# (Why) is the Bayes factor not sensitive to the choice of prior distribution on hypotheses?

I learned that the posterior odds is the ratio of the two posterior probabilities of hypothesis:

\begin{align} PO[H_1:H_2] &= \frac{P(H_1|\text{data})}{P(H_2|\text{data})} \\ &= \frac{P(\text{data}|H_1)}{P(\text{data}|H_2)} * \frac{P(H_1)}{P(H_2)} \\ &= \text{Bayes factor} * \text{prior odds} \end{align}

And $$\text{BF}[H_1:H_2]=\frac{P(\text{data}|H_1)}{P(\text{data}|H_2)}$$

The answer reads that because of the above equation it does not depend on any prior probability of $$H_1$$ or $$H_2$$.

But in my understanding, it seems that if we set $$H_1$$ and $$H_2$$ differently(we can set the two priors as any distribution pairs) the Bayes factor would also vary, hence making the Bayes factor sensitive to the choice of prior. I cannot get my head around why Bayes factor is insensitive to prior?

It is easier to check the impact of the prior when using Harold Jeffreys' decomposition of the prior measure: $$\pi(\theta) = \varpi_1 \pi_1(\theta)\mathbb I_{H_1}(\theta) + \varpi_2 \pi_2(\theta)\mathbb I_{H_2}(\theta)$$ (assimilating the hypotheses to subsets of the parameter space). In that case, the Bayes factor writes $$\mathfrak B_{12}(x) = \dfrac{\int_{H_1} \pi_1(\theta) f_1(x|\theta)\,\text d\theta}{\int_{H_2} \pi_2(\theta) f_2(x|\theta)\,\text d\theta}$$ which shows that

1. the prior weights $$\varpi_1$$ and $$\varpi_2$$ have no influence on $$\mathfrak B_{12}(x)$$
2. the submodel priors $$\pi_1$$ and $$\pi_2$$ influence the value of $$\mathfrak B_{12}(x)$$

In your query we see that:

$$\text{Bayes factor} = \frac{P(\text{data}|H_1)}{P(\text{data}|H_2)}$$

Examining this expression we see that it does not contain the prior distribution and hence can't be sensitive to it. The posterior odds are, however, a different matter.

To the extent that value of the Bayes factor is conditional on $$H_1$$ and $$H_2$$, yes, it varies. However in no sense do $$H_1$$ and $$H_2$$ have a distribution there, it's just a deterministic function of two arguments. It's the priors which state at what rate particular values of $$H_1$$ and $$H_2$$ get fed into that function.

• You're effectively asking why a function e.g. $f(x)=3x$ not depend on a second distinct function $g(x)$. The answer is not deep. Commented Jan 28, 2021 at 15:10