# Significant Test for Joint Probability Distributions

I am familiar with significant tests and distance measure between standard probability distributions. However, I am looking for a significance test between joint probability distributions (JPD) for a set of ecological data. (The data is discrete and categorical).

As an example, we have the following two joint probability distributions, $${P_1}$$ and $${P_2}$$. (Categories in rows; attributes in columns).

$${P_1}= \begin{pmatrix} 0.203 & 0.203 & 0.020 \\ 0.033 & 0.229 & 0.033 \\ 0.059 & 0.072 & 0.150 \\ \end{pmatrix}$$

$${P_2}= \begin{pmatrix} 0.159 & 0.051 & 0.025 \\ 0.080 & 0.239 & 0.051 \\ 0.040 & 0.188 & 0.167 \\ \end{pmatrix}$$

with the respective sample sizes being $${N_{P_1}=153}$$; $${N_{P_2}=276}$$

The respective, if not obvious, hypotheses are: $${{H_0}: {P_1}={P_2}}$$ and $${{H_A}: {P_1}\neq{P_2}}$$.

Is there a significance test for this situation? My assumption there is such a test (after all, there seems to be a test for everything) and that I have simply not come across yet. Any pointers would be most welcome.

(Note: I have already done a range of other tests (e.g. Chi-squared test, entropy) for each JPD. I have also calculated the Hellinger distance between each JPD. This question relates explicitly to a significance test between two JPD).