IWSM 2013, the University of Georgia in Athens, Georgia, USA 18-21 - PowerPoint PPT Presentation

IWSM 2013, the University of Georgia in Athens, Georgia, USA 18-21 July, 2013

Presentation structure 1. Introduction 2. Constrained Bayesian problems of the multiple comparisons 3. Properties of the hypotheses acceptance regions 4. The method of sequential analysis of Bayesian type 5. Investigation of the method of sequential analysis of Bayesian type 6. Experimental research

1. Introduction We will consider the following methods of sequential analysis: � Sequential methods of Wald (middle of forties of 20 th century); century); � Bayesian sequential procedures (end of forties of 20 th century); � Berger sequential tests (beginning of nineties of 20 th century); � The sequential analysis method of Bayesian type (2010).

2. Constrained Bayesian problems of the multiple comparisons To minimize the averaged value of probabilities of incorrectly rejected hypotheses ∑ ∑ ∑ ∑ S S ∫ ∫ x x = r p H p H d (1) ( ) ( | ) δ i i = = = = ≠ ≠ i i j j j j i i Γ Γ 1 1 1 1 , , j j subject to the averaged value of incorrectly accepted hypotheses ∑ S ∫ x x − ≤ α p H p H d . (2) 1 ( ) ( | ) i i = i Γ 1 i

θ = i H θ S Here is the number of tested hypotheses , : i x T = x ∈ n n x x R R supposed that the sample ( , is ( 1 ,..., ) n observation space) is generated by distribution x x p x p x i = = θ θ i θ θ i ≡ ≡ H H p p p p x x x x θ θ = = ( ( , , ) ) ( ( ,..., ,..., ; ; ,..., ,..., ) ) ( ( | | ) ) , , i i S S ; ; 1 1 ,..., ,..., n n k k i i 1 1 1 1 T = θ θ θ ( 1 ,..., ) is the vector of parameters of distribution, k i ∈ Θ k Θ k k ∀ : = θ ; i i S ; is -dimensional parametrical 1 ,..., p H n ≠ k ( ) space; in general, ; is the a priori probability of i Γ H hypothesis ; is the region of acceptance of hypothesis i i α H ; is the maximum allowed level of the averaged i value of the probabilities of incorrectly accepted hypotheses.

Task (1), (2) is one of possible formulations of the constrained Bayesian problem. In a similar manner, we can introduce and solve different constrained Bayesian tasks. Solutions of these tasks differ to each other but, generally, they can be written as follows: ∑ S , Γ = x x > x j j k p H k p H { : ( | ) ( | ) } j j j � � = ≠ j � � 1 , = j S , (3) 1 ,..., where = , . S ≤ j < +∞ k � � 1 ,..., 0

Investigation of these regions shows that they have identical properties, namely, in general case they can intersect and their union could not coincide with observation space. This fact allows us to use each of them for construction of the appropriate sequential method of testing of hypotheses. But, for simplicity, further we shall consider only the task (1), (2), though all results obtained below, after appropriate modifications, are true for all other other tasks tasks also. also. For concreteness let us rewrite the regions of acceptance of hypotheses (3) for the task (1), (2): { } ∑ S x x x Γ = < λ = p H p H p H p H j S , (4) : ( ) ( | ) ( ) ( | ) , 1 ,..., j i i j j i = i ≠ j 1 , λ < λ < +∞ 0 where ( ) the same scalar value for all regions, is determined so that in (2) the equality takes place.

3. Properties of the hypotheses acceptance regions It is known that, in traditional statements of the problem of statistical hypotheses testing, their Γ Γ = ∅ i � acceptance regions are not intersected, i.e. , j i ≠ i ≠ j j , and the union of all regions of acceptance of , and the union of all regions of acceptance of hypotheses coincides with the observation space, i.e. S Γ = n � 1 R . i = i These conditions break down at consideration of above-formulated constrained Bayesian task of hypotheses testing.

α < α < Namely, for any value of ( 0 1 ) in restriction λ < λ < +∞ (2) (that is the same, for any value of ( )), in 0 n R the observation space exist: the regions of unambiguous acceptance of the tested hypotheses, the regions of the suspicion on the validity of several (more regions of the suspicion on the validity of several (more than one) tested hypotheses (corresponding to sub- regions of the intersection of the regions of acceptance of corresponding hypotheses (4)) and the region of impossibility of acceptance of the tested hypotheses n R (corresponding to the region of the space , which do not belong to any of the regions of acceptance of hypotheses (4)).

x Accordingly, for any concrete observation result , on the basis of which the decision is made, in the interval x x α ≤ α * ( ) ( ) ( 0 ; 1 ) there are such values that for * x x α x α ∈ α * the observation result belongs to only one [ ( ); ( )] * of the regions of acceptance of hypotheses (4) and the corresponding hypothesis is accepted, respectively. At x * x α < α , the observation result appears in a sub-region of ( ) intersection of two or several regions of acceptance of intersection of two or several regions of acceptance of hypotheses (4), and it is impossible to make a unique decision. In that case, the appropriate hypotheses are * x α > α suspected on the validity. At , the observation result ( ) x n R appears in the region of the space which does not belong to any of the regions of acceptance of hypotheses (4). In this case, it is impossible to make the decision on the basis x of the set observation result .

Thus, the situation is similar to the sequential analysis when, on the basis of present observation results, it could be impossible to make a decision (with the given probabilities of errors) about the validity of one of the hypotheses from the considered set. Therefore, in the considered tasks, if there is the situation of impossibility considered tasks, if there is the situation of impossibility of making an unambiguous decision for the given significance level, we shall continue the observations until such an opportunity appears. Thus, on the basis of above-considered constrained Bayesian task, let us determine the method of sequential analysis for multiple testing problems. For clarity let us call this method the sequential analysis method of Bayesian type .

4. The method of sequential analysis of Bayesian type Let us suppose that there is an opportunity of n R obtaining repeated observations. Let be the m m sampling space of all possible samples of indepen- x n T = x x dent -dimensional observation vectors . ( 1 ,..., ) n + + n n S S R R n n R R 1 1 Let us split Let us split into into disjoint sub-regions disjoint sub-regions , , m m , 1 + S 1 n n n n = n R R , R R � R ,..., , such that . m m S , + m S m m i , 2 1 = , i 1

Let us determine the following decision rule. If the x = x x m 1 matrix of observation results ( ,..., ) belongs to the H sub-region n , = , then hypothesis is accepted, R , i S 1 ,..., i m i x = x x m 1 ( ,..., ) and, if belongs to the sub-region n R , the decision is not made, and the observations go , + m S 1 on until one of the tested hypotheses is accepted.

R n = + i S Regions , are determined in the , 1 ,..., 1 m i , following way: R n = , is such a part of acceptance i S , 1 ,..., m i , H Γ m region of hypothesis which does not belong to any i i Γ m = − + n j i i S R other region ; is such a part , 1 ,..., 1 , 1 ,..., , + j m S 1 n R of sampling space which belongs simultaneously to m Γ m , = i S more than one region , or it does not belong 1 ,..., i i to any of these regions. Here the index m = m ( ) points to the fact that the regions are 1 , 2 ,... m determined on the basis of sequential observation results. On the basis of the properties described in Item 3, hypotheses acceptance regions R n = + , could be i S , 1 ,..., 1 m i , determined as follows.

Let us designate the population of sub-regions of Γ m intersections of acceptance regions of hypotheses i H i = S in constrained Bayesian task of hypotheses ( 1 ,..., ) i testing with the regions of acceptance of other hypotheses H m = ≠ I j S j i , , by . By 1 ,..., ; j i S , we designate the population of regions n = = n − − = Γ = Γ m E E R R � i � m m m m i i i 1 1 n of space R which do not belong to any of hypotheses m acceptance regions. Then the hypotheses acceptance regions in the method of sequential analysis of Bayesian type are determined in the following way: ( ) � S n + = m n R � I E n = Γ m m = R I i S ; . (6) / , 1 ,..., m S i m m i i i , 1 = i , 1 m n = Γ m i S I E Here regions , are defined on the , , 1 ,..., , i i m basis of hypotheses acceptance regions (4) from Item 2.