I don't know what I am missing in following in my understanding. Whether it is my mathematica code that is incorrect or my mathematical skill is short.
Gamma distribution is the conjugate prior when the likelihood function is Poisson distribution. By this, I understood the following:
$$\overbrace{f(x|y)}^{\rm Gamma(\alpha+x, \beta + 1)}=\frac{\overbrace{f(y|x)}^{\rm Poisson(\lambda)} \cdot \overbrace{f(x)}^{\rm Gamma(\alpha, \beta)}}{\underbrace{f(y)}_{NB(\alpha, \frac{1}{1+\beta}\:)}} $$
If my understanding above is correct then why the LHS is not equal to RHS in following Mathematica code ?
Remove["Global`*"]
f = FullSimplify[(PDF[PoissonDistribution[y],
x])*(PDF[GammaDistribution[a, b], y])/(PDF[
NegativeBinomialDistribution[a, 1/(1 + b)], x])];
g = PDF[GammaDistribution[a + x, (b + 1)], y];
a = 1;
b = 2;
x = 5;
Plot[{f, g}, {y, 1, 10}, PlotLegends -> "Expressions"]
Correction on Wikipedia page conjugate prior
$$\overbrace{f(\lambda|x)}^{\text{Gamma}(k+x,\frac{\theta}{\theta + 1})}=\frac{\overbrace{f(x|\lambda)}^{\text{Poisson}(\lambda)} \cdot \overbrace{f(\lambda)}^{\text{Gamma}(k, \theta)}}{\underbrace{f(y)}_{\text{NB}(k, \frac{1}{1+\theta}\:)}} $$
and
$$\overbrace{f(\lambda|x)}^{\text{Gamma}(\alpha+x,\beta+1)}=\frac{\overbrace{f(x|\lambda)}^{\text{Poisson}(\lambda)} \cdot \overbrace{f(\lambda)}^{\text{Gamma}(\alpha, \beta)}}{\underbrace{f(y)}_{\text{NB}(\alpha, \frac{\beta}{\beta +1}\:)}} $$
i.e. the two Negative Binomial distributions are mistakenly swapped on wiki page.