How can I minimize floating point error when multiplying normal distribution PDFs?

If you multiply two normal distribution PDFs with means $$\mu_1$$ and $$\mu_2$$ and variances $$v_1$$ and $$v_2$$, then according to this page, the new mean is

$$\mu = \frac{\mu_1 v_2 + \mu_2 v_1}{v_1 + v_2}$$

and the new variance is

$$v = \frac{v_1 v_2}{v_1 + v_2}.$$

However, if I use those expressions directly to combine several distributions sequentially, the floating point errors quickly pile up. Is there an algorithm for doing this stably?

• Can you give a concrete example when "the floating point errors quick pile up"? Dec 18 '18 at 18:29

Let $$\mathcal N(u, \sigma)=\frac{1}{\sqrt{2\pi\sigma^2} } e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$$ , the normal distribution with mean $$u$$ and standard derivation $$\sigma$$. Suppose we are given $$n$$ Gaussian normal distribution $$\mathcal N_i(u_i, \sigma_i)$$ . What is the product of $$n$$ of them? It turns out,
$$\prod_{i=1}^n \mathcal N(\mu_i,\sigma_i)= s\mathcal N (\mu,\sigma)$$ where $$\frac1{\sigma^{2}} = \sum_{i=1}^n \frac1{\sigma_i^{2}}\,, \quad \frac\mu{\sigma^2} = \sum_{i=1}^n \frac{\mu_i}{\sigma_i^{2}}$$ and $$s = (2\pi)^{\frac{1-n}2} \frac{\sigma}{\prod_{i=1}^n \sigma_i}\exp\left(\frac12 \left(\frac{\mu^2}{\sigma^2} - \sum_{i=1}^n \frac{\mu_i^2}{\sigma_i^2}\right) \right)$$
Exercise. Prove the above formula for the product of $$n$$ normal distribution.