Question

We now take the following energy function Eq. (1), which defines a joint distribution over x and y given by Eq. (2). In Eq. (1), the -?x_iy_i terms have the desired effect of giving a lower energy (thus higher probability) when x_i and y_i have the same sign and a higher energy when they have the opposite sign; the -?x_ix_j terms guarantee that the energy is lower when the ground truth pixels have the same sign than when they have the opposite sign; the hx_i terms have the effect of biasing the model towards pixel values that have one particular sign in preference to the other. E(x, y) = h ?_i x_i - ? ?_{{i,j}} x_ix_j - ? ?_i x_iy_i (1) p(x, y) = 1/Z exp{-E(x, y)} (2) (a) [2 mark] Consider the use of iterated conditional modes (ICM) to minimize the energy function given by Eq. (1). Write down an expression for the change in the values of the energy associated with the two states of a particular variable x_k, with all other variables held fixed, and show that it depends only on quantities that are local to x_k in the graph. (b) [1 mark] Consider a particular case of the energy function given by Eq. (1) in which the coefficients ? = h = 0. Show that the most probable configuration of the latent variables is given by x_i = y_i for all i.

          We now take the following energy function Eq. (1), which defines a joint distribution over x and y given by Eq. (2). In Eq. (1), the -?x_iy_i terms have the desired effect of giving a lower energy (thus higher probability) when x_i and y_i have the same sign and a higher energy when they have the opposite sign; the -?x_ix_j terms guarantee that the energy is lower when the ground truth pixels have the same sign than when they have the opposite sign; the hx_i terms have the effect of biasing the model towards pixel values that have one particular sign in preference to the other.

E(x, y) = h ?_i x_i - ? ?_{{i,j}} x_ix_j - ? ?_i x_iy_i (1)

p(x, y) = 1/Z exp{-E(x, y)} (2)

(a) [2 mark] Consider the use of iterated conditional modes (ICM) to minimize the energy function given by Eq. (1). Write down an expression for the change in the values of the energy associated with the two states of a particular variable x_k, with all other variables held fixed, and show that it depends only on quantities that are local to x_k in the graph.

(b) [1 mark] Consider a particular case of the energy function given by Eq. (1) in which the coefficients ? = h = 0. Show that the most probable configuration of the latent variables is given by x_i = y_i for all i.

We now take the following energy function Eq. (1), which defines a joint distribution over x and y given by Eq. (2). In Eq. (1), the -?xiyi terms have the desired effect of giving a lower energy (thus higher probability) when xi and yi have the same sign and a higher energy when they have the opposite sign; the -?xixj terms guarantee that the energy is lower when the ground truth pixels have the same sign than when they have the opposite sign; the hxi terms have the effect of biasing the model towards pixel values that have one particular sign in preference to the other.

E(x, y) = h ?i xi - ? ?i,j xixj - ? ?i xiyi (1)

p(x, y) = 1/Z exp-E(x, y) (2)

(a) [2 mark] Consider the use of iterated conditional modes (ICM) to minimize the energy function given by Eq. (1). Write down an expression for the change in the values of the energy associated with the two states of a particular variable xk, with all other variables held fixed, and show that it depends only on quantities that are local to xk in the graph.

(b) [1 mark] Consider a particular case of the energy function given by Eq. (1) in which the coefficients ? = h = 0. Show that the most probable configuration of the latent variables is given by xi = yi for all i.

Added by Linda W.

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