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Introduction to Bayesian Inference

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### The problem

An unknown process provides data samples z in [0,1]. The density of probability corresponding to this process is unknown. Nevertheless, let us model it with the following distribution :

The distribution depends on some parameter θ which is θ=0.3 in the above figure. The value of θ corresponding to the process is thuns unknown... and we would like to determine it, as we are provided with successive data samples. The distribution depicted above is called the model.

### Joint distribution

Let us consider the following a priori distribution for the parameter θ. Let us choose a parabolic one :

So in our world, the parameter θ is tossed according to the above density of probability, and then, the data samples are tossed according to the model parametrized by this θ value. The model is thus the probability density of the samples z, knowing the actual value of θ. In other words, the model is a conditional probability.

The probability of a sample z to be tossed depends finally on the two above densities of probability, which leads to a joint probability for the occurrence of (z,θ) as follows :

### Bayesian inference

Let us now toss the data samples z according to the model, with a parameter θ=0.7. This value is the unknown one that the inference principle is expected to discover. The following shows the update of the density of probability for the parameter θ while samples are provided. This is the Bayesian inference. This density of probability actually focuses around the value θ=0.7.

### Other examples

These movies are the same, except for the initial prior abount the θ distribution.