Formulation of the . . . Main Idea Let Us Describe This . . . Resulting Definition Why Beta Priors: Main Result and Its Proof Invariance-Based Proof (cont-d) How to Get a General . . . Explanation Acknowledgments Bibliography Olga Kosheleva 1 , Vladik Kreinovich 1 , and Home Page Kittawit Autchariyapanitkul 2 Title Page 1 University of Texas at El Paso ◭◭ ◮◮ El Paso, Texas 79968, USA ◭ ◮ olgak@utep.edu, vladik@utep.edu 2 Faculty of Economics, Maejo University Page 1 of 15 Chiang Mai, Thailand, kittar3@hotmail.com Go Back Full Screen Close Quit
Formulation of the . . . Main Idea 1. Formulation of the Problem Let Us Describe This . . . • In the Bayesian approach: Resulting Definition Main Result and Its Proof – when we do not know the probability p ∈ [0 , 1] of Proof (cont-d) some event, How to Get a General . . . – it is usually recommended to use a Beta prior dis- Acknowledgments tribution for p , with pdf Bibliography ρ ( x ) = c · x α − 1 · (1 − x ) β − 1 . Home Page Title Page • There have been numerous successful application of the use of the Beta distribution in the Bayesian approach. ◭◭ ◮◮ ◭ ◮ • How can we explain this success? Page 2 of 15 • Why not use some other family of distributions located on the interval [0 , 1]? Go Back Full Screen Close Quit
Formulation of the . . . Main Idea 2. Formulation of the Problem (cont-d) Let Us Describe This . . . • The need for such an explanation is especially impor- Resulting Definition tant now, when the statistician community is: Main Result and Its Proof Proof (cont-d) – replacing the traditional p-value techniques How to Get a General . . . – with more reliable hypothesis testing methods. Acknowledgments • One such method is the Minimum Bayesian Factor Bibliography (MBF) method. Home Page • This method is based on Beta priors ρ ( x ) = c · x a cor- Title Page responding to β = 1. ◭◭ ◮◮ • In this paper, we provide a natural explanation for ◭ ◮ these empirical successes. Page 3 of 15 Go Back Full Screen Close Quit
Formulation of the . . . Main Idea 3. Main Idea Let Us Describe This . . . • We want to find a natural prior distribution on the Resulting Definition interval [0 , 1]. Main Result and Its Proof Proof (cont-d) • This distribution should describe how frequently dif- How to Get a General . . . ferent probability values p appear. Acknowledgments • In determining this distribution, a natural idea to take Bibliography into account is that: Home Page – in practice, Title Page – all probabilities are, in effect, conditional probabil- ◭◭ ◮◮ ities. ◭ ◮ • We start with some class, and in this class, we find the Page 4 of 15 corresponding frequencies. Go Back Full Screen Close Quit
Formulation of the . . . Main Idea 4. Main Idea (cont-d) Let Us Describe This . . . • From this viewpoint: Resulting Definition Main Result and Its Proof – we can start with the original probabilities and with Proof (cont-d) their prior distribution, How to Get a General . . . – or we can impose additional conditions and con- Acknowledgments sider the resulting conditional probabilities. Bibliography • For example, in medical data processing, we may con- Home Page sider the probability that: Title Page – a patient with a certain disease ◭◭ ◮◮ – recovers after taking the corresponding medicine. ◭ ◮ • We can consider this original probability. Page 5 of 15 • Alternatively, we can consider the conditional proba- Go Back bility that a patient will recover. Full Screen • For example, the condition can be that the patient is at least 18 years old. Close Quit
Formulation of the . . . Main Idea 5. Main Idea (cont-d) Let Us Describe This . . . • We can impose many such conditions. Resulting Definition Main Result and Its Proof • We are looking for a universal prior, a prior that would Proof (cont-d) describe all possible situations. How to Get a General . . . • So, it makes sense to consider priors for which: Acknowledgments – after such a restriction, Bibliography Home Page – we will get the exact same prior for the correspond- ing conditional probability. Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 15 Go Back Full Screen Close Quit
Formulation of the . . . Main Idea 6. Let Us Describe This Main Idea in Precise Terms Let Us Describe This . . . • In general, the conditional probability P ( A | B ) has the Resulting Definition form Main Result and Its Proof P ( A | B ) = P ( A & B ) . Proof (cont-d) P ( B ) How to Get a General . . . • Crudely speaking, this means that: Acknowledgments – when we transition from the original probabilities Bibliography Home Page to the new conditional ones, – we limit ourselves to the original probabilities which Title Page do not exceed some value p 0 = P ( B ), and ◭◭ ◮◮ – we divide each original probability by p 0 . ◭ ◮ • In these terms, the above requirement takes the follow- Page 7 of 15 ing form: for each p 0 ∈ (0 , 1), Go Back – if we limit ourselves to the interval [0 , p 0 ], Full Screen – then the ratios p/p 0 should have the same distribu- tion as the original one. Close Quit
Formulation of the . . . Main Idea 7. Resulting Definition Let Us Describe This . . . • Let us assume that we have a probability distribution Resulting Definition with probability density ρ ( x ) on the interval [0 , 1] . Main Result and Its Proof Proof (cont-d) • We say that this distribution is invariant if: How to Get a General . . . – for each p 0 ∈ (0 , 1) , Acknowledgments – the ratio x/p 0 (restricted to the values x ≤ p 0 ) has Bibliography the same distribution, i.e.: Home Page Title Page ρ ( x/p 0 : x ≤ p 0 ) = ρ ( x ) . ◭◭ ◮◮ ◭ ◮ Page 8 of 15 Go Back Full Screen Close Quit
Formulation of the . . . Main Idea 8. Main Result and Its Proof Let Us Describe This . . . • Proposition. A probability distribution is invariant if Resulting Definition and only if it has a form Main Result and Its Proof ρ ( x ) = c · x a for some c and a. Proof (cont-d) How to Get a General . . . • Proof. The conditional probability density has the Acknowledgments form Bibliography ρ ( x/p 0 : x ≤ p 0 ) = C ( p 0 ) · ρ ( x/p 0 ) . Home Page Title Page • Here, C is an appropriate constant depending on p 0 . ◭◭ ◮◮ • Thus, the invariance condition has the form ◭ ◮ C ( p 0 ) · ρ ( x/p 0 ) = ρ ( x ) . Page 9 of 15 • By moving the term C ( p 0 ) to the right-hand side and Go Back def denoting λ = 1 /p 0 (so that p 0 = 1 /λ ), we get Full Screen ρ ( λ · x ) = c ( λ ) · ρ ( x ) . Close Quit
Formulation of the . . . Main Idea 9. Proof (cont-d) Let Us Describe This . . . • We get ρ ( λ · x ) = c ( λ ) · ρ ( x ), where we denoted Resulting Definition Main Result and Its Proof def c ( λ ) = 1 /C (1 /λ ) . Proof (cont-d) How to Get a General . . . • The probability density function is an integrable func- Acknowledgments tion – its integral is equal to 1. Bibliography • Known: all integrable solutions of the above functional Home Page equation has the form ρ ( x ) = c · x a for some c , a . Title Page • The proposition is thus proven. ◭◭ ◮◮ • Reminder: these distributions – corr. to β = 1 – are ◭ ◮ used in the Bayesian approach to hypothesis testing. Page 10 of 15 Go Back Full Screen Close Quit
Formulation of the . . . Main Idea 10. How to Get a General Prior Distribution Let Us Describe This . . . • The above proposition describes the case when: Resulting Definition Main Result and Its Proof – we have a single distribution Proof (cont-d) – corresponding to a single piece of prior information. How to Get a General . . . • In practice, we may have many different pieces of in- Acknowledgments formation: Bibliography – some of these pieces are about the probability p of Home Page the corresponding event E , Title Page – some may be about the probability p ′ = 1 − p of ◭◭ ◮◮ the opposite event ¬ E . ◭ ◮ • According to the above Proposition, each piece of in- formation about p can be described by the pdf c i · x a i . Page 11 of 15 • Similarly, each piece of information about p ′ = 1 − p Go Back can be described by the probability density Full Screen c ′ j · x a ′ j . Close Quit
Formulation of the . . . Main Idea 11. General Case (cont-d) Let Us Describe This . . . • In terms of the original probability p = 1 − p ′ , this Resulting Definition probability density has the form Main Result and Its Proof Proof (cont-d) c ′ j · (1 − x ) a ′ j . How to Get a General . . . Acknowledgments • All these piece of information are independent. Bibliography • So, a reasonable idea is to multiply these probability Home Page density functions. Title Page • After multiplication, we get a distribution of the type ◭◭ ◮◮ c · x a · ( a − x ) a ′ , where a = a i and a ′ = � � a ′ j . ◭ ◮ i j Page 12 of 15 • This is exactly the Beta distribution – for α = a + 1 Go Back and β = a ′ + 1. Full Screen • Thus, we have indeed justified the use of Beta priors. Close Quit
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