To strengthen my understanding of fundamental probability theory, I am working my way through Professor Aaron Hertzmann's Introduction to Bayesian Learning course notes. Section 3.8 of these course notes includes the following exercise.
Original Problem
Derive a formula for $P(\mathbf{A})$, assuming you know $P(\mathbf{A}|\mathbf{B}_1,\mathbf{C})$ and $P(\mathbf{A}|\mathbf{B}_2,\mathbf{C})$ and $P(\mathbf{B}_1|\mathbf{C})+P(\mathbf{B}_2|\mathbf{C})=1$.
My Attempt
So, my goal is to express $P(\mathbf{A})$ in terms of known quantities.
$$ \begin{align} P(\mathbf{B}_1|\mathbf{C}) &+ P(\mathbf{B}_2|\mathbf{C}) &=& 1 && \text{}\\ P(\mathbf{B}_1|\mathbf{C})P(\mathbf{C}) &+ P(\mathbf{B}_2|\mathbf{C})P(\mathbf{C}) &=& P(\mathbf{C}) && \text{Algebra}\\ P(\mathbf{B}_1,\mathbf{C}) &+ P(\mathbf{B}_2,\mathbf{C}) &=& P(\mathbf{C}) && \text{Product Rule}\\ P(\mathbf{B}_1,\mathbf{C}|\mathbf{A}) &+ P(\mathbf{B}_2,\mathbf{C}|\mathbf{A}) &=& P(\mathbf{C}|\mathbf{A}) && \text{Condition on } \mathbf{A}\\ \frac{P(\mathbf{A}|\mathbf{B}_1,\mathbf{C})P(\mathbf{B}_1,\mathbf{C})}{P(\mathbf{A})} &+ \frac{P(\mathbf{A}|\mathbf{B}_2,\mathbf{C})P(\mathbf{B}_2,\mathbf{C})}{P(\mathbf{A})} &=& P(\mathbf{C}|\mathbf{A}) && \text{Bayes Theorem} \\ \end{align} $$
$$ \begin{align} &\frac{P(\mathbf{A}|\mathbf{B}_1,\mathbf{C})P(\mathbf{B}_1,\mathbf{C}) + P(\mathbf{A}|\mathbf{B}_2,\mathbf{C})P(\mathbf{B}_2,\mathbf{C})}{P(\mathbf{A})} &=& P(\mathbf{C}|\mathbf{A}) && \text{Algebra}~~~~~~~~~~~~~~\\ &\frac{P(\mathbf{A}|\mathbf{B}_1,\mathbf{C})P(\mathbf{B}_1,\mathbf{C}) + P(\mathbf{A}|\mathbf{B}_2,\mathbf{C})P(\mathbf{B}_2,\mathbf{C})}{P(\mathbf{C}|\mathbf{A})} &=& P(\mathbf{A}) && \text{Algebra}~~~~~~~~~~~~~~\\ \end{align} $$
Here is where I get stuck, because I don't know $P(\mathbf{C}|\mathbf{A})$, $P(\mathbf{B}_1,\mathbf{C})$, or $P(\mathbf{B}_2,\mathbf{C})$. It is not clear to me how I can express these probabilities in terms of the quantities I know.
These course notes are very much a work in progress. The author frequently writes TO-DO notes to himself about things to add and remove. Is it possible that this exercise is actually under-determined? Or am I simply going about this derivation the wrong way?
Similar Questions
When searching StackExchange, I found this similar question. However, it was still not directly applicable to my problem.
Tagging as Homework
Although this question is not homework for me, I am tagging it as homework anyway because: (1) it is very introductory in nature; (2) a similar question could conceivably be homework for someone else; and (3) I would be happy with hints.
