The Bass Diffusion model
The Bass Diffusion model elaborates on how a new product diffusion occurs in society and, therefore, can help to analyze the innovation diffusion.
This model is very interesting. It uses some assumption that relates to consumer’s behavior, then, based on those assumptions, develops a model in functional form. Then, one can fit the real data on that functional form to solve the parameters.
This model has three assumptions:
Univariate Density Estimation
Parametric Desity Estimation
Draw from a normal distribution
Given \({{X}_{1}},{{X}_{2}},\ldots ,{{X}_{n}}\) \(i.i.d\) draw from a normal distribution with mean of \(\mu\) and variance of \({{\sigma }^{2}}\) the joint \(PDF\) can be expressed as:
\[f\left( {{X}_{1}},{{X}_{2}},\ldots {{X}_{3}} \right)=\prod\limits_{i=1}^{n}{\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{i}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}}\]
\[f\left( {{X}_{1}},{{X}_{2}},\ldots {{X}_{3}} \right)=\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{1}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}\times \frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{2}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}\times \cdots \times \frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{n}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}\]
The term \(\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}\) is a constant multiplying this term for \(n\) times gives \({{\left( \frac{1}{\sqrt{2\pi {{\sigma }^{2}}}} \right)}^{n}}=\frac{1}{{{\left( 2\pi \sigma \right)}^{\frac{n}{2}}}}\).
Potential Outcome Framework
There are at least three different school of thoughts regarding causality: 1) granger causality; 2) Rubin’s potential outcome framework and 3) Pearl’s causality. Each of these thoughts have their own pros and cons. I will quickly discuss about the Rubin’s potential outcome framework and show the fundamental problem of causal inference.
For an example, say a researcher wants to study impact of certain treatment (say performance enhancing drug say \(T\)) on some outcomes (say capacity to solve riddle quizzes say \(Y\)).
Theorem 1: Gaussian Tail Inequality
Given \({{x}_{1}},\cdots ,{{x}_{n}}\sim N\left( 0,1 \right)\) then, \(P\left( \left| X \right|>\varepsilon \right)\le \frac{2{{e}^{-{{{\varepsilon }^{2}}}/{2}\;}}}{\varepsilon }\) and \(P\left( \left| {{{\bar{X}}}_{n}} \right|>\varepsilon \right)\le \frac{2}{\sqrt{n}\varepsilon }{{e}^{-{n{{\varepsilon }^{2}}}/{2}\;}}\overset{l\arg e\ n}{\mathop{\le }}\,{{e}^{-{n{{\varepsilon }^{2}}}/{2}\;}}\).
Proof of Gaussian Tail Inequality
Consider a univariate \({{x}_{1}},\cdots ,{{x}_{n}}\sim N\left( 0,1 \right)\), then the probability density function is given as \(\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}{{e}^{-\frac{{{x}^{2}}}{2}}}\).
Let’s take the derivative w.r.t \(x\) we get:
\[\frac{d\phi \left( x \right)}{dx}={\phi }'\left( x \right)=\frac{d\left( \frac{1}{\sqrt{2\pi }}{{e}^{-\frac{{{x}^{2}}}{2}}} \right)}{dx}=\frac{1}{\sqrt{2\pi }}\frac{d\left( \,{{e}^{-\frac{{{x}^{2}}}{2}}} \right)}{dx}=\frac{1}{\sqrt{2\pi }}\frac{d\left( \,{{e}^{-\frac{{{x}^{2}}}{2}}} \right)}{d\left( -\frac{{{x}^{2}}}{2} \right)}\frac{d\left( -\frac{{{x}^{2}}}{2} \right)}{dx}=\frac{1}{\sqrt{2\pi }}{{e}^{-\frac{{{x}^{2}}}{2}}}\left( -x \right)=-x\phi \left( x \right)\]
Let’s define the gaussian tail inequality.