What is a deterministic model?
A mathematically deterministic model is a representation y = f(x) that allows you to make predictions of y based on x. The model is used like this: when x=3, then we predict that y=f(3).
For example, suppose y = 2+3x-.4x2. We can predict that if x=3, then y=7.4.
Note that this "prediction" does not necessarily occur in the past, future, or even the present. It is simply a hypothetical, "what-if" statement. It helps us identify what would be the outcome if we were to use a particular x. For example, what would be the maximum stress (y) that a dam could bear, if we were to use x=(thickness of concrete). If we can answer these types of "what-if" questions, then we can make plans accordingly.
This type of model is "deterministic" because y is completely determined if
you know x. In real life, it is extremely rare that we can completely determine
a y using an x,
and thus we must use probabilistic (stochastic) models. One possible
exception is y = "amount paid" and x = "price tag." Here we have y =
Round(x(1+.08), .01), if in fact the sales tax is 8%, and if the "Round"
function rounds to the nearest cent. But even in this example, due to human errors, the value of
y will sometimes differ from Round(x(1+.08), .01), so the relationship is not deterministic
in reality. Only if there are no errors
will this example be one where the relation is truly
deterministic; i.e., an example where knowing x means you know y exactly.
In the physical and engineering sciences, one might think that relationships are more often deterministic. However, this is not the case. An example where the function is clearly not deterministic: the maximum stress a dam can bear cannot possibly be predicted perfectly from the thickness of the concrete. It depends on the type and preparation of concrete, the shape of the dam and its environment, and on the specific physical and structural make-up of the given concrete sample, down to the atomic level. It is impossible to characterize this information so completely as to arrive at a deterministic prediction of the maximum stress level that the dam can bear.
In the social sciences, relationships are not even close to deterministic. Can you predict tomorrow's Dow Jones average precisely using any model? No. Can you predict what a person will do, precisely, using any model? No. On the other hand, you can predict such things probabilistically.
What is a purely probabilistic model?
A probability model is a representation Y ~ p(y). Note that we say "Y~p(y)" not "Y=p(y)". The notation "Y~p(y)" specifically means that y is generated at random from a probability distribution whose mathematical form is p(y). This model also allows you to make "what-if" predictions as to the value of y, but, unlike the deterministic model, it does not allow you to say precisely what the value of y will be.
For example, suppose you wish to predict whether the next customer will buy either a red car, a gray car, or a green car. The possible values of y are "red", "gray", or "green", and the distribution p(y) might have the form:
The model does not tell you precisely what the next customer will do, because the model simply says it is random: y could be either "red", "gray" or "green". However, the model does allow aggregate what-if predictions as follows: "What if we were to sell a car to next 100 customers? Then about 35 of them would buy a red car, 40 would buy a gray car, and 25 of them would buy a green car."
Again, this "prediction" does not necessarily occur in the past, future, or even the present. It is simply a hypothetical, "what-if" statement. The probability model allows us to predict aggregate outcomes if we were to observe a large number of y values.
What is a model that has both deterministic and probabilistic components?
This type of model is called a regression model. The regression model is the representation Y ~ p(y|x), which states that, for a given x, Y is generated at random from a probability distribution whose mathematical form is p(y|x). This model also allows you to make "what-if" predictions as to the value of Y. Like the deterministic model, these predictions will depend on the value of x. However, since it is also a stochastic model, it does not allow you to say precisely what the value of y will be; as shown in the example above, the stochastic models only allow you to make aggregate "what-if" predictions.
Take the car example above. Suppose x=age of customer. We are now saying that the distribution of color preference depends on age of customer. For example, when x=20, the distribution might be
But when x=60, the distribution might be
The model does not tell you precisely what the next customer will do, but does allow aggregate what-if predictions of the following type. "What if we were to sell a car to 100 20-year old customers? Then about 50 of them would buy a red car, 20 would buy a gray car, and 30 of them would buy a green car."
Also, we can make the following what-if prediction: "What if we were to sell a car to 100 60-year old customers? Then about 20 of them would buy a red car, 40 would buy a gray car, and 40 of them would buy a green car."