Unseen things can be part of empirical inquiry as well. Indeed they regularly are, and can be inferred from what is observable. Nyaya believed that the soul/self can be inferred from the observable properties of consciousness. That is where their disputation with Buddhists were.
Expert testimony is accepted in science and any branch of empirical inquiry as well. What makes someone an expert is under contention though. You are confusing empirical inquiry with atheism for some reason. Atheism may be or may not be the conclusion from empirical inquiry, its the methodology of inquiry that makes it empirical, not the conclusion. Nyaya philosophy follows an empirical and rational mode of inquiry in the same sense as Aristotle does in his philosophy and epistemology even though both schools do believe in God/Gods based on reasoning and inference.
No. I am not conflating atheism and empirical enquiry and atheism.
1. My response regarding atheism was to Shiva's patent assertion that Gautama muni did not accept Ishwara.
The Nyaya Sutra says:
BOOK IV,
“21. Since fruits are awarded by God, man's acts, we conclude, are not the sole cause thereof.—21”
So, it is obvious that Gautama muni takes Ishwara as axiomatic. How else is karma phala attributed?
2. Regarding empiricism
Nyaya accepts an intelligent atman. There are two implications of this.
a) Since mind is dependent upon atman and is a product, it cannot know/see atman. Any Hindu of any school will understand this. Nyaya teaches as below:
“106. The knowledge of truth is rendered habitual by & special practice of meditation.—38.”
“110. We are instructed to practise meditation in such places as a forest, a cave or a sand-bank.—42.
“113. And there is absence of a body in our release
“114. For that purpose there should be a purifying of our soul by abstinence from evil and observance of certain duties as well as by following the spiritual injunctions gleaned from the Yoga institute.—46.”
115. To secure release, it is necessary to study and follow this treatise on knowledge as well as to hold discussions”
b) Implication of a conscious Atman
Hinduism works with the concept of a 'whole' that is intelligent and distinct from the particulars. Shankara's advaita uses this in its basic premise of 'Adhyasa', the concept where a particular is mistaken for the whole and whole is attributed the property of a particular. Shankara employs an example of a heated iron ball to demonstrate this effect. In case, of a heated iron ball, some may think that heat has property of hardness (which comes from iron) and that the property of the ball is being 'hot'. Shankara called this Adhyasa.
Now, a knower of this intelligent whole (Atman-Brahman), is well nigh God. This is not acceptable to monotheists and even Buddha did not clarify it. This was brought into picture by Gaudapada in Mandukya Karika.
99 The Knowledge of the wise man, who is all light, is never related to any object. All the jivas, as well as Knowledge, are ever unrelated to objects. This is not the view of Buddha.