Exam Questions - Block 2: Proportionality factor

[SebastianVeuskens]

What is the proportionality factor that is so important in computing posterior distributions and why it is often considered not important?

[Fedeveronesi]

The proportionality factor is denoted by the symbol ∝ and it is used to link two expressions, when their ratio is a constant value. In general, it allows to eliminate constant terms from an expression, or to retrieve from it useful constant quantities for what we want to demonstrate. In particular, in Bayesian Statistics it's useful in order to pass from the prior distribution to the posterior (in particular, the posterior is always proportional to the product of the likelihood and the prior, since the denominator of the Bayes theorem can be computed from the data and , therefore, is constant). In other applications we can extract from the ∝ symbol some quantities in order to recognize the kernel of a certain distribution for the posterior.

[SebastianVeuskens]

I think she means with proportionality factor in this case:

$p(x)$

This is part of the formula of the Bayesian theorem:

$h(\theta|x) = \frac{p(x|\theta) g(\theta)}{p(x)}$

The denominator is a constant that only depends on the data.

I think she means maybe that this proportionality is not so important because it is not part of the kernel of the posterior distribution. Thus, it can often be omitted.

[Fedeveronesi]

Yes i think it's referred to the denominator of the bayes theorem. Thank you for the remark!