Plate captions overlapping

Hadn't seen that library before, perhaps because it is a third-party library, and not discussed in the TikZ manual. The only documentation seems to be a short readme file, so I had a look at the code of the library itself.

The library file can be found by running kpsewhich tikzlibrarybayesnet.code.tex in a terminal, on my system it is found in /usr/local/texlive/2015/texmf-dist/tex/latex/tikz-bayesnet/tikzlibrarybayesnet.code.tex.

The \plate command is defined there as follows:

\newcommand{\plate}[4][]{ %
  \node[wrap=#3] (#2-wrap) {}; %
  \node[plate caption=#2-wrap] (#2-caption) {#4}; %
  \node[plate=(#2-wrap)(#2-caption), #1] (#2) {}; %
}

The caption is created by the second node, which uses the style plate caption. This style in turn is defined by

\tikzstyle{plate caption} = [caption, node distance=0, inner sep=0pt,
below left=5pt and 0pt of #1.south east]

So you can see that it is positioned relative to the south east corner of a node (#1), defined by the argument given to the style, i.e. #2-wrap in the definition of \plate. #2 here is turn the first mandatory argument to the \plate command, i.e. the first one after the options. The node #2-wrap is a node defined with the fit option, and is a node that encompasses all the nodes given in the second mandatory argument of \plate.

The actual frame of the \plate is drawn by the third \node in its definition, which is a node that encompasses the caption and the list of nodes in the \plate (also via the fit library).

One way of moving the caption around is to redefine the plate caption style. As the library is written that has to be done before each \plate command with \tikzset. That is, add something like

\tikzset{plate caption/.append style={below right=0pt and 0pt of #1.north west}

before the \plate. Using append style as above you don't have to include the rest of the style definition, only the position will be overwritten. If you want to modify this for just one \plate, put a pair of braces around \tikzset and \plate, i.e.

{ % opening brace
\tikzset{..}
\plate{..}{..}{..}
} % closing brace

To late in the evening for me to try to modify everything, so in the code below I've only moved the C classes caption. Hopefully you can figure out the rest yourself.

\documentclass[border=3mm]{standalone} % For LaTeX2e
\usepackage{tikz}
\usetikzlibrary{bayesnet}

\begin{document}

\begin{tikzpicture}

  % Define nodes
  \node[obs]             (y)      {$y_{i}^{(j)}$};
  \node[obs, right=of y]  (f)      {$f_{ik}^{(j)}$};
  \factor[above=of y,yshift=0.7cm]   {yf}    {Cat} {} {} ; %
  \factor[above=of f,yshift=0.7cm]   {rf}    {Cat} {} {} ; 


  \node[latent, left=of yf]  (pi)      {$\pi_{c}^{(j)}$};
  \node[latent, above=of pi, yshift=0.5cm]  (beta)   {$\beta_{c}^{(j)}$};
  \factor[above=of pi, yshift=0.4cm] {pi-f} {left:Dir} {} {} ; %

  \node[latent, above=of yf, xshift=0.8cm, yshift=3cm ]  (t)   {$t_i$};
  \node[latent, above=of t, yshift=0.5cm]  (p)   {$p$};
  \node[latent, right=of p, xshift=0.8cm]  (alpha)   {$\alpha$};
  \factor[above=of t, yshift=0.5cm] {t-f} {left:Dir} {} {} ; %
  \factor[right=of p, xshift=0.5cm] {p-f} {Dir} {} {} ; %

  \node[latent, right=of rf]  (d)      {$z_{k,c}^{(i)}$};
  \node[latent, above=of d, yshift=0.5cm]  (gamma)   {$\gamma_{0,c}^{(i)}$};
  \factor[above=of d, yshift=0.4cm] {d-f} {right:Dir} {} {} ; %


  % Connect the nodes
  \factoredge {gamma}  {d-f}  {d} ;
  \factoredge {alpha}  {p-f}  {p}; %
  \factoredge {p}  {t-f}  {t}; %
  \factoredge {beta}  {pi-f}   {pi} ;
  \factoredge {pi}  {yf}   {y} ;
  \factoredge {d}  {rf}   {f} ;

 \gate {y-gate} {(yf)(yf-caption)} {t}
 \gate {f-gate} {(rf)(rf-caption)} {t}

% Plates
\plate [inner sep=0.3cm, xshift=0.1cm, yshift=-0.2cm,color=red] {pA} {(f)(rf)(d)(d-f)} {$k \in K_i$}; %
{\tikzset{plate caption/.append style={above=5pt of #1.north west}}
\plate [inner sep=0.3cm, xshift=0.2cm, color=blue] {pC} {(beta)(pi)(d)(gamma)} {C classes};}
\plate [inner sep=0.7cm, xshift=0.3cm, yshift=-0.4cm, color=brown] {pT} {(t)(gamma)(d)(yf)(y)(rf)(f)} {N objects};
\plate [inner sep=0.2cm, xshift=-0.2cm] {pM} {(beta)(pi)(yf)(y)(rf)(f)} {M workers} ;
\plate [inner sep=0.2cm, xshift=-0.05cm, yshift=0.15cm] {pJi} {(y)(f)} {$J_i$} 

%minimum height=1.5cm, minimum width=3.0cm, yshift=0.4cm,
\end{tikzpicture}
 \end{document}