# Does my solution to an incorrect MAP problem mean anything?

I was doing maximum a posteriori estimation of the variance $$v\equiv \sigma^2$$ using samples $$\{x_n\}$$ from a normally distributed random variable $$X \sim \mathcal{N}(0,1)$$, where the variance has an inverse Gamma prior distribution.

The posterior for $$N$$ samples is proportional to this:

$$\begin{split} p(v|\{x_n\},a,b) &\propto \prod_{n=1}^N \mathcal{N}(x_n | 0, v) \Gamma^{-1}(v | a/2, b/2)\\ &= \prod_{n=1}^N \left[ (2\pi)^{-1/2} v^{-1/2} e^{-\frac{x_n^2}{2v}} \times \frac{(b/2)^{a/2}}{\Gamma(a/2)} v^{-a/2-1} e^{-\frac{b}{2v}} \right] \end{split}$$

Wait, this is wrong! It should actually be:

$$\begin{split} p(v|\{x_n\},a,b) &\propto \left[ \prod_{n=1}^N \mathcal{N}(x_n | 0, v) \right] \times \Gamma^{-1}(v | a/2, b/2)\\ &= \left[ \prod_{n=1}^N (2\pi)^{-1/2} v^{-1/2} e^{-\frac{x_n^2}{2v}} \right] \times \frac{(b/2)^{a/2}}{\Gamma(a/2)} v^{-a/2-1} e^{-\frac{b}{2v}} \end{split}$$

Essentially, I thought the product $$\prod_{n=1}^N$$ was over products of the Normal and inverse gamma distributions, but that's incorrect. When I took the log of the first (incorrect!) posterior and maximized it anyway, I got this:

$$\hat{\sigma}^2 \equiv \hat{v} = \frac{ \sum_n x_n^2 + N \cdot b }{(a + 3) N} \neq \frac{ \sum_n x_n^2 + b }{a + 2 + N}$$

The last expression is the correct MAP, and it's of course not the same as what I got.

Question: is the estimate that I got by making this mistake in the posterior absolute nonsense?

As far as I can tell, I unintentionally did this:

1. Mistakenly assumed this probabilistic model for pairs of random variables $$(x_n, v)$$, where $$x_n$$ are my data, but $$v$$ is not observed: $$p(x_n, v | a,b) = \mathcal{N}(x_n | 0,v) \times \Gamma^{-1}(v | a/2,b/2)$$
2. Wrote the likelihood (assuming independence of all pairs): $$p\left( \left\{(x_n,v)\right\}_{n=1}^N \middle| a,b \right) = \prod_{n=1}^N p(x_n, v | a,b)$$ Same as the first formula in my question, but now this is not a posterior, but rather a joint distribution.
3. And maximized the (log-)likelihood w.r.t that $$v$$ to obtain: $$\hat{v} = \arg\max_v \prod_{n=1}^N p(x_n, v | a,b)$$

Is such a process even valid? Did I estimate any meaningful quantity by doing this? Since the variance $$v$$ is not observed, this seems similar in spirit to compound distributions, but here I'm not integrating over $$v$$ and instead maximizing with respect to it...