Bayesian Statistics - Basic question about priorI try to get an understanding of bayesian statistics. My intuition tells me that in the expression for the posterior p ( ϑ | x ) = p ( x | ϑ ) p ( ϑ ) ∫ Θ p ( x | θ ) p ( θ ) d θ the term p ( ϑ ) is the marginal distribution of the likelihood-function p ( ϑ , x ). It is obtained by p ( ϑ ) = ∫ X p ( ϑ | x ) p X ( x ) d xwhere p X ( x ) should be the marginal distribution of the Observable data. Does that make sense?To this point it makes sense with this example: Offering somebody a car insurance without knowing the person's style of driving (determined by ϑ ∈ Θ) to feed some statistical model, we still can make use of the nation's car-crash statistics as our prior, which is a pdf on Θ. That would be the marginal distribution of the "driving styles" across the population.Maybe I am just oversimplifying here, because my resources did not mention this.

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Bayesian Statistics - Basic question about priorI try to get an understanding of bayesian statistics. My intuition tells me that in the expression for the posterior  p  (  ϑ      |    x  )  =            p      (      x              |            ϑ      )      p      (      ϑ      )                      ∫        Θ            p      (      x              |            θ      )      p      (      θ      )      d      θ      the term   p  (  ϑ  ) is the marginal distribution of the likelihood-function   p  (  ϑ  ,  x  ). It is obtained by  p  (  ϑ  )  =      ∫    X    p  (  ϑ      |    x  )      p    X    (  x  )  d  xwhere       p    X    (  x  ) should be the marginal distribution of the Observable data. Does that make sense?To this point it makes sense with this example: Offering somebody a car insurance without knowing the person&#039;s style of driving (determined by   ϑ  ∈  Θ) to feed some statistical model, we still can make use of the nation&#039;s car-crash statistics as our prior, which is a pdf on   Θ. That would be the marginal distribution of the &quot;driving styles&quot; across the population.Maybe I am just oversimplifying here, because my resources did not mention this.

Ismael Wilkinson · Accepted Answer

Step 1In the Bayesian way of thinking, the prior distribution has no dependence on the data, so trying to get   p      (    ϑ    )   by integrating over x is an incorrect way to think about it. The distribution   p      (    ϑ    )   exists first -- it represents what you believe about the   ϑ parameter (i.e., which values it is more or less likely to have) before having seen any data. Then, you observe x and update your beliefs to the conditional distribution   p      (    ϑ              |            x    )  .Step 2But from which distribution do you observe x? This distribution will be different depending on what   ϑ is. So, it only makes sense to talk about p(x) as an expectation over different   θ values, as in the denominator of the Bayesian update.

bruinhemd3ji · Accepted Answer

Step 1I suggest you to change the point of view in order to enter in the bayesian way of thinking:Think at the posterior in the following way:  π  (  θ      |        x    )  ∝  π  (  θ  )  ×  p  (      x        |    θ  )Step 2Where the prior is a distribution that include all the information you have on your parameter multiplied (corrected) by the likelihood which include all the information given by the data.As per the fact that   π  (  θ  )  ×  p  (      x        |    θ  ) could not be a distribution (because its integral could be   ≠  1) you have to normalize it multiplying the previous product by a constant.

Bayesian Statistics - Basic question about prior. I try to get an understanding of bayesian statistics. My intuition tells me that in the expression for the posterior p(vartheta|x)=(p(x|vartheta)p(vartheta))/(int_{Theta}p(x|theta)p(theta)d theta)

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