

![0_image_0.png](0_image_0.png)

column marginals. Hence, one can approximately sample from this distribution by running this Markov chain for a sufficiently long time (although there are no theoretical results on convergence rates). The swap operation has later been generalized to deal with real-valued databases as well (Ojala et al, **2008), and** we will get back to this in Sec. 4.

The models we will develop in this paper are based on exactly these invariants of the row and column sums, be it in a somewhat relaxed form: we will assume that the expected values **of the row and column sums are equal to specified** values. Mathematically, this can be expressed as:

$$\begin{array}{r c l}{{\sum_{\mathbf{D}\in\mathcal{D}^{m\times n}}P(\mathbf{D})\left(\sum_{j}\mathbf{D}(i,j)\right)}}&{{=}}&{{d_{i}^{r},}}\\ {{}}&{{}}&{{}}\\ {{\sum_{\mathbf{D}\in\mathcal{D}^{m\times n}}P(\mathbf{D})\left(\sum_{i}\mathbf{D}(i,j)\right)}}&{{=}}&{{d_{j}^{c},}}\end{array}$$

where d r i is the i**'th expected row sum and** d c j the j**'th expected column sum.**
Although they have been suggested for binary databases (Gionis et al, 2007), and later extended to real-valued databases (Ojala et al, 2008), we will explore the consequences of these constraints in broader generality, for various choices for D.