# Help with the prior distribution

The question is as follows:

Consider an SDOF mass-spring system. The value of the mass is known and is equal to 1 kg.
The value of the spring stiffness is unknow and based on the experience and judgement the following is assumed. Value of stiffness is in the following range [0.5, 1.5] N/m.

To have a more accurate estimate of the value of the stiffness an experiment is performed where in the natural frequency of the system is observed. The following observation are made:

  Observation 1     Freq = 1.021 rad/sec
Observation 2     Freq = 1.015 rad/sec
Observation 3     Freq = 0.994 rad/sec
Observation 4     Freq = 1.005 rad/sec
Observation 5     Freq = 0.989 rad/sec

1. Based on the information provided write the functional form of prior PDF.
2. Plot the likelihood function with different number of observations.
3. Based on the information provided write the functional form of the posterior PDF.
4. Plot the posterior distribution.

My work so far:

spring constant $$k = \sqrt{{w}/{m}}$$ m = 1kg, so $$w = k^{2}$$.

$$k \sim Uniform(0.5, 1.5)$$,

so pdf of w = $$f(w) = 2w$$

where $$w\ \epsilon\ [\sqrt{0.5},\sqrt{1.5}]$$

So prior distribution is linear in the range root(0.5), root(1.5).

$$Likelihood = L = 2^{5}(1.021*1.015..*0.989) \approx 2.04772$$

This is what I have done so far. I am new to Bayesian inference and I am not sure how to proceed after this or if what I have done so far is correct. Pleas advice on how to find the posterior function.

• @Xi'an I don't want the distribution for $\sqrt{k}$. That is not even in the picture Aug 20, 2020 at 7:01
• I gather you want an analytical solution, which I don't think I'm able to help with, but just wanted to note that w is deterministic w.r.t. k, and therefore doesn't contribute to the posterior - see here for a discussion on how to handle it. Sep 6, 2020 at 4:02

I gave up my reputation for a bounty so unable to comment.

The posterior is the prior multiplied by the likelihood. If you use a conjugate prior, these types of problems will work out nicely.

What is the sampling distribution in this case? Normal?

Not sure how to handle the bounds but you could use a normal prior for $$k$$ w/ infinite variance to resemble a uniform distribution or you could just do a normal distribution centered at 1 w/ some large variance out to 0.5 and 1.5.

You say you are not interested in $$k$$ though? Can you work back to it?