Solved – Apply Bayes rule sequentially

bayesianprobability

I have

$$\pi(a) =P(a|b) = \frac{P(b|a)P(a)}{P(b)}$$

I would like to now update given some new information, $C=c$

Is it possible to write:

$$P(a|b,c) = \frac{P(b,c|a)P(a)}{P(b,c)} \stackrel{?}{=} \frac{P(c|a)\pi(a)}{P(c)} $$

You can write:

$$P(a,b,c) = P(a \vert b,c)P(b,c) = P(a \vert b,c)P(c \vert b)P(b)$$

or, also valid:

$$P(a,b,c) = P(c \vert a,b)P(a,b) = P(c \vert a,b)P(b \vert a)P(a)$$

Putting together both expressions:

$$P(a \vert b,c) = \frac{P(c \vert a,b)P(b \vert a)P(a)}{P(c \vert b)P(b)} = \frac{P(c \vert a,b) \pi(a)}{P(c \vert b)} $$

And if this new observation $c$ does not depend on the previous observation $b$ (i.e. $P(b,c) = P(b)P(c)$), you can write:

$$P(a \vert b,c) = \frac{P(c \vert a) \pi(a)}{P(c)} $$