?݂??????j
beta.cdf?F ?x?[?^???z?̗ݐϕ??z???̒l?????߂?
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?Q?l????
?ߓ??_?q, ???{?_??, ?Y???I?q. (2011). ?????Z?@?Ƃ??Ẵ^?b?`?̎҂Ǝ{?s?҂ɂ???????ʂɊւ??錤??. ??????Õی???w?I?v, 7(1), 1-10.
Jeffreys, H. (1961). Theory of probability(3 ed.). Oxford: Oxford University Press.
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