How to get regularization parameter given by πœ† = 𝛼/𝛽 in MAP - pattern-recognition

maximizing the posterior distribution is equivalent to minimizing the regularized sum-of-squares error function with a regularization parameter given by πœ† = 𝛼/𝛽.
Maximizing the posterior distribution is equivalent to minimizing the regularized sum-of-squares error function
𝐸 ̃(𝐰)= 1/2 βˆ‘(𝑛=1)^𝑁▒{𝑦〖(π‘₯〗𝑛, 𝐰)βˆ’π‘‘_𝑛 }^2 +πœ†/2 ‖𝐰‖^2
where
‖𝐰‖^2 ≑𝐰^T 𝐰= 𝑀_0^2+𝑀1^2+…+𝑀𝑀^2
with a regularization parameter given by πœ† = 𝛼/𝛽.

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MLE of Gamma Distribution from mgcv::GAM and fitdistrplus

I've fitted a GAM to some data. However, I'm having trouble understanding how the information about the returned distribution from mgcv::GAM relates to that fitted by fitdistrplus.
From the output of summary(GAMObject), I deduce that the (Dispersion?) 'Scale est': 0.0020408 is approximately the inverse of Alpha.
>GAMObject$family
Family: Gamma
Link function: identity
>MASS::gamma.shape(GAMObject)
Alpha=486.660679
SE = 3.060256
>fitdistrplus::fitdist(Data,"gamma","mle")
Fitting of the distribution ' gamma ' by maximum likelihood
Parameters:
estimate Std. Error
shape 13.823388 0.08592579
rate 1.796375 0.01137114
>plot(fit.gamma)
Here's the output from the GAM.
>mgcv::gam.check(GAMObject)

How to use the gamma distribution equation

I am using R to fit a GLM with Gamma distribution (link inverse). I would like to use the equation of the model to get other values of my predictors, knowing the response value. I know that the equation of a gamma distribution with link inverse is 1/ΞΌ = b0 + b1x1i, can you confirm that I should substitute to ΞΌ the mean of my response value?

How to represent hypothesis function in logistic regression cost function

Below is logistic regression cost function with features(x) , training examples(y)
How should the hypotheses function (circled red) be represented ? :
I'm attempting to implement this function but unsure what value (or function) the hypothesis should take ?

The activation function in logistic regression is the sigmoid function (https://en.wikipedia.org/wiki/Sigmoid_function), defined as
which is also probability of y taking on a 1 value for a given x and parameter theta's to be determined (sigmoid is always between 0 and 1)
The cost function you mentioned comes from maximum likelihood estimation (https://en.wikipedia.org/wiki/Maximum_likelihood) of training (X, y) pairs. The log-likelihood of any (X, y) pair is exactly
The final loss function is precisely the sum of all log-likelihood for all (X, y) training pairs.
Thus, the ''hypothesis" you are talking about is simply sigmoid, 1/(1+exp(-theta * x)) (actually I am not familiar with the term hypothesis used in this context, but the expression resembles any standard expression involving sigmoid and MLE)

Negative binomial dispersion parameter in Matlab

The matlab function nbinfit returns the values r and p for the negative binomial. Is there an equivalent MLE function in matlab that instead returns the values for mu (mean) and theta (the dispersion parameter) for the "ecological" or "Polya" parametrization of the negative binomial?
(something like fitdistr in R)

Cost function in cv. glm for a fitted logistic model when cutoff value of the model is not 0.5

I have a logistic model fitted with the following R function:
glmfit<-glm(formula, data, family=binomial)
A reasonable cutoff value in order to get a good data classification (or confusion matrix) with the fitted model is 0.2 instead of the mostly used 0.5.
And I want to use the cv.glm function with the fitted model:
cv.glm(data, glmfit, cost, K)
Since the response in the fitted model is a binary variable an appropriate cost function is (obtained from "Examples" section of ?cv.glm):
cost <- function(r, pi = 0) mean(abs(r-pi) > 0.5)
As I have a cutoff value of 0.2, can I apply this standard cost function or should I define a different one and how?

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