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Add more notation explaination

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Markus Kaiser 2 years ago
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      dynamic_dirichlet_deep_gp.pdf
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      dynamic_dirichlet_deep_gp.tex

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dynamic_dirichlet_deep_gp.pdf

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dynamic_dirichlet_deep_gp.tex

@ -127,7 +127,7 @@ Each data point is generated by evaluating one of the $K$ latent functions and a
The assignment of the $\nth{n}$ data point to one of the functions is specified by the indicator vector $\mat{a_n} \in \Set*{0, 1}^K$, which has exactly one non-zero entry.
Our goal is to formulate simultaneous Bayesian inference on the functions $f^{\pix{k}}$ and the assignments $\mat{a_n}$.
For notational conciseness, we collect all $N$ inputs as $\mat{X} = \left(\mat{x_1}, \ldots, \mat{x_N}\right)$ and all outputs as $\mat{Y} = \left(\mat{y_1}, \ldots, \mat{y_N}\right)$.
For notational conciseness, we follow the GP related notation in \parencite{hensman_scalable_2015} and collect all $N$ inputs as $\mat{X} = \left(\mat{x_1}, \ldots, \mat{x_N}\right)$ and all outputs as $\mat{Y} = \left(\mat{y_1}, \ldots, \mat{y_N}\right)$.
We further denote the $\nth{k}$ latent function value associated with the $\nth{n}$ data point as $\rv{f_n^{\pix{k}}} = \Fun{f^{\pix{k}}}{\mat{x_n}}$ and collect them as $\mat{F^{\pix{k}}} = \left( \rv{f_1^{\pix{k}}}, \ldots, \rv{f_N^{\pix{k}}} \right)$ and $\mat{F} = \left( \mat{F^{\pix{1}}}, \ldots, \mat{F^{\pix{K}}} \right)$.
We refer to the $\nth{k}$ entry in $\mat{a_n}$ as $a_n^{\pix{k}}$ and denote $\mat{A} = \left(\mat{a_1}, \ldots, \mat{a_N}\right)$.
@ -149,7 +149,7 @@ Given this notation, the marginal likelihood of DAGP can be separated into the l
where $\sigma^{\pix{k}}$ is the noise level of the $\nth{k}$ Gaussian likelihood and $\Ind$ is the indicator function.
Since we assume the $K$ processes to be independent given the data and assignments, we place independent GP priors on the latent functions
$\Prob*{\mat{F} \given \mat{X}} = \prod_{k=1}^K \Gaussian*{\mat{F^{\pix{k}}} \given \Fun*{\mu^{\pix{k}}}{\mat{X}}, \Fun*{\K^{\pix{k}}}{\mat{X}, \mat{X}}}$.
$\Prob*{\mat{F} \given \mat{X}} = \prod_{k=1}^K \Gaussian*{\mat{F^{\pix{k}}} \given \Fun*{\mu^{\pix{k}}}{\mat{X}}, \Fun*{\K^{\pix{k}}}{\mat{X}, \mat{X}}}$ with mean function $\mu^{\pix{k}}$ and kernel $\K^{\pix{k}}$.
Our prior on the assignment process is composite.
First, we assume that the $\mat{a_n}$ are drawn independently from multinomial distributions with logit parameters $\mat{\alpha_n} = \left( \alpha_n^{\pix{1}}, \ldots, \alpha_n^{\pix{K}} \right)$.
One approach to specify $\mat{\alpha_n}$ is to assume them to be known a priori and to be equal for all data points~\parencite{lazaro-gredilla_overlapping_2012}.

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