Amathuba e-Axiomatic: Ukuqonda Izisekelo Zezibalo

Informatec Digital » Izihloko » Amathuba e-Axiomatic: Ukuqonda Izisekelo Zezibalo

Amathuba e-Axiomatic: Incazelo Nomongo

I-Axiomatic probability , eyaziwa nangokuthi i-measure theory , iyindlela yezibalo esebenzisa isethi yama-axioms ukuchaza nokufunda amathuba emicimbi. Leli gatsha lavela ekuqaleni kwekhulu lama-20, ngenxa yomsebenzi wezazi zezibalo ezifana no-Andrey Kolmogorov no-Émile Borel, ababeka isisekelo se-theory ehambisanayo neqinile yamathuba.

Kulo mongo, amathuba aqondwa njengesilinganiso esijwayelekile sethuba lomcimbi othize owenzeka phakathi kwesampula yesikhala. I-axioms of probability axiomatic iqinisekisa ukuthi lesi silinganiso senelisa izici ezithile ezibalulekile, njengokungasho lutho, okungeziwe, kanye nokujwayelekile.

I-Axiom 1: Akukho Okubi

I-axiom yokuqala yamathuba ithi amathuba anoma yimuphi umcimbi ( A ), oshiwo ngokuthi ( P(A) ), kumelwe ahlale makhulu noma alingane noziro. Ngamanye amazwi, awekho amathuba amabi. Le axiom ivezwa ngokwezibalo kanje:

$$ P(A) \geq 0 $$

Lesi simiso siyaqondakala, njengoba kungenangqondo ukukhuluma ngamathuba angemahle emhlabeni wangempela. Isibonelo, asikwazi ukusho ukuthi amathuba okuthola amakhanda lapho upheqa uhlamvu lwemali ngu (-0.5).

I-Axiom 2: Amathuba Wesikhala Sesampula

I-axiom yesibili ithi amathuba aso sonke isikhala sesampula, esichazwa ngokuthi ( \Omega), ahlala alingana no-1. Isikhala sesampula simele yonke imiphumela engaba khona yokuhlolwa okungahleliwe. Ngokwezibalo, le axiom ivezwa kanje:

$$ P(\Omega) = 1 $$

Le-axiom isho ukuthi isamba samathuba azo zonke izehlakalo ezingaba khona esikhaleni sesampula kufanele silingane no-1. Isibonelo, uma ugoqa idayi efanelekayo enezinhlangothi eziyisithupha, isamba samathuba okuthola inombolo ngayinye (1, 2, 3, 4, 5, kanye no-6) ilingana no-1.

I-axiom 3: Ukwengeza

I-axiom yesithathu yamathuba e-axiomatic ithi, kunoma yikuphi ukulandelana kwemicimbi ekhethekile (( A_1, A_2, \lddots, A_n )), amathuba okuhlangana kwalezi zenzakalo alingana nesamba samathuba azo ngamanye. Ngokwezibalo, le axiom ivezwa kanje:

$$ P(A_1 \inkomishi A_2 \inkomishi \ amachashazi \ inkomishi A_n) = P(A_1) + P(A_2) + \ldots + P(A_n) $$

Izehlakalo ezimbili azifani uma zingeke zenzeke ngesikhathi esisodwa. Isibonelo, lapho ugingqa idayizi, imicimbi "ukweqa inombolo elinganayo" kanye "nokweqa inombolo eyinqaba" ayihlanganisi, njengoba inombolo ayikwazi ukuba kokubili ngokulinganayo futhi ibe yinqaba.

I-Basic Theory of Axiomatic Probability

Kusuka kuma-axiom ayisisekelo, amathuba e-axiomatic athola uchungechunge lwemibono evumela amathuba okubala kanye nokukhohlisa ezimweni eziyinkimbinkimbi kakhulu. Ezinye ze-theorems ezibaluleke kakhulu yilezi:

1. I-Conditional Probability Theorem

Amathuba anemibandela abhekisela emathubeni okuthi umcimbi ( A ) uzokwenzeka, njengoba esinye isenzakalo ( B ) sesenzekile kakade. Le theorem ivezwa ngokwezibalo kanje:

$$ P(A|B) = \frac{P(A \cap B)}{P(B)} $$

Lapho ( P(A|B) ) imele amathuba okuthi ( A ) anikezwe ( B ), ( P(A \cap B) ) amathuba okuthi u-( A ) no ( B ) uhlangane, futhi ( P(B) ) amathuba okuthi ( B ).
Isibonelo samathuba anemibandela kungaba ukubala amathuba okuthi umuntu unesifo esithile, uma kubhekwa ukuthi uhlolwe ukuthi unalo yini uhlolo lokuxilonga.

2. I-Theorem ye-Bayes

Ithiyori ye-Bayes isandiso samathuba anemibandela avumela amathuba omcimbi ukuthi abuyekezwe ngokusekelwe kulwazi olusha. Le theorem ivezwe kanje:

$$ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} $$

Lapho u-P(A|B) okungamathuba ka-A anikeziwe, u-P(B|A) amathuba okuthi B anikezwe u-A, u-P(A) amathuba angaphambili okuthi A, futhi u-P(B) amathuba okuthi B.
Le theorem isetshenziswa kakhulu emikhakheni efana nemithi, ubuhlakani bokwenziwa kanye ukwenza izinqumo, njengoba ivumela ukuvuselela izinkolelo zokuqala njengoba kutholakala ubufakazi obusha.

3. I-Theorem Yamathuba Ephelele

Ingqikithi ye-theoremu yamathuba isivumela ukuba sibale amathuba omcimbi ( A ), sicabangela zonke izehlakalo ezingaba khona ( B_i ) ezakha ukuhlukaniswa kwesampula yesikhala. Ngokwezibalo, ivezwa kanje:

$$ P(A) = P(A|B_1) \cdot P(B_1) + P(A|B_2) \cdot P(B_2) + \ldots + P(A|B_n) \cdot P(B_n) $$

Lapho ( P(A|B_i) ) okungamathuba okuthi ( A ) anikezwe ( B_i ), futhi ( P(B_i) ) amathuba okuthi ( B_i ).
Isibonelo sokusebenzisa le theory kungaba ukubala amathuba okuthi umfundi aphase isivivinyo, kucatshangelwa izindlela ezahlukene abengazifunda (eyedwa, eqenjini, nomfundisi, njll.).

I-athikili ehlobene:

Izingxenye Eziyi-10 Ezithakazelisayo Zalokho Okufundwa Isayensi Ehlelekile

Izinhlelo zokusebenza ze-Axiomatic Probability

Amathuba e-Axiomatic anezinhlobonhlobo zezinhlelo zokusebenza emikhakheni ehlukahlukene yesayensi nengokoqobo. Ezinye zezindawo lapho leli gatsha lezibalo linomthelela omkhulu yilezi:

1. I-Statistical Physics

Ku-physics yezibalo, amathuba e-axiomatic asetshenziselwa ukuchaza nokubikezela ukuziphatha kwezinhlelo eziyinkimbinkimbi ezakhiwe inombolo enkulu yezinhlayiya. Izimiso zamathuba zisivumela ukuthi senze imodeli yezenzakalo ezifana nokusatshalaliswa kwesivinini kugesi, ukudonsa kazibuthe kwezinto, kanye noshintsho lwesigaba.

2. Ezezimali kanye Nomnotho

Emkhakheni wezezimali nezomnotho, amathuba e-axiomatic abalulekile ekuhlaziyeni ubungozi, ukulinganiswa kwempahla kanye nokuthatha izinqumo zokutshalwa kwezimali. Amamodeli angenzeka asetshenziselwa ukufunda ukuntengantenga kwezimakethe, ukubikezela ukuziphatha kwentengo nokuhlola inzuzo yamasu ahlukene okutshala imali.

3. I-Artificial Intelligence kanye nokufunda ngomshini

Amathuba e-Axiomatic iyithuluzi elibalulekile ekuthuthukisweni kobuhlakani bokwenziwa kanye nama-algorithms okufunda komshini. Amamodeli angenzeka, njengamanethiwekhi e-Bayesian namamodeli e-Markov afihliwe, avumela imishini ukuthi ifunde kudatha futhi yenze izinqumo ezisekelwe ekungaqinisekini. Lawa masu asetshenziswa ezindaweni ezifana nokuqashelwa kwezwi, umbono wekhompyutha kanye nokuncoma umkhiqizo.

4. Imithi kanye ne-Epidemiology

Emkhakheni wezokwelapha, amathuba e-axiomatic asetshenziselwa ukuhlaziya ukusebenza kahle kokwelashwa, ukubikezela ukusabalala kwezifo, nokuhlola ukunemba kokuhlolwa kokuxilonga. Amamodeli angenzeka avumela ukulinganisa ubungozi bokuthuthukisa izimo ezithile zezokwelapha, kanye nokuklama amasu okuvimbela nokulawula ubhubhane.