A Bayesian nonparametric method for the LR assessment in case of - PowerPoint PPT Presentation

Random partitions of [ n ] Let [ n ] denote the set [ n ] = { 1 , 2 , ..., n } . A partition of the set [ n ] will be denoted as π [ n ] . Giulia Cereda () Short title October 8, 2015 9 / 33

Random partitions of [ n ] Let [ n ] denote the set [ n ] = { 1 , 2 , ..., n } . A partition of the set [ n ] will be denoted as π [ n ] . Random partitions on the set [ n ] will be denoted as Π [ n ] . Giulia Cereda () Short title October 8, 2015 9 / 33

DNA database can be reduced DATABASE of size 10 Person 1 (4 − 10 − 6 − 7) Person 2 (3 − 5 − 6 − 8) Person 3 (3 − 7 − 8 − 10) Person 4 (10 − 1 − 4 − 5) Person 5 (3 − 7 − 8 − 10) Person 6 (3 − 7 − 8 − 10) Person 7 (1 − 5 − 7 − 2) Person 8 (3 − 7 − 8 − 10) Person 9 (3 − 5 − 6 − 8) Person 10 (3 − 7 − 8 − 10) Giulia Cereda () Short title October 8, 2015 10 / 33

DNA database can be reduced DATABASE of size 10 Person 1 (4 − 10 − 6 − 7) Person 2 (3 − 5 − 6 − 8) Person 3 (3 − 7 − 8 − 10) Person 4 (10 − 1 − 4 − 5) Person 5 (3 − 7 − 8 − 10) Person 6 (3 − 7 − 8 − 10) Person 7 (1 − 5 − 7 − 2) Person 8 (3 − 7 − 8 − 10) Person 9 (3 − 5 − 6 − 8) Person 10 (3 − 7 − 8 − 10) Giulia Cereda () Short title October 8, 2015 11 / 33

DNA database can be reduced DATABASE of size 10 Person 1 (4 − 10 − 6 − 7) Person 2 (3 − 5 − 6 − 8) Person 3 (3 − 7 − 8 − 10) Person 4 (10 − 1 − 4 − 5) Person 5 (3 − 7 − 8 − 10) Person 6 (3 − 7 − 8 − 10) Person 7 (1 − 5 − 7 − 2) Person 8 (3 − 7 − 8 − 10) Person 9 (3 − 5 − 6 − 8) Person 10 (3 − 7 − 8 − 10) Assumption 2 → data can be replaces by the equivalence classes on the indices of the relation “to have the same DNA type”. This is a partition of the set [ n ] : {{ 1 } , { 2 , 9 } , { 3 , 5 , 6 , 8 , 10 } , { 4 } , { 7 }} Giulia Cereda () Short title October 8, 2015 11 / 33

Reduced data Giulia Cereda () Short title October 8, 2015 12 / 33

Reduced data Data D is made of the database + 2 new observations Giulia Cereda () Short title October 8, 2015 12 / 33

Reduced data Data D is made of the database + 2 new observations D = π [ n +2] partition of the set { 1 , 2 , ..., n + 2 } Giulia Cereda () Short title October 8, 2015 12 / 33

Reduced data Data D is made of the database + 2 new observations D = π [ n +2] partition of the set { 1 , 2 , ..., n + 2 } Example: Database → π [10] = {{ 1 } , { 2 , 9 } , { 3 , 5 , 6 , 8 , 10 } , { 4 } , { 7 }} Giulia Cereda () Short title October 8, 2015 12 / 33

Reduced data Data D is made of the database + 2 new observations D = π [ n +2] partition of the set { 1 , 2 , ..., n + 2 } Example: Database → π [10] = {{ 1 } , { 2 , 9 } , { 3 , 5 , 6 , 8 , 10 } , { 4 } , { 7 }} D → π [12] = {{ 1 } , { 2 , 9 } , { 3 , 5 , 6 , 8 , 10 } , { 4 } , { 7 } , { 11 , 12 }} Giulia Cereda () Short title October 8, 2015 12 / 33

Reduced data Data D is made of the database + 2 new observations D = π [ n +2] partition of the set { 1 , 2 , ..., n + 2 } Example: Database → π [10] = {{ 1 } , { 2 , 9 } , { 3 , 5 , 6 , 8 , 10 } , { 4 } , { 7 }} D → π [12] = {{ 1 } , { 2 , 9 } , { 3 , 5 , 6 , 8 , 10 } , { 4 } , { 7 } , { 11 , 12 }} We can see the data as a random variable. In that case, D = Π [ n +2] . Giulia Cereda () Short title October 8, 2015 12 / 33

The distribution of D = Π [ n +2] depends on p . However, it does not depend on the order of the p i . Giulia Cereda () Short title October 8, 2015 13 / 33

The distribution of D = Π [ n +2] depends on p . However, it does not depend on the order of the p i . ↓ We can consider directly the ordered vector p ∈ ∇ ∞ = { ( p 1 , p 2 , .... ) , p 1 ≥ p 2 ≥ ... > 0 , � p i = 1 } . Giulia Cereda () Short title October 8, 2015 13 / 33

The distribution of D = Π [ n +2] depends on p . However, it does not depend on the order of the p i . ↓ We can consider directly the ordered vector p ∈ ∇ ∞ = { ( p 1 , p 2 , .... ) , p 1 ≥ p 2 ≥ ... > 0 , � p i = 1 } . For instance, p 3 = the frequency of the third most frequent DNA type in Nature. Giulia Cereda () Short title October 8, 2015 13 / 33

Prior distribution on p ∈ ∇ ∞ Bayesian nonparametrics: we need a prior for the parameter p . Giulia Cereda () Short title October 8, 2015 14 / 33

Prior distribution on p ∈ ∇ ∞ Bayesian nonparametrics: we need a prior for the parameter p . Two parameter Poisson Dirichlet distribution . Giulia Cereda () Short title October 8, 2015 14 / 33

Prior distribution on p ∈ ∇ ∞ Bayesian nonparametrics: we need a prior for the parameter p . Two parameter Poisson Dirichlet distribution . Parameters: 0 < α < 1, θ > − α Giulia Cereda () Short title October 8, 2015 14 / 33

The model (first part) Giulia Cereda () Short title October 8, 2015 15 / 33

The model (first part) A , Θ ∇ ∞ ∋ P

The model (first part) A , Θ P | α, θ ∼ PD( α, θ ) ∇ ∞ ∋ P

The model (first part) A , Θ P | α, θ ∼ PD( α, θ ) ∇ ∞ ∋ P ... N ∋ X 1 X 2 X n X i = j → the i-th observation has the j th most common type in Nature. Giulia Cereda () Short title October 8, 2015 15 / 33

The model (first part) A , Θ P | α, θ ∼ PD( α, θ ) ∇ ∞ ∋ P ... N ∋ X 1 X 2 X n X i = j → the i-th observation has the j th most common type in Nature. Giulia Cereda () Short title October 8, 2015 15 / 33

The model (first part) ( A , Θ) ∼ f A , Θ P | α, θ ∼ PD( α, θ ) ∇ ∞ ∋ P ... N ∋ X 1 X 2 X n

The model (first part) ( A , Θ) ∼ f A , Θ P | α, θ ∼ PD( α, θ ) ∇ ∞ ∋ P ... X n +1 N ∋ X 1 X 2 X n Suspect

The model (first part) ( A , Θ) ∼ f A , Θ P | α, θ ∼ PD( α, θ ) ∇ ∞ ∋ P ... X n +1 N ∋ X 1 X 2 X n X 1 , ..., X n +1 | p ∼ i.i.d p Suspect

The model (first part) ( A , Θ) ∼ f A , Θ P | α, θ ∼ PD( α, θ ) ∇ ∞ ∋ P { H p , H d } ∋ H ... X n +1 X n +2 N ∋ X 1 X 2 X n X 1 , ..., X n +1 | p ∼ i.i.d p Suspect Crime stain

The model (first part) ( A , Θ) ∼ f A , Θ P | α, θ ∼ PD( α, θ ) ∇ ∞ ∋ P { H p , H d } ∋ H ... X n +1 X n +2 N ∋ X 1 X 2 X n X 1 , ..., X n +1 | p ∼ i.i.d p Suspect Crime stain � δ x n +1 if H = H p X n +2 | p , H , x n +1 ∼ if H = H d p Giulia Cereda () Short title October 8, 2015 16 / 33

The model (first part) ( A , Θ) ∼ f A , Θ P | α, θ ∼ PD( α, θ ) ∇ ∞ ∋ P { H p , H d } ∋ H ... X n +1 X n +2 N ∋ X 1 X 2 X n X 1 , ..., X n +1 | p ∼ i.i.d p Suspect Crime stain � δ x n +1 if H = H p X n +2 | p , H , x n +1 ∼ if H = H d p Giulia Cereda () Short title October 8, 2015 16 / 33

The model (first part) A , Θ P H ... X n +1 X n +2 X 1 X 2 X n Giulia Cereda () Short title October 8, 2015 17 / 33

Random partitions Some notation: Given X 1 , ..., X n ∈ N , random variables, Π [ n ] ( X 1 , X 2 , ..., X n ) is the random partition defined by the equivalence classes of i ∼ j iff X i = X j . Giulia Cereda () Short title October 8, 2015 18 / 33

Random partitions Some notation: Given X 1 , ..., X n ∈ N , random variables, Π [ n ] ( X 1 , X 2 , ..., X n ) is the random partition defined by the equivalence classes of i ∼ j iff X i = X j . Π [ n ] = π Db X 1 , ..., X n − → [ n ] Π [ n +1] = π Db+ X 1 , ..., X n , X n +1 − → [ n +1] Π [ n +2] = π Db++ X 1 , ..., X n , X n +1 , X n +2 − → [ n +2] Giulia Cereda () Short title October 8, 2015 18 / 33

Random partitions Some notation: Given X 1 , ..., X n ∈ N , random variables, Π [ n ] ( X 1 , X 2 , ..., X n ) is the random partition defined by the equivalence classes of i ∼ j iff X i = X j . Π [ n ] = π Db X 1 , ..., X n − → [ n ] Π [ n +1] = π Db+ X 1 , ..., X n , X n +1 − → [ n +1] Π [ n +2] = π Db++ X 1 , ..., X n , X n +1 , X n +2 − → [ n +2] X 1 , ..., X n are not observed, but generates the same partition as the original database. Data can be defined as D = Π [ n +2] . Giulia Cereda () Short title October 8, 2015 18 / 33

The complete model A , Θ P H X 1 X 2 X n X n +1 X n +2 D Giulia Cereda () Short title October 8, 2015 19 / 33

Pitman sampling formula Giulia Cereda () Short title October 8, 2015 20 / 33

Pitman sampling formula P ∼ PD( α, θ ) X 1 , X 2 , ..., X n | P = p ∼ i.i.d p Giulia Cereda () Short title October 8, 2015 20 / 33

Pitman sampling formula P ∼ PD( α, θ ) X 1 , X 2 , ..., X n | P = p ∼ i.i.d p then Π [ n ] = Π [ n ] ( X 1 , ..., X n ) has the following distribution: Giulia Cereda () Short title October 8, 2015 20 / 33

Pitman sampling formula P ∼ PD( α, θ ) X 1 , X 2 , ..., X n | P = p ∼ i.i.d p then Π [ n ] = Π [ n ] ( X 1 , ..., X n ) has the following distribution: k α,θ ( π [ n ] ) = [ θ + α ] k − 1; α � Pr(Π [ n ] = π [ n ] | α, θ ) = P n [1 − α ] n i − 1;1 , [ θ + 1] n − 1;1 i =1 Giulia Cereda () Short title October 8, 2015 20 / 33

Pitman sampling formula P ∼ PD( α, θ ) X 1 , X 2 , ..., X n | P = p ∼ i.i.d p then Π [ n ] = Π [ n ] ( X 1 , ..., X n ) has the following distribution: k α,θ ( π [ n ] ) = [ θ + α ] k − 1; α � Pr(Π [ n ] = π [ n ] | α, θ ) = P n [1 − α ] n i − 1;1 , [ θ + 1] n − 1;1 i =1 In our model � P n +2 α,θ ( π Db++ [ n +2] ) if h = H d Pr( D | α, θ, h ) = Pr(Π [ n +2] = π Db ++ [ n +2] | α, θ, h ) = P n +1 α,θ ( π Db+ [ n +1] ) if h = H p Giulia Cereda () Short title October 8, 2015 20 / 33

The model, simplified A , Θ H D Giulia Cereda () Short title October 8, 2015 21 / 33

The model, simplified A , Θ H D D = Π [ n +2] . Giulia Cereda () Short title October 8, 2015 21 / 33

Lemma Giulia Cereda () Short title October 8, 2015 22 / 33

Lemma A H X Y Lemma Given four random variables A , H , X and Y , as above, the likelihood function for h , given X = x and Y = y , satisfies lik ( h | x , y ) ∝ E ( p ( y | x , A , h ) | X = x ) . Giulia Cereda () Short title October 8, 2015 22 / 33

Lemma A , Θ H Π [ n +1] Π [ n +2] Giulia Cereda () Short title October 8, 2015 23 / 33

Lemma A , Θ H Π [ n +1] Π [ n +2] lik ( h | π [ n +1] , π [ n +2] ) ∝ E ( p ( π [ n +2] | π [ n +1] , A , Θ , h ) | Π [ n +1] = π [ n +1] ) . Giulia Cereda () Short title October 8, 2015 23 / 33

Likelihood ratio LR = p ( π [ n +2] | H p ) p ( π [ n +2] | H d ) = p ( π [ n +1] , π [ n +2] | H p ) p ( π [ n +1] , π [ n +2] | H d ) = lik( H p | π [ n +1] , π [ n +2] ) lik( H d | π [ n +1] , π [ n +2] ) Giulia Cereda () Short title October 8, 2015 24 / 33

Likelihood ratio LR = p ( π [ n +2] | H p ) p ( π [ n +2] | H d ) = p ( π [ n +1] , π [ n +2] | H p ) p ( π [ n +1] , π [ n +2] | H d ) = lik( H p | π [ n +1] , π [ n +2] ) lik( H d | π [ n +1] , π [ n +2] ) Lemma allows to write 1 � �� � LR = E ( p ( π [ n +2] | π [ n +1] , A , Θ , H p ) | Π [ n +1] = π [ n +1] ) E ( p ( π [ n +2] | π [ n +1] , A , Θ , H d ) | Π [ n +1] = π [ n +1] ) � �� � 1 − A n +1+Θ Giulia Cereda () Short title October 8, 2015 24 / 33

Likelihood ratio LR = p ( π [ n +2] | H p ) p ( π [ n +2] | H d ) = p ( π [ n +1] , π [ n +2] | H p ) p ( π [ n +1] , π [ n +2] | H d ) = lik( H p | π [ n +1] , π [ n +2] ) lik( H d | π [ n +1] , π [ n +2] ) Lemma allows to write 1 � �� � LR = E ( p ( π [ n +2] | π [ n +1] , A , Θ , H p ) | Π [ n +1] = π [ n +1] ) E ( p ( π [ n +2] | π [ n +1] , A , Θ , H d ) | Π [ n +1] = π [ n +1] ) � �� � 1 − A n +1+Θ 1 = � . � 1 − A n +1+Θ | Π [ n +1] = π [ n +1] E Giulia Cereda () Short title October 8, 2015 24 / 33

Likelihood ratio LR = p ( π [ n +2] | H p ) p ( π [ n +2] | H d ) = p ( π [ n +1] , π [ n +2] | H p ) p ( π [ n +1] , π [ n +2] | H d ) = lik( H p | π [ n +1] , π [ n +2] ) lik( H d | π [ n +1] , π [ n +2] ) Lemma allows to write 1 � �� � LR = E ( p ( π [ n +2] | π [ n +1] , A , Θ , H p ) | Π [ n +1] = π [ n +1] ) E ( p ( π [ n +2] | π [ n +1] , A , Θ , H d ) | Π [ n +1] = π [ n +1] ) � �� � 1 − A n +1+Θ 1 = � . � 1 − A n +1+Θ | Π [ n +1] = π [ n +1] E Giulia Cereda () Short title October 8, 2015 24 / 33

1 LR = � � 1 − A n +1+Θ | Π [ n +1] = π [ n +1] E Giulia Cereda () Short title October 8, 2015 25 / 33

1 LR = � � 1 − A n +1+Θ | Π [ n +1] = π [ n +1] E 1 − A By defining the random variable Φ = n n +1+Θ we can write the LR as Giulia Cereda () Short title October 8, 2015 25 / 33

1 LR = � � 1 − A n +1+Θ | Π [ n +1] = π [ n +1] E 1 − A By defining the random variable Φ = n n +1+Θ we can write the LR as n LR = E (Φ | Π [ n +1] = π [ n +1] ) . Giulia Cereda () Short title October 8, 2015 25 / 33

1 LR = � � 1 − A n +1+Θ | Π [ n +1] = π [ n +1] E 1 − A By defining the random variable Φ = n n +1+Θ we can write the LR as n LR = E (Φ | Π [ n +1] = π [ n +1] ) . We are interested in the distribution of Φ , Θ | Π [ n +1] Giulia Cereda () Short title October 8, 2015 25 / 33

1 LR = � � 1 − A n +1+Θ | Π [ n +1] = π [ n +1] E 1 − A By defining the random variable Φ = n n +1+Θ we can write the LR as n LR = E (Φ | Π [ n +1] = π [ n +1] ) . We are interested in the distribution of Φ , Θ | Π [ n +1] p ( φ, θ | π [ n +1] ) ∝ p ( π [ n +1] | φ, θ ) f ( φ, θ ) Giulia Cereda () Short title October 8, 2015 25 / 33

1 LR = � � 1 − A n +1+Θ | Π [ n +1] = π [ n +1] E 1 − A By defining the random variable Φ = n n +1+Θ we can write the LR as n LR = E (Φ | Π [ n +1] = π [ n +1] ) . We are interested in the distribution of Φ , Θ | Π [ n +1] p ( φ, θ | π [ n +1] ) ∝ p ( π [ n +1] | φ, θ ) f ( φ, θ ) Giulia Cereda () Short title October 8, 2015 25 / 33

Log likelihood with φ and θ log 10 p ( π [ n +1] | φ, θ ) 0 −80 − 1 0 0 − 6 − 4 0 0.55 −20 − 4 . 6 0 5 1 7 − 2 . 9 9 5 7 3 2 0.50 ( φ MLE , θ MLE ) φ ● 0.45 −20 0.40 −40 −80 −60 −100 150 200 250 θ Dutch Y-STR database, 7 loci, N=18,925 Giulia Cereda () Short title October 8, 2015 26 / 33

Log likelihood with φ and θ log 10 p ( π [ n +1] | φ, θ ) logL( φ , θ | π [n + 1] ) −100 0 −80 −80 −60 − 1 0 − 1 ( φ MLE , θ MLE )) 0 logN (( φ MLE , θ MLE ) , I obs − 6 − 4 0 −40 95% confidence interval 99% confidence interval 0.55 −20 −20 − 4 . 6 0 5 1 7 − 2 . 9 9 5 7 3 2 0.50 ( φ MLE , θ MLE ) φ ● −2.995732 0.45 −4.60517 −20 −20 0.40 −40 −40 −80 −60 −80 −60 −100 −100 150 200 250 θ Dutch Y-STR database, 7 loci, N=18,925 Giulia Cereda () Short title October 8, 2015 27 / 33

Log likelihood as a function of φ and θ logL( φ , θ | π [n + 1] ) 0 0 0 0 −80 −80 −60 − 1 1 − − 1 ( φ MLE , θ MLE )) logN (( φ MLE , θ MLE ) , I obs − 6 0 −40 −40 95% confidence interval 99% confidence interval 0.55 −20 −20 −4.60517 −2.995732 0.50 ( φ MLE , θ MLE ) φ ● −2.995732 0.45 −4.60517 − 2 0 −20 0.40 − −40 4 0 −80 −60 −60 −80 −100 1 0 0 − 150 200 250 θ p ( π [ n +1] | φ, θ ) ≈ N (( φ MLE , θ MLE ) , I − 1 MLE ) Giulia Cereda () Short title October 8, 2015 28 / 33

Log likelihood as a function of φ and θ logL( φ , θ | π [n + 1] ) 0 0 0 0 −80 −80 −60 − 1 1 − − 1 ( φ MLE , θ MLE )) logN (( φ MLE , θ MLE ) , I obs − 6 0 −40 −40 95% confidence interval 99% confidence interval 0.55 −20 −20 −4.60517 −2.995732 0.50 ( φ MLE , θ MLE ) φ ● −2.995732 0.45 −4.60517 − 2 0 −20 0.40 − −40 4 0 −80 −60 −60 −80 −100 1 0 0 − 150 200 250 θ p ( π [ n +1] | φ, θ ) ≈ N (( φ MLE , θ MLE ) , I − 1 MLE ) p ( φ, θ | π [ n +1] ) ∝ p ( π [ n +1] | φ, θ ) f ( φ, θ ) Giulia Cereda () Short title October 8, 2015 28 / 33

Log likelihood as a function of φ and θ logL( φ , θ | π [n + 1] ) 0 0 0 0 −80 −80 −60 − 1 − 1 − 1 ( φ MLE , θ MLE )) logN (( φ MLE , θ MLE ) , I obs − 6 0 −40 −40 95% confidence interval 99% confidence interval 0.55 −20 −20 −4.60517 −2.995732 0.50 ( φ MLE , θ MLE ) φ ● −2.995732 0.45 −4.60517 − 2 0 −20 0.40 −40 − 4 0 −80 −60 −60 −80 −100 1 0 0 − 150 200 250 θ p ( π [ n +1] | φ, θ ) ≈ N (( φ MLE , θ MLE ) , I − 1 MLE ) p ( φ, θ | π [ n +1] ) ∝ p ( π [ n +1] | φ, θ ) f ( φ, θ ) If the prior is smooth around the MLE then Giulia Cereda () Short title October 8, 2015 28 / 33

Log likelihood as a function of φ and θ logL( φ , θ | π [n + 1] ) 0 0 0 0 −80 −80 −60 − 1 1 − − 1 ( φ MLE , θ MLE )) logN (( φ MLE , θ MLE ) , I obs − 6 0 −40 −40 95% confidence interval 99% confidence interval 0.55 −20 −20 −4.60517 −2.995732 0.50 ( φ MLE , θ MLE ) φ ● −2.995732 0.45 −4.60517 − 2 0 −20 0.40 − −40 4 0 −80 −60 −60 −80 −100 1 0 0 − 150 200 250 θ p ( π [ n +1] | φ, θ ) ≈ N (( φ MLE , θ MLE ) , I − 1 MLE ) p ( φ, θ | π [ n +1] ) ∝ p ( π [ n +1] | φ, θ ) f ( φ, θ ) If the prior is smooth around the MLE then p ( φ, θ | π [ n +1] ) ≈ N (( φ MLE , θ MLE ) , I − 1 MLE ) . Giulia Cereda () Short title October 8, 2015 28 / 33

Log likelihood as a function of φ and θ logL( φ , θ | π [n + 1] ) 0 0 0 0 −80 −80 −60 − 1 1 − − 1 ( φ MLE , θ MLE )) logN (( φ MLE , θ MLE ) , I obs − 6 0 −40 −40 95% confidence interval 99% confidence interval 0.55 −20 −20 −4.60517 −2.995732 0.50 ( φ MLE , θ MLE ) φ ● −2.995732 0.45 −4.60517 − 2 0 −20 0.40 − −40 4 0 −80 −60 −60 −80 −100 1 0 0 − 150 200 250 θ p ( π [ n +1] | φ, θ ) ≈ N (( φ MLE , θ MLE ) , I − 1 MLE ) p ( φ, θ | π [ n +1] ) ∝ p ( π [ n +1] | φ, θ ) f ( φ, θ ) If the prior is smooth around the MLE then p ( φ, θ | π [ n +1] ) ≈ N (( φ MLE , θ MLE ) , I − 1 MLE ) . It follows that E (Φ | Π [ n +1] = π [ n +1] ) ≈ φ MLE . That is Giulia Cereda () Short title October 8, 2015 28 / 33

p ( φ, θ | π [ n +1] ) ≈ N (( φ MLE , θ MLE ) , I − 1 MLE ) . Giulia Cereda () Short title October 8, 2015 29 / 33

p ( φ, θ | π [ n +1] ) ≈ N (( φ MLE , θ MLE ) , I − 1 MLE ) . It follows that E (Φ | Π [ n +1] = π [ n +1] ) ≈ φ MLE . Giulia Cereda () Short title October 8, 2015 29 / 33