How to combine normal distributions in two-dimensional data My goal is to estimate the click-ability (the percentage of viewers who clicked) of a specific advertisement in a new web-page based on historical data. The nature of my data is such that each web page is categorized by two parameters - The site the page is in (e.g. stackoverflow.com), and type of the page (sports, entertainment, etc..). I know the clickability of my ad in many pages from my historical data, and would like to predict it for a new page which I don't have enough data on.
Since each web page is characterized by two parameters, I can summarize my data in a two dimensional sample matrix M'(X,Y), where each data point in the matrix is the click-ability of the ad in the specific combination of website and page type. For example if I know that the average of the click-rates in pages of type Programming in Stackoverflow.com is 0.5, when I get a new page which fits those parameters I will "guess" 0.5 as the click-rate of my ad in that page. When I look at the distribution of click-rates in a specific combination of site and page type, it is distributed normally in a nice bell shape - So in fact my guess is the mean estimator of the normal distribution that is specific to those parameters.
My problem arises when I don't have enough information in a specific coordinate in the matrix M'(X,Y) to get a robust estimate of the mean estimator - For example when I have historical data only for 2 web pages of type entertainment in stackoverflow. In that case, and assuming that I have new page in Stackoverflow that is an entertainment page - I can estimate the normal distribution of that advertisement in stackoverflow and independently estimate the distribution in pages of type entertainment in all websites (Not just stackoverflow). The question is how to combine the two normal mean estimators?
Here is a concrete example: 
Each number in the matrix represents what is the click-rate of my advertisement in a specific webpage, base on previous data in which I counted views and clicks in that specific webpage. Each coordinate in the matrix represents the click rates in all the web pages in the specific combination of site and page type. For simplicity I made the number of web pages in each combination of parameters small, but usually it is of size between 0 and 1000. My goal is to obtain the mean and variance parameters that best describes a specific coordinate, even for coordinates with little or no data (such as stackoverflow/entertainment or cnn/programming). For example the mean estimation for stackoverflow and programming would be ~0.5 with a standard deviation of around 0.1.
How might this be achieved? Is there a "correct way" to combine the the estimators for each dimension? Most solutions I found on the internet related on how to combine two samples of the same normal distribution - for example the inverse variance method. This doesn't seem quite right in my case - since each normal  distribution describes a separate dimension of 2-dimensional data. 
Another possible solution which doesn't seem right is 
Combination of two Gaussians. If one dimension has data for 1 million web pages and the other has 1000 - The first one would dominate the combined result, despite the fact that 1000 web pages is enough to get a robust statistic and thus should get the same weight as the other dimension when combining them. 
Is there any sound solution to this problem?
 A: This is not really a full answer but maybe is easier to read than stuffing into comments
Problem Set up:
a) So your raw data is of form 
----------
Top-level Site | Theme| Page id | #Clicks| #Impressions
----------

and you collect all the pages for that (top-level site,theme) together ( disregarding the number of impressions on each page)
b) the data is gathered rather than from a proper experiment - so you suffer from issues of unbalanced design. This is especially severe because you are dealing with website data (eg certain themes/top level sites will have many orders more pages than others )
Problem Question:
How to find an estimate for click through rate (CTR) for those combinations of (Top level site , theme) that have few pages - namely by combining an estimate based on theme with an estimate based on top level site in particular dealing with the unbalanced design.
Problem Answer:
You should use a regression model - in particular logistic regression would be the natural choice to estimate a probability. A logistic regression model written (in R formula notation) as 
   CTR ~ top level site + theme
will provide you with just such an estimate.  You might be able to get away with a single model of the form
   CTR ~ top level site + theme + top level site x theme
this will use the interactions when there is enough data and the individual dimensions additively if not.
I am not sure whether the regression model will have any issues with the unbalanced design - maybe more experienced members can pipe in. 
Predicting clicks: Estimating the click-through rate for new ads
Web-Scale Bayesian Click-Through Rate Prediction for Sponsored Search Advertising in Microsoft’s Bing Search Engine
the second paper aims to address more specifically your issue of developing a general model to fill in the gaps where you have missing data points. 
however a logistic regression model of CTR~ site + theme + site x theme might work OK 
(ie the model  tries to predict CTR with site and theme separately )
Some remarks
a standard assumption for click data is that it follows a binomial distribution with parameters (N =number of trials ie impressions, p =theoretical proportion of successes ie clicks ), so make sure you are comfortable with this. 
even if your data is generated by the same theoretical process (equal p) your estimate (from data) of p will vary - with amount of variation dependent on number of samples ( and value of p) (see eg binomial proportion confidence interval). 
You might want to review your different samples of click through rate based on this ( ie for each combination of  site/theme eg {0.3,0.25} is the variation in sample ctr just due to statistical variation ( too few impressions).
If you really do want to model the CTR as random then the beta distribution is a better choice than the normal distribution
(Normal distribution for CTR doesn't take account variability due to different sample sizes, and that ctr must be between zero and 1) 
you could then create a linear regression model ( or other statistical model) for each parameter (eg mu ~ site + theme +site x theme)
see eg Bid Landscape Forecasting in Online Ad Exchange Marketplace
as I hope is clear from the linked papers, this is a pretty standard approach for web advertising companies - so its definitely 'web scale'. 
Overfitting with 1000 x 1000 dummy variables
over fitting is not a problem, precisely because most of the inputs are zero.
you control over fitting with say L2 regularisation (which puts a penalty on the coefficients - with the effect that the a coefficient will be nonzero only if it reduces the overall error sufficiently: by playing with this regularisation you can ensure combinations with few impressions are ignored.
the high dimensional (but sparse) representation of dummy variables does require you to pass your input to logistic regression in a sparse format ( as a sparse matrix) or as a dictionary - otherwise you run out of memory.
A: Unless I completely misunderstood your question, it seems to me that you could approach this as a problem of missing values (incomplete data). If this is the case, the problem can be solved by using corresponding methods, such as multiple imputation. Depending on whether your data is normal or non-normal, you could use an R package to impute missing data, such as Amelia / Amelia II (http://cran.r-project.org/web/packages/Amelia) or mice (http://cran.r-project.org/web/packages/mice). Both packages are very flexible and support a variety of data types and imputation/regression models. If your data is normal, but you're not comfortable with programming in R, Amelia has an interactive GUI version, which might be sufficient for your purposes.
