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I'm trying to align the equal and greater than signs underneath each other but can't seem to get them aligned properly (see image attached). How can I align them correctly? I have included my lines of code below.

begin{equation}

begin{aligned}

Ebig[widehat{V}_i^{j_1...j_i}|X_i^{j_1...j_i}big] = EBigg[maxBigg{h_i(X_i^{j_1...j_i}),frac{1}{b}sum_{j=1}^{b} widehat{V}_{i+1}^{j_1...j_ij}Bigg}|X_i^{j_1...j_i}Bigg] \



geq maxBigg{h_i(X_i^{j_1...j_i}),EBigg[frac{1}{b}sum_{j=1}^{b} widehat{V}_{i+1}^{j_1...j_ij}|X_i^{j_1...j_i}Bigg]Bigg} \ 



= maxBig{h_i(X_i^{j_1...j_i}),Ebig[widehat{V}_{i+1}^{j_1...j_ij}|X_i^{j_1...j_i}big]Big} \ 



geq maxBig{h_i(X_i^{j_1...j_i}),Ebig[V_{i+1}big(X_{i+1}^{j_1...j_i1}big)|X_i^{j_1...j_i}big]Big} \ 



= V_ibig(X_i^{j_1...j_i}big).

end{aligned}

end{equation}

In addition to providing & alignments points in five places, you should also (a) write max instead of max and (b) use large | symbols (by writing bigmvert and biggmvert). Do please replace all instances of ... with dots. I would also like to suggest that you use upright-E letters to denote the expectations operator. Do write bigl( and bigr), etc rather than just big( and big); that way, the spacing around the opening and closing "fence" symbols can be optimized by LaTeX. Last but not least, don't over-size the fence symbols; e.g., don't write Biggl[ when biggl[ does just as well -- or, arguably, even better!

documentclass{article}

usepackage{amsmath}

DeclareMathOperator{E}{E}

begin{document}



begin{equation}

begin{aligned}

Ebigl[,widehat{V}_i^{j_1dots j_i}bigmvert X_i^{j_1dots j_i}bigr] 

&= Ebiggl[maxbiggl{h_i(X_i^{j_1dots j_i}),frac{1}{b}sum_{j=1}^{b} 

   widehat{V}_{i+1}^{j_1dots j_ij}biggr} biggmvert X_i^{j_1dots j_i}biggr] \

&ge maxbiggl{h_i(X_i^{j_1dots j_i}),Ebiggl[frac{1}{b}sum_{j=1}^{b} 

   widehat{V}_{i+1}^{j_1dots j_ij}biggmvert X_i^{j_1dots j_i}biggr]biggr} \ 

&= maxbigl{h_i(X_i^{j_1dots j_i}),Ebigl[,widehat{V}_{i+1}^{j_1dots j_ij}

   bigmvert X_i^{j_1dots j_i}bigr]bigr} \ 

&ge maxbigl{h_i(X_i^{j_1dots j_i}),

   Ebigl[,V_{i+1}bigl(X_{i+1}^{j_1dots j_i1}bigr)

   bigmvert X_i^{j_1dots j_i}bigr] bigr} \ 

&= V_ibigl(X_i^{j_1dots j_i}bigr),.

end{aligned}

end{equation}

end{document}

you forgot on ampersands ... and max is probably operator max:

documentclass{article}

usepackage{mathtools}



begin{document}

    begin{equation}

begin{aligned}

Ebig[widehat{V}_i^{j_1...j_i}|X_i^{j_1...j_i}big] 

    & = EBigg[maxBigg{h_i(X_i^{j_1...j_i}),frac{1}{b}sum_{j=1}^{b}

            widehat{V}_{i+1}^{j_1...j_ij}Bigg}|X_i^{j_1...j_i}Bigg] \

    & geq maxBigg{h_i(X_i^{j_1...j_i}),EBigg[frac{1}{b}sum_{j=1}^{b}

            widehat{V}_{i+1}^{j_1...j_ij}|X_i^{j_1...j_i}Bigg]Bigg} \

    & = maxBig{h_i(X_i^{j_1...j_i}),E

            big[widehat{V}_{i+1}^{j_1...j_ij}|X_i^{j_1...j_i}big]Big} \

    & geq maxBig{h_i(X_i^{j_1...j_i}),E

            big[V_{i+1}big(X_{i+1}^{j_1...j_i1}big)|X_i^{j_1...j_i}big]Big} \

    & = V_ibig(X_i^{j_1...j_i}big).

end{aligned}

    end{equation}

end{document}

Another solution with DeclarePairedDelimiter and its variants, from mathtools: this allows us to declare the E and Set command which can take an optional argument, chosen in big, Big, bigg, Bigg, which implicitly adds a pair of bigl ... bigr, &c., around the delimiters. Alternatively, there is a star version, which adds a pair of left ... right. Furthermore, the vertical line is translated into a given command, which adapts its size to the delimiters size. Note that in the first two lines, I used smaller delimiters, which I thought more convenient.

documentclass{article}

usepackage[utf8]{inputenc}%

usepackage{mathtools}



providecommandgiven{}



DeclarePairedDelimiterXPPE[1]{mathrm E}{}{%

renewcommandgiven{nonscript:delimsizevertnonscript:mathopen{}}

#1}

DeclarePairedDelimiterXSet[1]{}{%

renewcommandgiven{nonscript:delimsizevertnonscript:mathopen{}}

#1}

begin{document}



begin{equation}

begin{aligned}

E[big]{widehat{V}_i^{j_1...j_i}given X_i^{j_1...j_i}} & = E[bigg]{maxSet[bigg]{{h_i(X_i^{j_1...j_i}),frac{1}{b}sum_{j=1}^{b} widehat{V}_{i+1}^{j_1...j_ij}}given X_i^{j_1...j_i}} \

 & geq maxSet[bigg]{h_i(X_i^{j_1...j_i}), E[bigg]{frac{1}{b}sum_{j=1}^{b} widehat{V}_{i+1}^{j_1...j_ij}given X_i^{j_1...j_i}}} \

 & = maxSet[Big]{h_i(X_i^{j_1...j_i}),E[big]{widehat{V}_{i+1}^{j_1...j_ij}given X_i^{j_1...j_i}}} \

 & geq maxSet[Big]{h_i(X_i^{j_1...j_i}),E[big]{V_{i+1}bigl(X_{i+1}^{j_1...j_i1}bigr)given X_i^{j_1...j_i}}} \

 & = V_ibigl(X_i^{j_1...j_i}bigr).

end{aligned}

end{equation}



end{document}