# Is it possible to derive joint probabilities from marginals with assumptions about the conditionals?

I understand the title is too generic. I tried to look for similar questions and although there were a few that were seemingly about the same issue, either they provided answers in the negative or had no convincing answers or they suggested the use of copulas.

Since I have no working knowledge of copulas if they are actually the answer to my problem I am going to have to invest some time in getting acquainted with them, but before I do I would like to know if I should indeed invest the time, in the first place. Hence this question.

I have a population of individuals with a certain number of characteristics eg unemployed persons over some period of time; I know how many of them are located in a certain district (characteristic #1) also I know how many of them have achieved a certain education level eg MSc or relevant level (characteristic #2) but I don't have data on location and education for the same individual.

Given that the available info is something like the following table (for simplicity I don't include all the relevant characteristics-just 'location' (rows) and 'education' (columns)):

              | "MSc or higher"   "other edu"  |   sum
___________|________________________________|_______________________________________
"Region A" |       x               a        |   n_A    (unemployed in region A)
|                                |
"rest regs"|       y               b        | n_U-n_A  (unemployed in other regions)
___________|________________________________|_______________________________________
sum     |     n_MSc          n_U-n_MSc   |  n_U     (unemployed persons)
| (unemployed       (unemployed  |
|  with MSc)         with other  |
|                    education)  |

1. is it warranted to claim that eg $\frac{n_{MSc}}{n_{U}}$ is a measure of the risk of unemployment that a person with an education level equivalent or better than a MSc degree faces? Similarly, is eg $\frac{n_{A}}{n_{U}}$ a measure of the risk of unemployment for a person situated in Region A?
2. If the table above is reinterpreted as representing the unemployment risk associated with the relevant cell each time (ie if we divide the rightmost column and bottom row with $n_U$ to obtain marginal prrobabilities for the corresponding rows/columns and replace $x,y,a$ and $b$ with $p_x,p_y,p_a$ and $p_b$-the respective--unknown--joint probabilities) is there a way to retrieve those joint probabilities using only what information is contained in the tables presented above?
3. Are there plausible assumptions/restrictions that would assist or facilitate the calculations for finding the joint probabilities (eg some proposed/assumed relation between conditional frequencies) within reasonable bounds and for the purpose of having a rough estimate of what the actual figures about the joint instances of characteristics would be eg if more refined data sources (eg data sources detailing those joint frequencies) are considered?

( I apologize for the crude table layout but I was unable to use latex properly )

(edit: below this line I provide additional information in order to address considerations raised in the answer of @Ben which might be relevant to future discussants too)

"[...] Since you don't specify your sampling frame in the question, it is a bit hard to tell what exactly these data represent"

The data are a collection of (quarterly) time series documenting the absolute numbers of population, labour force, employed, unemployed and inactives for (eight) categories of educational attainment, (thirteen) districts and (six) 5-year age slots from 15-65+.

They are contained in three tables with eight, thirteen and six rows respectively and five columns each (per quarter).

In all cases, total population is decomposed into disjoint subsets of sums of individuals (eg persons with post-graduate qualification in total population or persons from District A in inactives, or unemployed aged 20-24 years old, respectively).

"[...]Since you have the counts inside the 2×2 contingency table, and not just their marginal totals, you already have a perfectly good basis for estimating the joint and conditional probabilities for the covariates in your data"

There are no observations for the number of people eg in District A who have finished primary education; that is, the number of people denoted by $x, y, a$ and $b$ in the simplified table I have included in the question, are not given;

the problem I'm facing is how to derive those joint frequencies, denoted in the question as $p_x, p_y, p_a$ and $p_b$.

"[...] If you want to find probabilities of unemployment conditional on these covariates then you will also need some data for employed people with these covariates."

I interpret this as in the following example:

Let UA denote "unemployed in District A" and UB denote "unemployed with a university degree"; also let U denote "Beeing unemployed", A denote "in District A" and B denote "with a university degree".

Then, $Pr(UA \cap UB)=Pr(U|A) \times Pr(A) + Pr(U|B) \times Pr(B)$,

where $Pr(A)$ equals "persons in District A / population" and $Pr(B)$ equals "persons with a university degree / population".

If my interpretation is correct, how should one calculate the conditional probabilities?

My understanding is that my data tables provide figures for the joint distribution of characteristics (eg number of people unemployed AND with a university degree) not conditional distributions.

1. is there a way to retrieve those joint probabilities using only what information is contained in the tables presented above?

You can not obtain the joint probabilities from the marginal probabilities. See the following table:

$$\begin{array}{cc|c} x_{11}-y & x_{12}+y & R_1 \\ x_{21}+y & x_{22}-y & R_2 \\ \hline C_1 & C_2 & N \\ \end{array}$$

You could derive some values for the inner cells like:

$$x_{ij} = \frac{R_iC_j}{N}$$

however multiple different joint probabilities (obtained by varying the value of $y$) can give the same marginals.

Thus you can not go in the opposite direction and derive the joint probabilities based on the marginal probabilities because there are multiple options.

2. Are there plausible assumptions/restrictions that would assist or facilitate the calculations for finding the joint probabilities

Possibly. For instance in the 2 x 2 table this can be some measure for the degree of dependence. However in the case of larger tables this becomes more and more complicated. There will be $(n_R-1)(n_C-1)$ parameters that you need to fix in order to be able to derive the joint probabilities. And for every extra dimension there will be more factors.

Since you have the counts inside the $2 \times 2$ contingency table, and not just their marginal totals, you already have a perfectly good basis for estimating the joint and conditional probabilities for the covariates in your data. Assuming that the raw data values for these people were exchangeable (i.e., their order is uninformative) you have the standard multinomial model for a contingency table:

$$(X,Y,A,B) \sim \text{Multinomial}\Big(n_U, (p_X, p_Y, p_A, p_B) \Big),$$

where $\boldsymbol{p} = (p_X, p_Y, p_A, p_B)$ is the vector of unknown cell probabilities. The joint distribution is:

$$p(x,y,a,b| n_U, \boldsymbol{p}) = {n_U \choose x \quad y \quad a \quad b} \cdot p_X^x \cdot p_Y^y \cdot p_A^a \cdot p_B^b.$$

You can estimate the joint probabilities of these covariates in this distribution directly from the individual cell counts, and you can also estimate the marginal probabilities from the row and column totals, just as you have done. The MLEs for the parameters are:

$$\hat{p}_X = \frac{x}{n_U} \quad \quad \hat{p}_Y = \frac{y}{n_U} \quad \quad \hat{p}_A = \frac{a}{n_U} \quad \quad \hat{p}_B = \frac{b}{n_U}.$$

The corresponding MLEs for the marginal probabilities are:

$$\hat{p}_X + \hat{p}_Y = \frac{n_{MSc}}{n_U} \quad \quad \hat{p}_X + \hat{p}_A = \frac{n_A}{n_U} \quad \cdots$$

All of the joint, marginal and conditional probabilities of the covariates can be estimated from the cell counts using the standard MLEs from the multinomial model. However, you need to be careful with your interpretation, and avoid the confusion of the inverse. Since you don't specify your sampling frame in the question, it is a bit hard to tell what exactly these data represent, but I am assuming from your description that they are a census or a sample of unemployed people over an area encompassing multiple regions. It appears that your data is for all unemployed people over all regions, which would mean that the status of unemployment is the condition behind all the probabilities. If that is a correct explanation of how the sampling worked, then the probability $p_A+p_A$ is the conditional probability of being in Region A conditional on being unemployed; that is not the same as the risk of being unemployed if you are in Region A.

It appears that you want to estimate the risk of unemployment, conditional on some covariate state (e.g., being in Region A, or having an MSc). However, your data appears to be only for unemployed people, and thus unemployment is a conditioning variable. This kind of data allows you to find probabilities of the covariates in your data, conditional on unemployment. If you want to find probabilities of unemployment conditional on these covariates then you will also need some data for employed people with these covariates.

(1) It is unclear from your post how this data was sampled. However, assuming that this is data for a random sample of unemployed people, your interpretations are not warranted. The mathematical formulae you give are correct estimators for the corresponding probabilities, but these are probabilities of covariates conditional on unemployment, not the other way around. You appear to be committing the confusion of the inverse and instead interpreting them as the probability of unemployment conditional on the covariate of interest. Without data on employed people, you don't have enough information to estimate the latter.

(2) The joint probabilities can be estimated from the data (as shown above), but as with (1), it is not correct to interpret these as probabilities of unemployment. If you would like to estimate the latter, you will need more information than is contained in data that conditions on unemployment.

(3) See above.