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tisPrimeTime
tisPrimeTime
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Suppose I have the joint distribution:

\begin{align} p(\mathbf{x}) = p(x_1, ... , x_n) \end{align}

The maximum a posteriori (MAP) solution is given by:

\begin{align} \mathbf{x}_{MAP} = \arg \min p(\mathbf{x}) \end{align}\begin{align} \mathbf{x}_{MAP} = \arg \max p(\mathbf{x}) \end{align}

Intuitively I believe that finding the maximums of each marginal should not always correspond to $\mathbf{x}_{MAP}$. That is,

\begin{align} \mathbf{x}_{MAP_1} \neq \max_{x_1}(p(x_1))\\ \mathbf{x}_{MAP_2} \neq \max_{x_2}(p(x_2)) \\ \vdots \\ \mathbf{x}_{MAP_n} \neq \max_{x_n}(p(x_n)) \end{align}

I believe this should be the case, HOWEVER I can't convince myself fully. Every joint distribution I imagine, tends to satisfy this trivially (e.g. the mode of a multi-variate Gaussian, can be found by considering the mode for each of the marginals).

  1. Is there/are there counter-examples to prove this point?
  2. Or perhaps is my intuition wrong? That is, the mode of the marginals always corresponds to the MAP mode.
  3. Are there sources on the internet which explore this issue in depth (since no matter what I Google I cannot find an answer on this)?
  4. Are there any conditions to declare when this is / isn't true?

Suppose I have the joint distribution:

\begin{align} p(\mathbf{x}) = p(x_1, ... , x_n) \end{align}

The maximum a posteriori (MAP) solution is given by:

\begin{align} \mathbf{x}_{MAP} = \arg \min p(\mathbf{x}) \end{align}

Intuitively I believe that finding the maximums of each marginal should not always correspond to $\mathbf{x}_{MAP}$. That is,

\begin{align} \mathbf{x}_{MAP_1} \neq \max_{x_1}(p(x_1))\\ \mathbf{x}_{MAP_2} \neq \max_{x_2}(p(x_2)) \\ \vdots \\ \mathbf{x}_{MAP_n} \neq \max_{x_n}(p(x_n)) \end{align}

I believe this should be the case, HOWEVER I can't convince myself fully. Every joint distribution I imagine, tends to satisfy this trivially (e.g. the mode of a multi-variate Gaussian, can be found by considering the mode for each of the marginals).

  1. Is there/are there counter-examples to prove this point?
  2. Or perhaps is my intuition wrong? That is, the mode of the marginals always corresponds to the MAP mode.
  3. Are there sources on the internet which explore this issue in depth (since no matter what I Google I cannot find an answer on this)?
  4. Are there any conditions to declare when this is / isn't true?

Suppose I have the joint distribution:

\begin{align} p(\mathbf{x}) = p(x_1, ... , x_n) \end{align}

The maximum a posteriori (MAP) solution is given by:

\begin{align} \mathbf{x}_{MAP} = \arg \max p(\mathbf{x}) \end{align}

Intuitively I believe that finding the maximums of each marginal should not always correspond to $\mathbf{x}_{MAP}$. That is,

\begin{align} \mathbf{x}_{MAP_1} \neq \max_{x_1}(p(x_1))\\ \mathbf{x}_{MAP_2} \neq \max_{x_2}(p(x_2)) \\ \vdots \\ \mathbf{x}_{MAP_n} \neq \max_{x_n}(p(x_n)) \end{align}

I believe this should be the case, HOWEVER I can't convince myself fully. Every joint distribution I imagine, tends to satisfy this trivially (e.g. the mode of a multi-variate Gaussian, can be found by considering the mode for each of the marginals).

  1. Is there/are there counter-examples to prove this point?
  2. Or perhaps is my intuition wrong? That is, the mode of the marginals always corresponds to the MAP mode.
  3. Are there sources on the internet which explore this issue in depth (since no matter what I Google I cannot find an answer on this)?
  4. Are there any conditions to declare when this is / isn't true?
Source Link
tisPrimeTime
tisPrimeTime
  • 555
  • 4
  • 15

MAP versus Component-Wise Maximum Marginal

Suppose I have the joint distribution:

\begin{align} p(\mathbf{x}) = p(x_1, ... , x_n) \end{align}

The maximum a posteriori (MAP) solution is given by:

\begin{align} \mathbf{x}_{MAP} = \arg \min p(\mathbf{x}) \end{align}

Intuitively I believe that finding the maximums of each marginal should not always correspond to $\mathbf{x}_{MAP}$. That is,

\begin{align} \mathbf{x}_{MAP_1} \neq \max_{x_1}(p(x_1))\\ \mathbf{x}_{MAP_2} \neq \max_{x_2}(p(x_2)) \\ \vdots \\ \mathbf{x}_{MAP_n} \neq \max_{x_n}(p(x_n)) \end{align}

I believe this should be the case, HOWEVER I can't convince myself fully. Every joint distribution I imagine, tends to satisfy this trivially (e.g. the mode of a multi-variate Gaussian, can be found by considering the mode for each of the marginals).

  1. Is there/are there counter-examples to prove this point?
  2. Or perhaps is my intuition wrong? That is, the mode of the marginals always corresponds to the MAP mode.
  3. Are there sources on the internet which explore this issue in depth (since no matter what I Google I cannot find an answer on this)?
  4. Are there any conditions to declare when this is / isn't true?