Bayesian Perceptron: Why to marginalize over neuron's output instead of it's weights?

Question

I found a very interesting paper on the internet that tries to apply Bayesian inference with a gradient-free online-learning approach: Bayesian Perceptron: Towards fully Bayesian Neural Networks.

I would love to understand this work, but unfortunately I am reaching my limits with my Bayesian knowledge. Let us assume that we have the weights $\mathcal{w}$ of our model and observed the data $\mathcal{D}$. Using the Bayes rule, we obtain the posterior according to $$p(\mathcal{w}|D)=\frac{p(D|\mathcal{w})p(\mathcal{w})}{p(D)}$$.

In words: we update our prior belief over our weights by multiplying the prior with the likelihood and divide everything by the evidence. In order to calculate the true posterior, we would need to calculate the evidence by marginalizing over (intergrating out) our unknown parameters. This gives the integral $$p(D) = \int p(D|\mathbf{w})p(\mathbf{w})dw$$.

So far so good. Now I refer to the paper mentioned above. Here, the approach is presented exemplarily on a neuron whose weighted sum is called $a$, which is then given to the activation function $f(.)$. Moreover it is assumed that $\mathbf{w}\sim N (\mu_w, \mathbf{C}_w)$. Because of the linearity it can be exploited that also $\mathbf{a}\sim N (\mu_a, \mathbf{C}_a)$.

What I am confused about now is formula (14), which seems to show the compute the true posterior: $$p(w) = \int p(a, w|D_i)da = \int p(w|a, D_i)p(a|D_i)da$$

Why is $a$ integrated out here and not $w$? We want a distribution over $w$, don't we? But without marginalization over $w$ there is still uncertainty inside $w$

Glad about any help and food for thought;)

Answer 1

We want a distribution over $w$, don't we?

Yes. You want to obtain a distribution over the parameters, which models the uncertainty about the parameters. This distribution over the parameters can induce a probability distribution over the possible functions consistent with your data.

Why is $a$ integrated out here and not $w$?

This is just the definition of marginalization.

For simplicity, consider 2 discrete random variables $X$ and $Y$, which can take values $x_i$ and $y_i$, respectively, and for which you can have (at least conceptually) a joint distribution $p_{(X, Y)}(x, y)$. Then the marginal probability that $X$ takes the value $x_i$ independently of any outcome of the r.v. $Y$ (and this is the intuition behind the marginal distribution!) is given by

$${\displaystyle p_{X}(x_{i})=\sum _{j}p(x_{i},y_{j})}, \tag{1}\label{1}$$

So, in the case of equation 14 of the paper, there's the assumption that you have $p(w, a)$. Then $p(w)$ is a marginal distribution over $w$ (the parameters or weights of the neural network), independently of the values of the activations. Of course, $w$ and $a$ can be dependent or correlated, and that's why you also have $p(w, a)$, i.e. the probability distribution that describes how these two r.v.s vary together, but what you want to know is the probability distribution over $w$ independently of $a$, so you "marginalize out" the possible values of $a$ (not $w$).

Remember that $p(x, y) = p(x \mid y) p(y) = p(y \mid x) p(x)$ (this is known as the chain rule of probability for two r.v.s, which leads to the Bayes' theorem), so you could write equation (\ref{1}) as follows

$${\displaystyle p_{X}(x_{i})=\sum _{j}p(x_{i},y_{j})} = \sum _{j}p(x_{i} \mid y_{j}) p(y_j), \tag{2}\label{2}$$

In the case of real-valued r.v.s, you will have an integral rather than a summation, but the idea is roughly the same.

But without marginalization over $w$ there is still uncertainty inside $w$.

What we want to know is $p(w)$, i.e. a distribution over $w$, which models the uncertainty about the possible values that $w$ can take. So, once you know $p(w)$, depending on its shape, we will have more or less uncertainty about the possible values of $w$. For example, let's say $p(w)$ is a Gaussian distribution. Then, depending on the variance, you will have more or less uncertainty about the possible values of $w$.

Your doubt seems to be that you thought that marginalization is a way to not have uncertainty, but that's not true. So, the goal in Bayesian machine/deep learning is not to be certain about the values of the parameters, but to model the uncertainty about the values of the parameters.