Why is posterior density proportional to prior density times likelihood function?

According to Bayes’ theorem, P(y|θ)P(θ)=P(θ|y)P(y). But according to my econometric text, it says that P(θ|y)P(y|θ)P(θ). Why is it like this? I don’t get why P(y) is ignored.

Answer

Pr(y), the marginal probability of y, is not “ignored.” It is simply constant. Dividing by Pr(y) has the effect of “rescaling” the Pr(y|θ)P(θ) computations to be measured as proper probabilities, i.e. on a [0,1] interval. Without this scaling, they are still perfectly valid relative measures, but are not restricted to the [0,1] interval.

Pr(y) is often “left out” because Pr(y)=Pr(y|θ)Pr(θ)dθ is often difficult to evaluate, and it is usually convenient enough to indirectly perform the integration via simulation.

Attribution
Source : Link , Question Author : bayes-problem , Answer Author : Sycorax

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