Mdthbs

Here there is my proposal to split your equation.

documentclass{article}

usepackage[margin=2cm]{geometry}

usepackage{mathtools,amssymb}



begin{document}



begin{equation}

begin{split}

Delta (h)_{T+S} & =prod_{j=1}^{N} left[frac {P(x^j,y^j|theta_{mathrm{paired}})} 

{P(x^j,y^j|theta_{mathrm{unpaired}})}right]cdot\

& cdot left[frac{P(bp)_{x^j,y^j} 

P(bp_{x^j,y^j}|pbp_{x^{j-1},y^{j+1}}) P(dl_{x^j}) P(dl_{x^j}|pdl_{x^{j-1}})P(dr_{y^j})P(dr_{y^j}|pdr_{y^{j+1}})}{P(sl) 

P(dl|sl) P(sr) P(dr|sr)}right]

end{split}

end{equation}

end{document}

The brackets are surplus here, the fraction is a group by itself, so you don't need extra grouping. If I use brackets, I will enclose the whole formula to the right of the = sign into a pair of brackets. I also used biggl[ and biggr] for easy splitting. Finally, as @Sebastiano suggested, use mathrm for text.

Note: you didn't tell us how thick the margins are, so I assumed 1in. If my answer is still too wide, uncomment the small block.

documentclass{article}

usepackage{amsmath}

usepackage[margin=1in]{geometry}



begin{document}



%{small

begin{equation}

begin{split} Delta (h)_{T+S} &= 

prod_{j=1}^{N} biggl[frac {P(x^j,y^j|theta_mathrm{paired})} 

{P(x^j,y^j|theta_mathrm{unpaired})} times{} \

&phantom{{}={}}frac{P(bp)_{x^j,y^j} 

P(bp_{x^j,y^j}|pbp_{x^{j-1},y^{j+1}}) P(dl_{x^j})  

P(dl_{x^j}|pdl_{x^{j-1}})P(dr_{y^j})P(dr_{y^j}|pdr_{y^{j+1}})}{P(sl) 

P(dl|sl) P(sr) P(dr|sr)}biggr]

end{split}

end{equation}

%}



end{document}

one more alternative, with use of mathtools and nccmath:

documentclass{article}

usepackage{geometry}

usepackage{mathtools, nccmath}



%---------------- show page layout. don't use in a real document!

usepackage{showframe}

renewcommandShowFrameLinethickness{0.15pt}

renewcommand*ShowFrameColor{color{red}}

%---------------------------------------------------------------%

begin{document}

    begin{equation}

medmath{

begin{multlined}

Delta (h)_{T+S}

    = prod_{j=1}^{N} Biggl[frac{P(x^j,y^jmidtheta_{mathrm{paired}})}

                                 {P(x^j,y^jmidtheta_{mathrm{unpaired}})}Biggr]\

    cdot Biggl[frac{P(bp)_{x^j,y^j} P(bp_{x^j,y^j}mid pbp_{x^{j-1},y^{j+1}}) P(dl_{x^j}) P(dl_{x^j}mid pdl_{x^{j-1}})P(dr_{y^j})P(dr_{y^j}|pdr_{y^{j+1}})}

                     {P(sl) P(dlmid sl) P(sr) P(drmid sr)}Biggr]

end{multlined}

        }

    end{equation}

end{document}

A way to put the equation on a single line, with the splitfrac command from mathtools``, themedmathcommand fromnccmath` and some adjustment for the delimiters:

documentclass{article}

usepackage[margin=2cm]{geometry}

usepackage{mathtools, nccmath, amssymb}



begin{document}



begin{equation}

Delta (h)_{T+S} =prod_{j=1}^{N} left[frac {P(x^j,y^jmidtheta_{mathrm{paired}})}

{P(x^j,y^jmidtheta_{mathrm{unpaired}})}right]cdot

 left[frac{medmath{splitfrac{P(bp)_{x^j,y^j}

Pbigl(bp_{x^j,y^j}mid pbp_{x^{j-1},y^{j+1}}bigr)P(dl_{x^j}) }{ P(dl_{x^j}mid pdl_{x^{j-1}})Pbigl(dr_{y^j}bigr)Pbigl(dr_{y^j}mid pdr_{y^{j+1}}bigr)rule[-1.5ex]{0pt}{1ex}}}}{P(sl)

P(dl|sl) P(sr) P(dr| sr)^{strut} }right]

end{equation}



end{document} 

With multiline one can insert \ at the locations where the line should be broken.

Multiletter subscripts ought not be set in math italics, the kerning is wrong. I suggest to warp them in mathrm{}

The fontsize change needs to be outside the equation and without {} following it. small was still a bit to large, but with footnotesize the equation fits even without cheating and changing the margin sizes.

documentclass{article}

usepackage{amsmath}



begin{document}



{

scriptsize

begin{multline}

Delta (h)_{T+S} =

prod_{j=1}^{N} left[frac {P(x^j,y^j|theta_{mathrm{paired}})} 

{P(x^j,y^j|theta_{mathrm{unpaired}})}right]cdot\

cdotleft[frac{P(bp)_{x^j,y^j} 

P(bp_{x^j,y^j}|pbp_{x^{j-1},y^{j+1}}) P(dl_{x^j})  

P(dl_{x^j}|pdl_{x^{j-1}})P(dr_{y^j})P(dr_{y^j}|pdr_{y^{j+1}})}{P(sl) 

P(dl|sl) P(sr) P(dr|sr)}right]

end{multline}

}



end{document}

Here's a multlined-based solution. It also uses mid rather than | to denote conditioning, and it "snugs up" the subscript terms to dr.

documentclass{article}

usepackage[a4paper,margin=2.5cm]{geometry} % make suitable page and margin choices

usepackage{mathtools} % for 'multlined' environment



begin{document}



begin{equation}

renewcommand{!}{mkern-2mu} % default: mkern-3mu

begin{multlined}

Delta(h)_{T+S} = prod_{j=1}^{N} biggl[

frac {P(x^j!,y^jmid theta_{mathrm{paired}})}{%

P(x^j!,y^jmid theta_{mathrm{unpaired}})}\

times

frac{P(bp)_{x^j!,y^j}

P(bp_{x^j!,y^j}mid pbp_{x^{j-1}!,y^{j+1}}) 

P(dl_{x^j}) P(dl_{x^j}mid pdl_{x^{j-1}})

P(dr_{!y^j})P(dr_{!y^j}mid pdr_{!y^{j+1}})}{%

P(sl) P(dlmid sl) P(sr) P(drmid sr)} biggr]

end{multlined}

end{equation}

end{document}