How to introduce different costs by class in a binary logistic regression?
What is the form of the Negative Log-Likelihood Goal function in Logistic regression if we introduce different costs per class (e.g. the cost of erring class 1 is errc1, and class 2 is errc2)?
logistic-regression log-likelihood
asked Nov 19 '18 at 0:34
MrT77
114
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What is the form of the Negative Log-Likelihood Goal function in Logistic regression if we introduce different costs per class (e.g. the cost of erring class 1 is errc1, and class 2 is errc2)?
logistic-regression log-likelihood
asked Nov 19 '18 at 0:34
MrT77
114
add a comment |
What is the form of the Negative Log-Likelihood Goal function in Logistic regression if we introduce different costs per class (e.g. the cost of erring class 1 is errc1, and class 2 is errc2)?
logistic-regression log-likelihood
asked Nov 19 '18 at 0:34
MrT77
114
What is the form of the Negative Log-Likelihood Goal function in Logistic regression if we introduce different costs per class (e.g. the cost of erring class 1 is errc1, and class 2 is errc2)?
logistic-regression log-likelihood
logistic-regression log-likelihood
asked Nov 19 '18 at 0:34
MrT77
114
asked Nov 19 '18 at 0:34
MrT77
114
asked Nov 19 '18 at 0:34
MrT77
114
asked Nov 19 '18 at 0:34
MrT77
114
asked Nov 19 '18 at 0:34
MrT77
114
114
add a comment |
add a comment |
1 Answer
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The logistic regression hypothesis (sigmoid) is defined as:
$$h_w(x) = sigma(w^T x)$$
where $sigma(z) = frac{1}{1 + exp(-z)}$.
Logistic regression is fit by maximum likelihood, i.e. we want to maximize the probability of the observed data:
$$ arg,max ({mathbf{w}}) ;p(x^{(1)},y^{(1)}),p(x^{(2)},y^{(2)}),...,p(x^{(N)},y^{(N)})$$
which is equivalent to (by eliminating the constant priors):
$$ arg,max ({mathbf{w}}) ;p(y^{(1)}|x^{(1)})cdot p(y^{(2)}|x^{(2)})cdot...cdot p(y^{(N)}|x^{(N)})$$
By assumption, these conditional probabilities are the sigmoid, so our goal function (through MLE, and assuming independence) is:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{y_i}cdot (p(y_i=0|x_i))^{1-y_i}$$
If we introduce different costs, this is equivalent to weighting according to the correspondent class probabilities, e.g. if the cost of erring $C_2$ is twice the cost of erring $C_1$, this is like having another $C_2$ case in the training data. Thus, it's equivalent to doubling it's probability:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{C_1/C_0*y_i}cdot (p(y_i=0|x_i))^{(C_0/C_1)*(1-y_i)}$$
answered Dec 18 '18 at 13:35
MrT77
114
add a comment |
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1 Answer
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The logistic regression hypothesis (sigmoid) is defined as:
$$h_w(x) = sigma(w^T x)$$
where $sigma(z) = frac{1}{1 + exp(-z)}$.
Logistic regression is fit by maximum likelihood, i.e. we want to maximize the probability of the observed data:
$$ arg,max ({mathbf{w}}) ;p(x^{(1)},y^{(1)}),p(x^{(2)},y^{(2)}),...,p(x^{(N)},y^{(N)})$$
which is equivalent to (by eliminating the constant priors):
$$ arg,max ({mathbf{w}}) ;p(y^{(1)}|x^{(1)})cdot p(y^{(2)}|x^{(2)})cdot...cdot p(y^{(N)}|x^{(N)})$$
By assumption, these conditional probabilities are the sigmoid, so our goal function (through MLE, and assuming independence) is:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{y_i}cdot (p(y_i=0|x_i))^{1-y_i}$$
If we introduce different costs, this is equivalent to weighting according to the correspondent class probabilities, e.g. if the cost of erring $C_2$ is twice the cost of erring $C_1$, this is like having another $C_2$ case in the training data. Thus, it's equivalent to doubling it's probability:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{C_1/C_0*y_i}cdot (p(y_i=0|x_i))^{(C_0/C_1)*(1-y_i)}$$
answered Dec 18 '18 at 13:35
MrT77
114
add a comment |
The logistic regression hypothesis (sigmoid) is defined as:
$$h_w(x) = sigma(w^T x)$$
where $sigma(z) = frac{1}{1 + exp(-z)}$.
Logistic regression is fit by maximum likelihood, i.e. we want to maximize the probability of the observed data:
$$ arg,max ({mathbf{w}}) ;p(x^{(1)},y^{(1)}),p(x^{(2)},y^{(2)}),...,p(x^{(N)},y^{(N)})$$
which is equivalent to (by eliminating the constant priors):
$$ arg,max ({mathbf{w}}) ;p(y^{(1)}|x^{(1)})cdot p(y^{(2)}|x^{(2)})cdot...cdot p(y^{(N)}|x^{(N)})$$
By assumption, these conditional probabilities are the sigmoid, so our goal function (through MLE, and assuming independence) is:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{y_i}cdot (p(y_i=0|x_i))^{1-y_i}$$
If we introduce different costs, this is equivalent to weighting according to the correspondent class probabilities, e.g. if the cost of erring $C_2$ is twice the cost of erring $C_1$, this is like having another $C_2$ case in the training data. Thus, it's equivalent to doubling it's probability:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{C_1/C_0*y_i}cdot (p(y_i=0|x_i))^{(C_0/C_1)*(1-y_i)}$$
answered Dec 18 '18 at 13:35
MrT77
114
add a comment |
The logistic regression hypothesis (sigmoid) is defined as:
$$h_w(x) = sigma(w^T x)$$
where $sigma(z) = frac{1}{1 + exp(-z)}$.
Logistic regression is fit by maximum likelihood, i.e. we want to maximize the probability of the observed data:
$$ arg,max ({mathbf{w}}) ;p(x^{(1)},y^{(1)}),p(x^{(2)},y^{(2)}),...,p(x^{(N)},y^{(N)})$$
which is equivalent to (by eliminating the constant priors):
$$ arg,max ({mathbf{w}}) ;p(y^{(1)}|x^{(1)})cdot p(y^{(2)}|x^{(2)})cdot...cdot p(y^{(N)}|x^{(N)})$$
By assumption, these conditional probabilities are the sigmoid, so our goal function (through MLE, and assuming independence) is:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{y_i}cdot (p(y_i=0|x_i))^{1-y_i}$$
If we introduce different costs, this is equivalent to weighting according to the correspondent class probabilities, e.g. if the cost of erring $C_2$ is twice the cost of erring $C_1$, this is like having another $C_2$ case in the training data. Thus, it's equivalent to doubling it's probability:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{C_1/C_0*y_i}cdot (p(y_i=0|x_i))^{(C_0/C_1)*(1-y_i)}$$
answered Dec 18 '18 at 13:35
MrT77
114
The logistic regression hypothesis (sigmoid) is defined as:
$$h_w(x) = sigma(w^T x)$$
where $sigma(z) = frac{1}{1 + exp(-z)}$.
Logistic regression is fit by maximum likelihood, i.e. we want to maximize the probability of the observed data:
$$ arg,max ({mathbf{w}}) ;p(x^{(1)},y^{(1)}),p(x^{(2)},y^{(2)}),...,p(x^{(N)},y^{(N)})$$
which is equivalent to (by eliminating the constant priors):
$$ arg,max ({mathbf{w}}) ;p(y^{(1)}|x^{(1)})cdot p(y^{(2)}|x^{(2)})cdot...cdot p(y^{(N)}|x^{(N)})$$
By assumption, these conditional probabilities are the sigmoid, so our goal function (through MLE, and assuming independence) is:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{y_i}cdot (p(y_i=0|x_i))^{1-y_i}$$
If we introduce different costs, this is equivalent to weighting according to the correspondent class probabilities, e.g. if the cost of erring $C_2$ is twice the cost of erring $C_1$, this is like having another $C_2$ case in the training data. Thus, it's equivalent to doubling it's probability:
$$ J(w) = arg,max ({mathbf{w}}) prod_{i=1}^{N} ;(p(y_i=1|x_i))^{C_1/C_0*y_i}cdot (p(y_i=0|x_i))^{(C_0/C_1)*(1-y_i)}$$
answered Dec 18 '18 at 13:35
MrT77
114
answered Dec 18 '18 at 13:35
MrT77
114
answered Dec 18 '18 at 13:35
MrT77
114
answered Dec 18 '18 at 13:35
MrT77
114
114
add a comment |
add a comment |
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