With Philosophy, a good starting point is always the work of Bertrand Russell. There's no doubt that you'd find sections in Russell's History of Western Philosophy that cover the philosophy of causation/causal inference, but given its size and broad scope, it would be difficult for me to pin down for you exactly where to look in this book. Taking the longer term view, though, this is the book to start with if you want to deepen your knowledge of philosophy - its evolution - and philosophers themselves.
A second book by Bertrand Russell that's worth consulting is Human Knowledge. Part V of this book covers Probability while Part VI is about the Postulates of Scientific Inference. Both of these topics are discussed from the philosopher's standpoint. To give you a taste of the book, I've added two extracts from the Introduction below.
In the Introduction to the book, Bertrand tells us a little bit about Part V Probability:
Since it is admitted that scientific inferences, as rule, only confer
probability on the conclusions, Part V proceeds to the examination of
Probability. This term is capable of various interpretations , and has
been differently defined by different authors. These interpretations
and definitions are examined, and so are the attempts to connect
induction with probability. In this matter the conclusion reached is,
in the main, that advocated by Keynes: that inductions do not make
their conclusions probable unless certain conditions are fulfilled, and that
experience alone can never prove that these conditions are fulfilled.
And on Part VI Postulates of Scientific Inference, Bertrand says (again, from the Introduction):
Part VI, on the postulates of scientific inference, endeavours to
discover what are the minimum assumptions, anterior to experience,
that are required to justify us in inferring laws from a collection of
data; and further, to inquire in what sense, if any, we can be said to
know that these assumptions are valid. The main logical function that
the assumptions have to fulfill is that of conferring a high
probability on the conclusions and inductions that satisfy certain
conditions. For this purpose, since only probability is in question,
we do not need to assume that such-and-such a connection of events
occurs always, but only that it occurs frequently. For example, one of the
assumptions that appear necessary is that of separable causal chains,
such as are exhibited by light-rays or sound-waves. This assumption
can be stated as follows: when an event having a complex space-time
structure occurs, it frequently happens that it is one of a train of
events having the same or a very similar structure. (A more exact
statement will be found in Chapter 6 of this Part.) This is part of a
wider assumption of regularity, or natural law, which, however,
requires to be stated in more specific forms than is usual, for in its
usual form it turns out to be a tautology.
That scientific inference requires, for its validity, principles which
experience cannot render even probable, is, I believe, an inescapable
conclusion from the logic of probability. For empiricism, it is an
awkward conclusion.
But I think it can be rendered somewhat more palatable by the analysis of the
concept of "knowledge" undertaken in Part II. "Knowledge", in my
opinion, is a much less precise concept than is generally thought, and
has its roots more deeply embedded in unverbalized animal behaviour
than most philosophers have been willing to admit. The logically basic
assumptions to which our analysis leads us are psychologically the end
of a long series of refinements which start from habits of expectation
in animals, such as that what has a certain kind of smell will be good
to eat. To ask, therefore, whether we "know" the postulates of
scientific inference, is not so definite a question as it seems. The
answer must be: in one sense, yes, in another sense, no; but in the
sense in which "no" is the right answer we know nothing whatever, and
"knowledge" in this sense is a delusive vision. The perplexities of
philosophers are due, in a large measure, to their unwillingness to
awaken from this blissful dream.
If you decide to take things further (down the academic line), I'd also suggest searching "causal inference" in the Oxford Journal Mind. There is a search tool on the Journal's website.
