System and method for medical predictive models using likelihood gamble pricing
View Patent ↗A method for predicting survival rates of medical patients includes providing a set D of survival data for a plurality of medical patients, providing a regression model having an associated parameter vector β, providing an example x 0 of a medical patient whose survival probability is to be classified, calculating a parameter vector {circumflex over (β)} that maximizes a log-likelihood function of β over the set of survival data, l(β|D), wherein the log likelihood l(β|D) is a strictly concave function of β and is a function of the scalar xβ, calculating a weight w 0 for example x 0 , calculating an updated parameter vector β* that maximizes a function l(β|D∪{(y 0 ,x 0 ,w 0 )}), wherein data points (y 0 ,x 0 ,w 0 ) augment set D, calculating a fair log likelihood ratio λ f from {circumflex over (β)} and β* using λ f =λ(β*|x 0 )+sign(λ({circumflex over (β)}|x 0 )){l({circumflex over (β)}|D)−l(β*|D)}, and mapping the fair log likelihood ratio λ f to a fair price y 0 f , wherein said fair price is a probability that class label y 0 for example x 0 has a value of 1.
1. A method for predicting survival rates of medical patients, said method comprising the steps of:
providing a set D of survival data for a plurality of medical patients having a same condition;
providing a regression model, said model having an associated parameter vector β;
providing an example x 0 of a medical patient whose survival probability is to be classified;
calculating a parameter vector {circumflex over (β)} that maximizes a log-likelihood function of β over the set of survival data, l(β|D), wherein the log likelihood l(β|D) is a strictly concave function of β and is a function of the scalar xβ;
calculating a weight w 0 for example x 0 ;
calculating an updated parameter vector β* defined as the parameter vector β that maximizes a function l(β|D∪{(y 0 ,x 0 ,w 0 )}), wherein data points (y 0 ,x 0 ,w 0 ) augment said set D;
calculating a fair log likelihood ratio λ f from {circumflex over (β)} and β* using λ f =λ(β*|x 0 )+sign(λ({circumflex over (β)}|x 0 )){l({circumflex over (β)}|D)−l(β*|D)}; and
mapping the fair log likelihood ratio λ f to a fair price y 0 f , wherein said fair price is a probability that class label y 0 for example x 0 has a value of 1.
2. The method of claim 1 , wherein said weight w 0 is calculated from
w
◇
=
-
∂
∂
(
x
◇
β
)
λ
(
β
|
x
◇
)
∂
∂
(
x
◇
β
)
l
(
β
|
(
y
◇
,
x
◇
,
w
◇
=
1
)
)
|
β
=
β
^
,
wherein λ(β|x 0 ) is a log-likelihood ratio of a likelihood that a class label y 0 as a value of 1 over a likelihood that a class label y 0 has a value of 0, wherein said log-likelihood-ratio is an affine function of the scalar xβ.
3. The method of claim 2 , wherein said regression model is a logistic regression model with a probability of label y being 1 is
p
(
y
=
1
|
x
,
β
)
=
1
1
+
exp
(
-
λ
(
β
|
x
)
)
,
wherein
λ
(
β
/
x
)
=
x
β
.
4. The method of claim 3 , wherein said log-likelihood of β is
l
(
β
|
D
)
=
∑
i
=
1
N
w
i
{
(
y
i
-
1
)
x
i
β
-
log
(
1
+
exp
(
-
x
i
β
)
)
}
.
5. The method of claim 3 , wherein said weight w 0 is calculated from
w
◇
=
-
sign
(
x
◇
β
^
)
y
◇
-
1
+
1
1
+
exp
(
x
◇
β
^
)
.
6. The method of claim 3 , wherein said fair log likelihood ratio λ f is λ f =x 0 β*sign(x 0 {circumflex over (β)})└l({circumflex over (β)}|D)−l(β*|D)┘.
7. The method of claim 3 , wherein said fair price is
y
◇
f
=
1
1
+
exp
(
-
λ
f
)
.
8. The method of claim 2 , wherein said regression model is a Gaussian regression model with two clusters having a Gaussian distribution for either class, N(0,σ 2 ) and N(1,σ 2 ), wherein σ 2 is a standard deviation.
9. The method of claim 8 , wherein said log-likelihood of β is
l
(
β
|
D
)
=
-
∑
i
=
1
N
w
i
(
x
i
β
-
y
i
)
2
/
σ
2
.
10. The method of claim 8 , wherein said weight w 0 is calculated from
w
◇
=
-
sign
(
2
x
◇
β
^
-
1
)
x
◇
β
^
-
y
◇
.
11. The method of claim 8 , wherein said fair log likelihood ratio λ f is
λ
f
=
log
p
(
y
=
1
|
x
,
β
)
p
(
y
=
0
|
x
,
β
)
=
(
2
x
β
-
1
)
/
σ
2
.
12. The method of claim 8 , wherein said fair price is
y
♦
f
=
1
+
λ
f
σ
2
/
2
=
x
♦
β
*
+
sign
(
2
x
♦
β
^
-
1
)
2
(
l
(
β
^
|
D
)
-
l
(
β
*
|
D
)
)
.
13. The method of claim 1 , wherein said weight w 0 =2, and said updated parameter vector β* is determined by maximizing l(β|D∪{(1,x 0 ,1),(0,x 0 ,1)}), wherein (1,x 0 ,1),(0,x 0 ,1) are data points augmenting said set D.
14. A method for predicting survival rates of medical patients, said method comprising the steps of:
providing a set D of survival data for a plurality of medical patients having a same condition;
providing a regression model, said model having an associated parameter vector β;
providing an example x 0 of a medical patient whose survival probability is to be classified;
calculating a first parameter
l
1
=
max
β
l
(
β
|
D
⋃
{
(
1
,
x
♦
,
1
)
}
)
that maximizes a log-likelihood function of β over the set of survival data, l(β|D) augmented by a data point (1,x 0 ,1), wherein the log likelihood l(β|D) is a strictly concave function of β and is a function of the scalar xβ;
calculating an second parameter
l
0
=
max
β
l
(
β
|
D
⋃
{
(
0
,
x
♦
,
1
)
}
)
that maximizes a log-likelihood function of β over the set of survival data, l(β|D) augmented by a data point (0,x 0 ,1);
calculating a fair log likelihood ratio λ f from λ f =l 1 −l 0 ; and
mapping the fair log likelihood ratio λ f to a fair price y 0 f , wherein said fair price is a probability that class label y 0 for example x 0 has a value of 1.
15. A non-transitory program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform the method steps for predicting survival rates of medical patients, said method comprising the steps of:
providing a set D of survival data for a plurality of medical patients having a same condition;
providing a regression model, said model having an associated parameter vector fi;
providing an example x:, of a medical patient whose survival probability is to be classified;
calculating a parameter vector/) that maxirnizes a log-likelihood function of fi over the set of survival data, I(/?ID), wherein the log likelihood t(./]ID) is a strictly concave function of fl and is a function of the scalar aft;
calculating a weight w: for example x ; calculating an updated parameter vector fi* defined as the parameter vector fi that maximizes a function t(fll D t.9{(3′,:r˜, ˜1;:)}), wherein data points (y: ,x˜, ˜.;) augment said set D;
calculating a fair log likelihood ratio 2 from /˜ and fi* using
2c:=d.(fl* [ a′.)+sign(/˜(fl [ a.˜)˜l(/] I D)−l(fl* t D)}; and
mapping the fair log likelihood ratio 2f to a fair price Y;!l, wherein said fair price is a probability that class label y+ for example x:> has a value of 1.
16. The computer readable program storage device of claim 15 , wherein said weight w 0 is calculated from
w
♦
=
-
∂
∂
(
x
♦
β
)
λ
(
β
|
x
♦
)
∂
∂
(
x
♦
β
)
l
(
β
|
(
y
♦
,
x
♦
,
w
♦
=
1
)
)
|
β
=
β
^
,
wherein λ(β|x 0 ) is a log-likelihood ratio of a likelihood that a class label y 0 has a value of 1 over a likelihood that a class label y 0 has a value of 0, wherein said log-likelihood-ratio is an affine function of the scalar xβ.
17. The computer readable program storage device of claim 16 , wherein said regression model is a logistic regression model with a probability of label y being 1 is
p
(
y
=
1
|
x
,
β
)
=
1
1
+
exp
(
-
λ
(
β
|
x
)
)
,
wherein
λ
(
β
|
x
)
=
x
β
.
18. The computer readable program storage device of claim 17 , wherein said log-likelihood of β is
l
(
β
|
D
)
=
∑
i
=
1
N
w
i
{
(
y
i
-
1
)
x
i
β
-
log
(
1
+
exp
(
-
x
i
β
)
)
}
.
19. The computer readable program storage device of claim 17 , wherein said weight w 0 is calculated from
w
♦
=
-
sign
(
x
♦
β
^
)
y
♦
-
1
+
1
1
+
exp
(
x
♦
β
^
)
.
20. The computer readable program storage device of claim 17 , wherein said fair log likelihood ratio λ f is λ f =x 0 β*+sign(x 0 {circumflex over (β)})└l({circumflex over (β)}|D)−l(β*|D)┘.
21. The computer readable program storage device of claim 17 , wherein said fair price is
y
♦
f
=
1
1
+
exp
(
-
λ
f
)
.
22. The computer readable program storage device of claim 16 , wherein said regression model is a Gaussian regression model with two clusters having a Gaussian distribution for either class, N(0,σ 2 ) and N(1,σ 2 ), wherein σ 2 is a standard deviation.
23. The computer readable program storage device of claim 22 , wherein said log-likelihood of β is
l
(
β
|
D
)
=
-
∑
i
=
1
N
w
i
(
x
i
β
-
y
i
)
2
/
σ
2
.
24. The computer readable program storage device of claim 22 , wherein said weight w 0 is calculated from
w
♦
=
-
sign
(
2
x
♦
β
^
-
1
)
x
♦
β
^
-
y
♦
.
25. The computer readable program storage device of claim 22 , wherein said fair log likelihood ratio λ f is
λ
f
=
log
p
(
y
=
1
|
x
,
β
)
p
(
y
=
0
|
x
,
β
)
=
(
2
x
β
-
1
)
/
σ
2
.
26. The computer readable program storage device of claim 23 , wherein said fair price is
y
♦
f
=
1
+
λ
f
σ
2
/
2
=
x
♦
β
*
+
sign
(
2
x
♦
β
^
-
1
)
2
(
l
(
β
^
|
D
)
=
l
(
β
*
|
D
)
)
.
27. The computer readable program storage device of claim 15 , wherein said weight w 0 =2, and said updated parameter vector β* is determined by maximizing l(β|D∪{(1,x 0 ,1),(0,x 0 ,1)}), wherein (1,x 0 ,1),(0,x 0 ,1) are data points augmenting said set D.