Conditional independence

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In probability theory, two independent R and B independent events are given the third event Y exactly when the occurrence of R and the occurrence of B is an independent event in their conditional probability distribution given Y . In other words, R and B are provided in an independent condition Y if and only if, given the knowledge that Y occurs , the knowledge of whether R occurs does not provide information about the probability of occurrence of B , and the knowledge of whether B occurs does not provide information about the possibility of R happens.

Video Conditional independence

Definisi formal

Dalam notasi to stand theoretical probability, R dan B diberikan seizra condisional independently Y jika dan hanya jika

${\ displaystyle \ Pr (R \ cap B \ mid Y) = \ Pr (R \ mid Y) \ Pr (B \ mid Y), \, } Â Â$

Atau dengan kata lain,

${\ displaystyle \ Pr (R \ mid B \ cap Y) = \ Pr (R \ mid Y). \,} Â Â$

Two random variables X and Y are independent conditional given the third random variable Z if and only if they are independent in their conditional probability distribution given Z . That is, X and Y are provided in an independent condition Z if and only if, given any value Z , the distribution the probability of X is the same for all values ​​ Y and the probability distribution Y is the same for all values ​​ X .

Second peristiwa R dan B conditional drills independently diberican? -gebra? nice

${\ displaystyle \ Pr (R \ cap B \ mid \ Sigma) = \ Pr (R \ mid \ Sigma) \ Pr (B \ mid \ Sigma) \ as} Â Â$

di mana ${\ displaystyle \ Pr (A \ mid \ Sigma)}  ÂÂ$ menunjukkan ekspektasi bersyarat dari fungsi indikator acara ${\ displaystyle A}  ÂÂ$ , ${\ displaystyle \ chi _ {A}}  ÂÂ$ , diberikan sigma aljabar ${\ displaystyle \ Sigma}  ÂÂ$ . Itu adalah,

${\ displaystyle \ Pr (A \ mid \ Sigma): = \ operatorname {E} [\ chi _ {A} \ mid \ Sigma].}  ÂÂ$

Two random variables X and Y are independently conditioned given a -algebra ? if the above equation applies to all R in? ( X ) and B? ( Y ).

Dua variabel acak X dan Y secara kondisional independen diberikan variabel acak W jika mereka diberikan secara independen? ( W ):? -algebra yang dihasilkan oleh W . Ini biasanya ditulis:

${\ displaystyle X \ pel \! \! \! \ pelaku Y \ mid W}  ÂÂ$ atau

${\ displaystyle X \ perp Y \ mid W}  ÂÂ$

Feminine dependence "X tidak tergantung pada Y, diberikan W"; Pengkondisian berlaku untuk seluruh pernyataan: "(X tidak bergantung pada Y) diberikan W".

${\ displaystyle (X \ pel \! \! \! \ Tugas Y) \ pertengahan W} Â Â$

If W assumes a set of values ​​that can be calculated, this is equivalent to the conditional independence of X and Y for events in the form of > Ã, = Ã, w ]. Conditional independence of more than two events, or more than two random variables, is defined analogously.

The following two examples show that X ? Y not meant or implied by X ? Y | W . First, let W be 0 with a probability of 0.5 and 1 otherwise. When W Ã, = Ã, 0 takes X and Y becomes independent, each has a value of 0 with probability of 0.99 and a value of 1 otherwise. When W Ã, = Ã, 1, X and Y return independent, but this time they take a value of 1 with a probability of 0.99. Then X ? Y Ã, | Ã, W . But X and Y depend, because Pr ( X Ã, = Ã, 0) & lt; Pr ( X Ã, = Ã, 0 | Y Ã, = Ã, 0). This is because Pr ( X Ã, = Ã, 0) = 0,5, but if Y Ã, = Ã, 0 then most likely W = Ã, 0 and thus it is X Ã, = Ã, 0 also, so Pr ( X Ã, = Ã, 0 | Y Ã, = Ã, 0) Ã, & gt; 0.5. For the second example, say X ? Y , each taking a value of 0 and 1 with a probability of 0.5. Let W be the product X ÃÆ'â € " Y . Then when W Ã, = Ã, 0, Pr ( X Ã, = Ã, 0) Ã, = Ã, 2/3, but Pr ( X Ã, = Ã, 0 | Y Ã, = Ã, 0) Ã, = Ã, 1/2, so X Ã,? Y Ã,
W wrong. This is also an example of Explaining Remote. See Kevin Murphy's tutorial where X and Y take the "smart" and "sporty" grades.

Maps Conditional independence

Example

Discussions about StackExchange provide some useful examples.

1. Let these two happen to be an opportunity A and B go home in time for dinner, and the third event is the fact that a snowstorm hit the city. While A and B have a lower probability of getting home in time for dinner, the lower probability will remain independent of each other. That is, the knowledge that A is late does not tell you whether B will be late. (They may live in different environments, travel at different distances, and use different modes of transportation.) However, if you have information that they live in the same neighborhood, use the same transportation, and work in the same place, then both events are NOT conditional dependent.
2. Conditional independence depends on the nature of the third event. If you throw two dice, one can assume that two dice behave independently of each other. Viewing the results of 1 dice will not tell you about the second dice result. (That is, two dice are independent.) If, however, the result of 1 die is 3, and someone tells you about the third event - that the sum of the two results is even - then the extra unit of this information limits the choice for the second result to an odd number. In other words, two events can be independent, but NOT conditional free.
3. Height and vocabulary are not independent; but they are free conditionally if you add age.

src: www.researchgate.net

Usage in Bayesian inference

Let p be the proportion of voters who will vote "yes" in the upcoming referendum. In taking polls, a person selects a randomly selected n of the population. For i Ã, = Ã, 1, Ã,..., Ã, n , leave X _i = Ã, 1 or 0 corresponding, respectively, whether the selected voter or not will choose "yes".

In a frequentist approach to statistical inference one would not associate any probability distribution to p (unless the probability could somehow be interpreted as the relative frequencies of occurrence of some event or as the proportion of some population) and one would say that < X _n is an independent random variable.

In contrast, in the Bayesian approach to statistical inference, one would assign probability distributions to p regardless of the absence of such "frequency" interpretation, and one would interpret the probabilities as the degree of confidence p it is within any given interval of probability. In that model, the random variables X ₁ , Ã,..., Ã, X _n is not standalone, but they are freely conditional given p . In particular, if a large amount of X is observed equal to 1, it will imply a high conditional probability, given that observation, that p is close to 1, and thus the conditional probability is high, given that observation, that the next X to be observed would be equal to 1.

src: slideplayer.com

Conditional freedom rule

The set of rules governing the declaration of conditional independence has been derived from the basic definition.

Catatan: karena implikasi ini berlaku untuk ruang probabilitas apa pun, mereka akan tetap berlaku jika seseorang menganggap sub-jagad dengan mengkondisikan semuanya pada variabel lain, katakanlah K . Misalnya, ${\ displaystyle X \ pel \! \! \! \ tegak Y \ Rightarrow Y \ pelet \! \! \! \ perp X}  ÂÂ$ juga akan berarti bahwa ${\ displaystyle X \ pel \! \! \! \ tegak Y \ mid K \ Rightarrow Y \ pelet \! \! \! \ perp X \ pertengahan K}  ÂÂ$ .

Catatan: di bawah ini, koma dapat dibaca sebagai "DAN".

Simetri

${\ displaystyle X \ pel \! \! \! \ perp Y \ quad \ Rightarrow \ quad Y \ pel \! \! \! \ perp X}  ÂÂ$

Dekomposisi

${\ displaystyle X \ pel \! \! \! \ perp A, B \ quad \ Rightarrow \ quad {\ text {and}} {\ begin {cases} X \ pelakunya \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B \ end {cases}}}  ÂÂ$

Bukti serupa menunjukkan kemandirian X dan B .

Persatuan lemah

${\ displaystyle X \ pel \! \! \! \ perp A, B \ quad \ Rightarrow \ quad {\ text {and}} {\ begin {cases} X \ pelakunya \ \ \ \ \ \ \ \ \ A \ pertengahan B \ \ \ \ \ \ \ \ \ \ \ \ B \ pertengahan A \ end {cases}}}  ÂÂ$

Bukti:

• Menurut defined, ${\ displaystyle \ Pr (X) = \ Pr (X \ mid A, B)} Â Â$ .
• Karena properti decomposisi ${\ displaystyle X \ pen \! \! \! \ perp B} Â$ , ${\ displaystyle \ Pr (X) = \ Pr (X \ mid B)} Â Â$ .
• Menggabungkan second persuasion takes a member of ${\ displaystyle \ Pr (X \ mid B) = \ Pr (X \ mid A, B)} Â$ , yang menetapkan ${\ displaystyle X \ pel \! \! \! \ perp A \ mid B} Â Â$ .

Kondisi kedua bisa dibuktikan sama.

Kontraksi

${\ displaystyle \ left. {\ begin {aligned} X \ pel \! \! \! \ perp A \ mid B \\ X \ pel \! \! \! \ pel B \ end {aligned}} \ right \} {\ text {and}} \ quad \ Rightarrow \ quad X \ pel \! \! \! \ pelak A, B}  ÂÂ$

Evidence:

Properti ini dapat dibuktikan dengan memperhatikan $Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; ÂPrÂ\; Â\; Â\; Â\; Â\; Â\; Â(Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂX\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â|Â\; Â\; Â\; Â\; Â\; Â\; ÂAÂ\; Â\; Â\; Â\; Â\; Â\; Â,Â\; Â\; Â\; Â\; Â\; Â\; ÂBÂ\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; ÂPrÂ\; Â\; Â\; Â\; Â\; Â\; Â(Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂX\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â|Â\; Â\; Â\; Â\; Â\; Â\; ÂBÂ\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; ÂPrÂ\; Â\; Â\; Â\; Â\; Â\; Â(Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂX\; Â\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â}Â\; Â\; ÂÂ\; Â\; Â\{\backslash \; displaystyle\; \backslash \; Pr\; (X\; \backslash \; mid\; A,\; B)\; =\; \backslash \; Pr\; (X\; \backslash \; mid\; B)\; =\; \backslash \; Pr\; (X)\}\; Â\; Â$ , setiap persamaan yang ditegaskan oleh ${\ displaystyle X \ pel \! \! \! \ perp A \ mid B} Â$ dan ${\ displaystyle X \ pen \! \! \! \ perp B} Â$ , masing-masing.

Kontraksi-weak-union-decomposition

Untuk distribusi probabilitas sangat positif, berikut ini juga berlaku:

${\ displaystyle \ left. {\ begin {aligned} X \ pel \! \! \! \ perp A \ pertengahan C, B \ \ \ \ \ \ \ \ \! \! \ perp B \ mid C, A \ end {aligned}} \ right \} {\ text {and}} \ quad \ Rightarrow \ quad X \ pelet \! \! \! \ perp B, A \ mid C }  ÂÂ$

Lima aturan di atas disebut "Aksioma Graphoid" oleh Pearl dan Paz, karena mereka memegang grafik, jika ${\ displaystyle X \ pel \! \! \! \ perp A \ mid B}  ÂÂ$ ditafsirkan sebagai berarti: "Semua jalur dari X ke A dicegat oleh himpunan B ".

src: images.slideplayer.com

Lihat juga

• Graphoid
• Ketergantungan bersyarat
• teorema de Finetti
• Harapan bersyarat

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Referensi

Source of the article : Wikipedia