How can I estimate user-item purchase probabilities of a e-commerce website?
If there are users' session available I'd recommend to go for a session-based recommendation - when we make recommendation of next click/buy item based on the current user's session. That seems the most appropriate for your data and also very attractive way of doing recommendation systems in the e-commerce, when still most of users while searching for purchase or window-shopping is not logged in. 

While predicting the next action, few measures can be used. But from practical perspective and current publications, two are quite interesting: 


REC@N - Recall at top-N best recommended items. This measure gives the information about the overall engagement of the users and many times it's said that is correlates with CTR on live(production) use. 
MRR@N - Mean Reciprocal Rank at top-N ranked recommendation, which gives you the information how well is your personalized ranking for the current session prediction. 


I'd recommend reading two recent papers about session-based recommendation - one is about making the next item recommendation based on sequences of items [1]. The other one [2] is about making recommendations based on the current user's session when you have rich context for recommended items and session events. Which is mostly true for the e-commerce system, where you have items described by: title, attributes, text description, images, etc. And also in users session there are additional events like interaction which search engines, which gives lots of users preferences in data for the query, sorting by which attribute, etc. This can be used as a current session context. In both works you can find references to related works, which can be helpful for you.

[1]: Hidasi, B., Karatzoglou, A., Baltrunas, L., &amp; Tikk, D. (2015). Session-based Recommendations with Recurrent Neural Networks, 1–9. Retrieved from http://arxiv.org/abs/1511.06939

[2]: Twardowski, B. (2016). Modelling Contextual Information in Session-Aware Recommender Systems with Neural Networks, 0–3. http://doi.org/10.1145/2959100.2959162