Louis de Broglie was a critical link between the work of Albert Einstein and Max Born's statistical interpretation of quantum mechanics.

It was de Broglie who first argued that if light, which was thought to consist of waves, is actually discrete particles (Einstein' called light quanta, later called photons), then matter, which is thought to consist of discrete particles, might also have a wave nature.

The fundamental idea of [my 1924 thesis] was the following: The fact that, following Einstein's introduction of photons in light waves, one knew that light contains particles which are concentrations of energy incorporated into the wave, suggests that all particles, like the electron, must be transported by a wave into which it is incorporated... My essential idea was to extend to all particles the coexistence of waves and particles discovered by Einstein in 1905 in the case of light and photons."

Einstein had said that the light wave at some position is a measure of the probability of finding a light particle there, that is, the intensity of the light wave is proportional to the number of photons there. It may have been implicit in his 1905 light quantum hypothesis, as de Broglie seems to think.

Although Einstein had said this to colleagues as early as 1921, we don't have specific quotes from Einstein until 1927 at the fifth Solvay conference.

|ψ|² expresses the probability that there exists at the point considered a particular particle of the cloud, for example at a given point on the screen.

(Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, G. Bacciagaluppi and A. Valentini, 2009. pp.440-442)

Over a year earlier than the Solvay conference, in July of 1926, Max Born used de Broglie's matter waves, as described by Erwin Schrödinger's wave equation, to interpret the wave as the probability of finding an electron going off in a specific collision direction as proportional to the square of the probability amplitude wave function. Born gave full credit to Einstein, de Broglie, and Schrödinger for the idea, although the "statistical interpretation" itself is pure Einstein..

Collision processes not only yield the most convincing experimental proof of the basic assumptions of quantum theory, but also seem suitable for explaining the physical meaning of the formal laws of the so-called “quantum mechanics.”

A year before the introduction of the Heisenberg uncertainty principle and the "orthodox" Copenhagen Interpretation, Born already sees there are multiple interpretations of quantum mechanics

Indeed, as it seems, it always produces the correct term values of the stationary states and the correct amplitudes for the oscillations that are radiated by the transitions, but opinions are divided regarding the physical interpretation of the formulas. The matrix form of quantum mechanics that was founded by Heisenberg and developed by him and the author of this article starts from the thought that an exact representation of processes in space and time is quite impossible and that one must then content oneself with presenting the relations between the observed quantities, which can only be interpreted as properties of the motions in the limiting classical cases. On the other hand, Schrödinger (3) seems to have ascribed a reality of the same kind that light waves possessed to the waves that he regards as the carriers of atomic processes by using the de Broglie procedure; he attempts “to construct wave packets that have relatively small dimensions in all directions,” and which can obviously represent the moving corpuscle directly.

Here Born offers a third, "statistical" interpretation of quantum mechanics, and he gives credit to Einstein for the relation between waves and particles

Neither of these viewpoints seems satisfactory to me. Here, I would like to try to give a third interpretation and probe its utility in collision processes. I shall recall a remark that Einstein made about the behavior of the wave field and light quanta. He said that perhaps the waves only have to be wherever one needs to know the path of the corpuscular light quanta, and in that sense, he spoke of a “ghost field.” It determines the probability that a light quantum - viz., the carrier of energy and impulse – follows a certain path; however, the field itself is ascribed no energy and no impulse.

One would do better to postpone these thoughts, when coupled directly to quantum mechanics, until the place of the electromagnetic field in the formalism has been established. However, from the complete analogy between light quanta and electrons, one might consider formulating the laws of electron motion in a similar manner. This is closely related to regarding the de Broglie-Schrödinger waves as “ghost fields,” or better yet, “guiding fields.”

I would then like to pursue the following idea heuristically: The guiding field, which is represented by a scalar function ψ of the coordinates of all particles that are involved and time, propagates according to Schrödinger’s differential equation. However, impulse and energy will be carried along as when corpuscles (i.e., electrons) are actually flying around. The paths of these corpuscles are determined only to the extent that they are constrained by the law of energy and impulse; moreover, only a probability that a certain path will be followed will be determined by the function ψ. One can perhaps summarize this, somewhat paradoxically, as: The motion of the particle follows the laws of probability, but the probability itself propagates in accord with causal laws.

(Quantum mechanics of collision processes (Quantenmechanik der Stoßvorgänge), Zeitschrift für Physik, 38 (1926), 803-827)

That same year, Bohm wrote his classic book Causality and Chance in Modern Physics, and de Broglie wrote a preface, hoping for a return to a deeper level deterministic physics, which would explain the probabilistic and statistical properties of quantum mechanics.

He wrote...

(Causality and Chance in Modern Physics, p.ix)